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![2 Introduction to Python
• Python is easier to learn and produces more readable code than most languages.
• Python and its extensions are easy to install.
Development of Python has been clearly influenced by Java and C++, but there is
also a remarkable similarity to MATLABR
(another interpreted language, very popular
in scientific computing). Python implements the usual concepts of object-oriented
languages such as classes, methods, inheritance etc. We do not use object-oriented
programming in this text. The only object that we need is the N-dimensional array
available in the module numpy (this module is discussed later in this chapter).
To get an idea of the similarities and differences between MATLAB and Python,
let us look at the codes written in the two languages for solution of simultaneous
equations Ax = b by Gauss elimination. Do not worry about the algorithm itself (it
is explained later in the text), but concentrate on the semantics. Here is the function
written in MATLAB:
function x = gaussElimin(a,b)
n = length(b);
for k = 1:n-1
for i= k+1:n
if a(i,k) ˜= 0
lam = a(i,k)/a(k,k);
a(i,k+1:n) = a(i,k+1:n) - lam*a(k,k+1:n);
b(i)= b(i) - lam*b(k);
end
end
end
for k = n:-1:1
b(k) = (b(k) - a(k,k+1:n)*b(k+1:n))/a(k,k);
end
x = b;
The equivalent Python function is
from numpy import dot
def gaussElimin(a,b):
n = len(b)
for k in range(0,n-1):
for i in range(k+1,n):
if a[i,k] != 0.0:
lam = a [i,k]/a[k,k]
a[i,k+1:n] = a[i,k+1:n] - lam*a[k,k+1:n]
b[i] = b[i] - lam*b[k]
for k in range(n-1,-1,-1):
b[k] = (b[k] - dot(a[k,k+1:n],b[k+1:n]))/a[k,k]
return b](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-20-320.jpg)
![3 1.1 General Information
The command from numpy import dot instructs the interpreter to load the
function dot (which computes the dot product of two vectors) from the module
numpy. The colon (:) operator, known as the slicing operator in Python, works the
same way as it does in MATLAB and Fortran90—it defines a slice of an array.
The statement for k = 1:n-1 in MATLAB creates a loop that is executed with
k = 1, 2, . . . , n − 1. The same loop appears in Python as for k in range(n-1).
Here the function range(n-1) creates the sequence [0, 1, . . . , n − 2]; k then loops
over the elements of the sequence. The differences in the ranges of k reflect the native
offsets used for arrays. In Python all sequences have zero offset, meaning that the
index of the first element of the sequence is always 0. In contrast, the native offset in
MATLAB is 1.
Also note that Python has no end statements to terminate blocks of code (loops,
subroutines, and so on). The body of a block is defined by its indentation; hence in-
dentation is an integral part of Python syntax.
Like MATLAB, Python is case sensitive. Thus the names n and N would represent
different objects.
Obtaining Python
The Python interpreter can be downloaded from
http : //www.python.org/getit
It normally comes with a nice code editor called Idle that allows you to run programs
directly from the editor. If you use Linux, it is very likely that Python is already in-
stalled on your machine. The download includes two extension modules that we use
in our programs: the numpy module that contains various tools for array operations,
and the matplotlib graphics module utilized in plotting.
The Python language is well documented in numerous publications. A com-
mendable teaching guide is Python by Chris Fehly (Peachpit Press, CA, 2nd ed.). As a
reference, Python Essential Reference by David M. Beazley (Addison-Wesley, 4th ed.)
is highly recommended. Printed documentation of the extension modules is scant.
However, tutorials and examples can be found on various websites. Our favorite ref-
erence for numpy is
http://www.scipy.org/Numpy Example List
For matplotlib we rely on
http://matplotlib.sourceforge.net/contents.html
If you intend to become a serious Python programmer, you may want to acquire
A Primer on Scientific Programming with Python by Hans P. Langtangen (Springer-
Verlag, 2009).](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-21-320.jpg)
![4 Introduction to Python
1.2 Core Python
Variables
In most computer languages the name of a variable represents a value of a given type
stored in a fixed memory location. The value may be changed, but not the type. This
is not so in Python, where variables are typed dynamically. The following interac-
tive session with the Python interpreter illustrates this feature ( is the Python
prompt):
b = 2 # b is integer type
print(b)
2
b = b*2.0 # Now b is float type
print(b)
4.0
The assignment b = 2 creates an association between the name b and the in-
teger value 2. The next statement evaluates the expression b*2.0 and associates the
result with b; the original association with the integer 2 is destroyed. Now b refers to
the floating point value 4.0.
The pound sign (#) denotes the beginning of a comment—all characters between
# and the end of the line are ignored by the interpreter.
Strings
A string is a sequence of characters enclosed in single or double quotes. Strings are
concatenated with the plus (+) operator, whereas slicing (:) is used to extract a por-
tion of the string. Here is an example:
string1 = ’Press return to exit’
string2 = ’the program’
print(string1 + ’ ’ + string2) # Concatenation
Press return to exit the program
print(string1[0:12]) # Slicing
Press return
A string can be split into its component parts using the split command. The
components appear as elements in a list. For example,
s = ’3 9 81’
print(s.split()) # Delimiter is white space
[’3’, ’9’, ’81’]
A string is an immutable object—its individual characters cannot be modified
with an assignment statement, and it has a fixed length. An attempt to violate im-
mutability will result in TypeError, as follows:](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-22-320.jpg)
![5 1.2 Core Python
s = ’Press return to exit’
s[0] = ’p’
Traceback (most recent call last):
File ’’pyshell#1’’, line 1, in ?
s[0] = ’p’
TypeError: object doesn’t support item assignment
Tuples
A tuple is a sequence of arbitrary objects separated by commas and enclosed
in parentheses. If the tuple contains a single object, a final comma is required;
for example, x = (2,). Tuples support the same operations as strings; they are
also immutable. Here is an example where the tuple rec contains another tuple
(6,23,68):
rec = (’Smith’,’John’,(6,23,68)) # This is a tuple
lastName,firstName,birthdate = rec # Unpacking the tuple
print(firstName)
John
birthYear = birthdate[2]
print(birthYear)
68
name = rec[1] + ’ ’ + rec[0]
print(name)
John Smith
print(rec[0:2])
(’Smith’, ’John’)
Lists
A list is similar to a tuple, but it is mutable, so that its elements and length can be
changed. A list is identified by enclosing it in brackets. Here is a sampling of opera-
tions that can be performed on lists:
a = [1.0, 2.0, 3.0] # Create a list
a.append(4.0) # Append 4.0 to list
print(a)
[1.0, 2.0, 3.0, 4.0]
a.insert(0,0.0) # Insert 0.0 in position 0
print(a)
[0.0, 1.0, 2.0, 3.0, 4.0]
print(len(a)) # Determine length of list
5
a[2:4] = [1.0, 1.0, 1.0] # Modify selected elements
print(a)
[0.0, 1.0, 1.0, 1.0, 1.0, 4.0]](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-23-320.jpg)
![6 Introduction to Python
If a is a mutable object, such as a list, the assignment statement b = a does not
result in a new object b, but simply creates a new reference to a. Thus any changes
made to b will be reflected in a. To create an independent copy of a list a, use the
statement c = a[:], as shown in the following example:
a = [1.0, 2.0, 3.0]
b = a # ’b’ is an alias of ’a’
b[0] = 5.0 # Change ’b’
print(a)
[5.0, 2.0, 3.0] # The change is reflected in ’a’
c = a[:] # ’c’ is an independent copy of ’a’
c[0] = 1.0 # Change ’c’
print(a)
[5.0, 2.0, 3.0] # ’a’ is not affected by the change
Matrices can be represented as nested lists, with each row being an element of
the list. Here is a 3 × 3 matrix a in the form of a list:
a = [[1, 2, 3],
[4, 5, 6],
[7, 8, 9]]
print(a[1]) # Print second row (element 1)
[4, 5, 6]
print(a[1][2]) # Print third element of second row
6
The backslash () is Python’s continuation character. Recall that Python se-
quences have zero offset, so that a[0] represents the first row, a[1] the second row,
etc. With very few exceptions we do not use lists for numerical arrays. It is much more
convenient to employ array objects provided by the numpy module. Array objects are
discussed later.
Arithmetic Operators
Python supports the usual arithmetic operators:
+ Addition
− Subtraction
∗ Multiplication
/ Division
∗∗ Exponentiation
% Modular division
Some of these operators are also defined for strings and sequences as follows:
s = ’Hello ’
t = ’to you’](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-24-320.jpg)
![7 1.2 Core Python
a = [1, 2, 3]
print(3*s) # Repetition
Hello Hello Hello
print(3*a) # Repetition
[1, 2, 3, 1, 2, 3, 1, 2, 3]
print(a + [4, 5]) # Append elements
[1, 2, 3, 4, 5]
print(s + t) # Concatenation
Hello to you
print(3 + s) # This addition makes no sense
Traceback (most recent call last):
File pyshell#13, line 1, in module
print(3 + s)
TypeError: unsupported operand type(s) for +: ’int’ and ’str’
Python also has augmented assignment operators, such asa+ = b, that are famil-
iar to the users of C. The augmented operators and the equivalent arithmetic expres-
sions are shown in following table.
a += b a = a + b
a -= b a = a - b
a *= b a = a*b
a /= b a = a/b
a **= b a = a**b
a %= b a = a%b
Comparison Operators
The comparison (relational) operators return True or False. These operators are
Less than
Greater than
= Less than or equal to
= Greater than or equal to
== Equal to
!= Not equal to
Numbers of different type (integer, floating point, and so on) are converted to
a common type before the comparison is made. Otherwise, objects of different type
are considered to be unequal. Here are a few examples:
a = 2 # Integer
b = 1.99 # Floating point
c = ’2’ # String
print(a b)
True](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-25-320.jpg)
![9 1.2 Core Python
Loops
The while construct
while condition:
block
executes a block of (indented) statements if the condition is True. After execution
of the block, the condition is evaluated again. If it is still True, the block is exe-
cuted again. This process is continued until the condition becomes False. The else
clause
else:
block
can be used to define the block of statements that are to be executed if the condition
is false. Here is an example that creates the list [1, 1/2, 1/3, . . .]:
nMax = 5
n = 1
a = [] # Create empty list
while n nMax:
a.append(1.0/n) # Append element to list
n = n + 1
print(a)
The output of the program is
[1.0, 0.5, 0.33333333333333331, 0.25]
We met the for statement in Section 1.1. This statement requires a target and a
sequence over which the target loops. The form of the construct is
for target in sequence:
block
You may add an else clause that is executed after the for loop has finished.
The previous program could be written with the for construct as
nMax = 5
a = []
for n in range(1,nMax):
a.append(1.0/n)
print(a)
Here n is the target, and the range object [1, 2, . . . , nMax − 1] (created by calling
the range function) is the sequence.
Any loop can be terminated by the
break](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-27-320.jpg)
![10 Introduction to Python
statement. If there is an else cause associated with the loop, it is not executed. The
following program, which searches for a name in a list, illustrates the use of break
and else in conjunction with a for loop:
list = [’Jack’, ’Jill’, ’Tim’, ’Dave’]
name = eval(input(’Type a name: ’)) # Python input prompt
for i in range(len(list)):
if list[i] == name:
print(name,’is number’,i + 1,’on the list’)
break
else:
print(name,’is not on the list’)
Here are the results of two searches:
Type a name: ’Tim’
Tim is number 3 on the list
Type a name: ’June’
June is not on the list
The
continue
statement allows us to skip a portion of an iterative loop. If the interpreter encounters
the continue statement, it immediately returns to the beginning of the loop without
executing the statements that follow continue. The following example compiles a
list of all numbers between 1 and 99 that are divisible by 7.
x = [] # Create an empty list
for i in range(1,100):
if i%7 != 0: continue # If not divisible by 7, skip rest of loop
x.append(i) # Append i to the list
print(x)
The printout from the program is
[7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98]
Type Conversion
If an arithmetic operation involves numbers of mixed types, the numbers are
automatically converted to a common type before the operation is carried out.](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-28-320.jpg)
![12 Introduction to Python
It displays the prompt and then reads a line of input that is converted to a string. To
convert the string into a numerical value use the function
eval(string)
The following program illustrates the use of these functions:
a = input(’Input a: ’)
print(a, type(a)) # Print a and its type
b = eval(a)
print(b,type(b)) # Print b and its type
The function type(a) returns the type of the object a; it is a very useful tool in
debugging. The program was run twice with the following results:
Input a: 10.0
10.0 class ’str’
10.0 class ’float’
Input a: 11**2
11**2 class ’str’
121 class ’int’
A convenient way to input a number and assign it to the variable a is
a = eval(input(prompt))
Printing Output
Output can be displayed with the print function
print(object1, object2, . . .)
that converts object1, object2, and so on, to strings and prints them on the same line,
separated by spaces. The newline character ’n’ can be used to force a new line. For
example,
a = 1234.56789
b = [2, 4, 6, 8]
print(a,b)
1234.56789 [2, 4, 6, 8]
print(’a =’,a, ’nb =’,b)
a = 1234.56789
b = [2, 4, 6, 8]
The print function always appends the newline character to the end of a line.
We can replace this character with something else by using the keyword argument
end. For example,
print(object1, object2, ...,end=’ ’)
replaces n with a space.](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-30-320.jpg)
![14 Introduction to Python
Reading Data from a File
There are three methods for reading data from a file. The method
file object.read(n)
reads n characters and returns them as a string. If n is omitted, all the characters in
the file are read.
If only the current line is to be read, use
file object.readline(n)
which reads n characters from the line. The characters are returned in a string that
terminates in the newline character n. Omission of n causes the entire line to be
read.
All the lines in a file can be read using
file object.readlines()
This returns a list of strings, each string being a line from the file ending with the
newline character.
A convenient method of extracting all the lines one by one is to use the loop
for line in file object:
do something with line
As an example, let us assume that we have a file named sunspots.txt in the
working directory. This file contains daily data of sunspot intensity, each line having
the format (year/month/date/intensity), as follows:
1896 05 26 40.94
1896 05 27 40.58
1896 05 28 40.20
etc.
Our task is to read the file and create a list x that contains only the intensity. Since
each line in the file is a string, we first split the line into its pieces using the split
command. This produces a list of strings, such as[’1896’,’05’,’26’,’40.94’].
Then we extract the intensity (element [3] of the list), evaluate it, and append the
result to x. Here is the algorithm:
x = []
data = open(’sunspots.txt’,’r’)
for line in data:
x.append(eval(line.split()[3]))
data.close()](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-32-320.jpg)
![17 1.3 Functions and Modules
Note that atan is passed to derivatives as a parameter. The output from the
program is
First derivative = 0.799999999573
Second derivative = -0.639999991892
The number of input parameters in a function definition may be left arbitrary.
For example, in the following function definition
def func(x1,x2,*x3)
x1 and x2 are the usual parameters, also called positional parameters, whereas x3
is a tuple of arbitrary length containing the excess parameters. Calling this function
with
func(a,b,c,d,e)
results in the following correspondence between the parameters:
a ←→ x1, b ←→ x2, (c,d,e) ←→ x3
The positional parameters must always be listed before the excess parameters.
If a mutable object, such as a list, is passed to a function where it is modified, the
changes will also appear in the calling program. An example follows:
def squares(a):
for i in range(len(a)):
a[i] = a[i]**2
a = [1, 2, 3, 4]
squares(a)
print(a) # ’a’ now contains ’a**2’
The output is
[1, 4, 9, 16]
Lambda Statement
If the function has the form of an expression, it can be defined with the lambda
statement
func name = lambda param1, param2,...: expression
Multiple statements are not allowed.
Here is an example:
c = lambda x,y : x**2 + y**2
print(c(3,4))
25](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-35-320.jpg)
![19 1.4 Mathematics Modules
A module can also be made accessible under an alias. For example, the math
module can be made available under the alias m with the command
import math as m
Now the prefix to be used is m rather than math:
import math as m
print(m.log(m.sin(0.5)))
-0.735166686385
The contents of a module can be printed by calling dir(module). Here is how
to obtain a list of the functions in the math module:
import math
dir(math)
[’__doc__’, ’__name__’, ’acos’, ’asin’, ’atan’,
’atan2’, ’ceil’, ’cos’, ’cosh’, ’e’, ’exp’, ’fabs’,
’floor’, ’fmod’, ’frexp’, ’hypot’, ’ldexp’, ’log’,
’log10’, ’modf’, ’pi’, ’pow’, sign’, sin’, ’sinh’,
’sqrt’, ’tan’, ’tanh’]
Most of these functions are familiar to programmers. Note that the module in-
cludes two constants: π and e.
cmath Module
The cmath module provides many of the functions found in the math module, but
these functions accept complex numbers. The functions in the module are
[’__doc__’, ’__name__’, ’acos’, ’acosh’, ’asin’, ’asinh’,
’atan’, ’atanh’, ’cos’, ’cosh’, ’e’, ’exp’, ’log’,
’log10’, ’pi’, ’sin’, ’sinh’, ’sqrt’, ’tan’, ’tanh’]
Here are examples of complex arithmetic:
from cmath import sin
x = 3.0 -4.5j
y = 1.2 + 0.8j
z = 0.8
print(x/y)
(-2.56205313375e-016-3.75j)
print(sin(x))
(6.35239299817+44.5526433649j)
print(sin(z))
(0.7173560909+0j)](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-37-320.jpg)
![20 Introduction to Python
1.5 numpy Module
General Information
The numpy module2
is not a part of the standard Python release. As pointed out ear-
lier, it must be installed separately (the installation is very easy). The module intro-
duces array objects that are similar to lists, but can be manipulated by numerous
functions contained in the module. The size of an array is immutable, and no empty
elements are allowed.
The complete set of functions in numpy is far too long to be printed in its entirety.
The following list is limited to the most commonly used functions.
[’complex’, ’float’, ’abs’, ’append’, arccos’,
’arccosh’, ’arcsin’, ’arcsinh’, ’arctan’, ’arctan2’,
’arctanh’, ’argmax’, ’argmin’, ’cos’, ’cosh’, ’diag’,
’diagonal’, ’dot’, ’e’, ’exp’, ’floor’, ’identity’,
’inner, ’inv’, ’log’, ’log10’, ’max’, ’min’,
’ones’, ’outer’, ’pi’, ’prod’ ’sin’, ’sinh’, ’size’,
’solve’, ’sqrt’, ’sum’, ’tan’, ’tanh’, ’trace’,
’transpose’, ’vectorize’,’zeros’]
Creating an Array
Arrays can be created in several ways. One of them is to use the array function to
turn a list into an array:
array(list,type)
Following are two examples of creating a 2 × 2 array with floating-point elements:
from numpy import array
a = array([[2.0, -1.0],[-1.0, 3.0]])
print(a)
[[ 2. -1.]
[-1. 3.]]
b = array([[2, -1],[-1, 3]],float)
print(b)
[[ 2. -1.]
[-1. 3.]]
Other available functions are
zeros((dim1,dim2),type)
which creates a dim1 × dim2 array and fills it with zeroes, and
ones((dim1,dim2),type)
which fills the array with ones. The default type in both cases is float.
2 NumPy is the successor of older Python modules called Numeric and NumArray. Their interfaces
and capabilities are very similar. Although Numeric and NumArray are still available, they are no
longer supported.](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-38-320.jpg)
![21 1.5 numpy Module
Finally, there is the function
arange(from,to,increment)
which works just like the range function, but returns an array rather than a se-
quence. Here are examples of creating arrays:
from numpy import *
print(arange(2,10,2))
[2 4 6 8]
print(arange(2.0,10.0,2.0))
[ 2. 4. 6. 8.]
print(zeros(3))
[ 0. 0. 0.]
print(zeros((3),int))
[0 0 0]
print(ones((2,2)))
[[ 1. 1.]
[ 1. 1.]]
Accessing and Changing Array Elements
If a is a rank-2 array, then a[i,j] accesses the element in row i and column j,
whereas a[i] refers to row i. The elements of an array can be changed by assign-
ment as follows:
from numpy import *
a = zeros((3,3),int)
print(a)
[[0 0 0]
[0 0 0]
[0 0 0]]
a[0] = [2,3,2] # Change a row
a[1,1] = 5 # Change an element
a[2,0:2] = [8,-3] # Change part of a row
print(a)
[[ 2 3 2]
[ 0 5 0]
[ 8 -3 0]]
Operations on Arrays
Arithmetic operators work differently on arrays than they do on tuples and lists—the
operation is broadcast to all the elements of the array; that is, the operation is applied
to each element in the array. Here are examples:
from numpy import array
a = array([0.0, 4.0, 9.0, 16.0])](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-39-320.jpg)
![22 Introduction to Python
print(a/16.0)
[ 0. 0.25 0.5625 1. ]
print(a - 4.0)
[ -4. 0. 5. 12.]
The mathematical functions available in numpy are also broadcast, as follows:
from numpy import array,sqrt,sin
a = array([1.0, 4.0, 9.0, 16.0])
print(sqrt(a))
[ 1. 2. 3. 4.]
print(sin(a))
[ 0.84147098 -0.7568025 0.41211849 -0.28790332]
Functions imported from the math module will work on the individual elements,
of course, but not on the array itself. An example follows:
from numpy import array
from math import sqrt
a = array([1.0, 4.0, 9.0, 16.0])
print(sqrt(a[1]))
2.0
print(sqrt(a))
Traceback (most recent call last):
.
.
.
TypeError: only length-1 arrays can be converted to Python scalars
Array Functions
There are numerous functions in numpy that perform array operations and other use-
ful tasks. Here are a few examples:
from numpy import *
A = array([[4,-2,1],[-2,4,-2],[1,-2,3]],float)
b = array([1,4,3],float)
print(diagonal(A)) # Principal diagonal
[ 4. 4. 3.]
print(diagonal(A,1)) # First subdiagonal
[-2. -2.]
print(trace(A)) # Sum of diagonal elements
11.0
print(argmax(b)) # Index of largest element
1
print(argmin(A,axis=0)) # Indices of smallest col. elements
[1 0 1]](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-40-320.jpg)
![23 1.5 numpy Module
print(identity(3)) # Identity matrix
[[ 1. 0. 0.]
[ 0. 1. 0.]
[ 0. 0. 1.]]
There are three functions in numpy that compute array products. They are illus-
trated by the following program. For more details, see Appendix A2.
from numpy import *
x = array([7,3])
y = array([2,1])
A = array([[1,2],[3,2]])
B = array([[1,1],[2,2]])
# Dot product
print(dot(x,y) =n,dot(x,y)) # {x}.{y}
print(dot(A,x) =n,dot(A,x)) # [A]{x}
print(dot(A,B) =n,dot(A,B)) # [A][B]
# Inner product
print(inner(x,y) =n,inner(x,y)) # {x}.{y}
print(inner(A,x) =n,inner(A,x)) # [A]{x}
print(inner(A,B) =n,inner(A,B)) # [A][B_transpose]
# Outer product
print(outer(x,y) =n,outer(x,y))
print(outer(A,x) =n,outer(A,x))
print(outer(A,B) =n,outer(A,B))
The output of the program is
dot(x,y) =
17
dot(A,x) =
[13 27]
dot(A,B) =
[[5 5]
[7 7]]
inner(x,y) =
17
inner(A,x) =
[13 27]
inner(A,B) =
[[ 3 6]
[ 5 10]]
outer(x,y) =](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-41-320.jpg)
![24 Introduction to Python
[[14 7]
[ 6 3]]
outer(A,x) =
[[ 7 3]
[14 6]
[21 9]
[14 6]]
Outer(A,B) =
[[1 1 2 2]
[2 2 4 4]
[3 3 6 6]
[2 2 4 4]]
Linear Algebra Module
The numpy module comes with a linear algebra module called linalg that contains
routine tasks such as matrix inversion and solution of simultaneous equations. For
example,
from numpy import array
from numpy.linalg import inv,solve
A = array([[ 4.0, -2.0, 1.0],
[-2.0, 4.0, -2.0],
[ 1.0, -2.0, 3.0]])
b = array([1.0, 4.0, 2.0])
print(inv(A)) # Matrix inverse
[[ 0.33333333 0.16666667 0. ]
[ 0.16666667 0.45833333 0.25 ]
[ 0. 0.25 0.5 ]]
print(solve(A,b)) # Solve [A]{x} = {b}
[ 1. , 2.5, 2. ]
Copying Arrays
We explained earlier that if a is a mutable object, such as a list, the assignment state-
ment b = a does not result in a new object b, but simply creates a new reference to
a, called a deep copy. This also applies to arrays. To make an independent copy of an
array a, use the copy method in the numpy module:
b = a.copy()
Vectorizing Algorithms
Sometimes the broadcasting properties of the mathematical functions in the numpy
module can be used to replace loops in the code. This procedure is known as](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-42-320.jpg)
![Hamblen put it. It was typed in pencil. I think he meant is printed in
pencil.
I asked him to immediately set about to obtain copies of that, as
I was leaving town, and to get a copy of the message, and to do
that it would be necessary for him to see Mr. Pirtle, our bookkeeping
manager, and to get the message, put it in an envelope, and address
it to my personal attention while I was in Kansas City.
Mr. Liebeler. Did Mr. Hamblen tell you anything about what the
message said or to whom it was addressed at this first conversation?
Mr. Wilcox. No, he didn't. We left for Kansas City and was there
over the Thanksgiving holidays.
Mr. Liebeler. You and your family?
Mr. Wilcox. Yes. And while there, this article appeared in the
Kansas City Star, their evening paper; very much the same story as
appeared in the Dallas Times Herald.
Mr. Liebeler. I show you a copy of a clipping which purports to
have appeared in the Dallas Times Herald on November 30, 1963. I
ask you if this tells a story similar to the report you saw in the
Kansas City Star?
Mr. Wilcox. Yes, same story.
Mr. Liebeler. We will mark this clipping as Exhibit 3002 on the
deposition of Laurance R. Wilcox, at Dallas, Tex., March 31, 1964. I
have initialed the clipping in question, Mr. Wilcox, and ask you to do
the same.
Mr. Wilcox. [Initials.] Immediately when I read this story in the
Kansas City paper I recognized it as being the same story that Mr.
Hamblen had told me just a couple of days before in my office.
We returned to Dallas Sunday, and immediately on my return to
the office Monday——
Mr. Liebeler. Monday, I suggest would have been December 2,
1963?](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-59-320.jpg)
![Mr. Liebeler. I mark that letter as Exhibit 3003, on the deposition
of Mr. Laurance R. Wilcox, at Dallas, Tex., March 31, 1964. I have
initialed the copy in question, Mr. Wilcox, and would like to have you
do the same thing if you would.
Mr. Wilcox. [Initials.]
Mr. Liebeler. Before we go on, I call your attention to the fact
that this letter of December 3, 1963, starts out by saying This is to
supplement my report of December 2 in connection with newspaper
article regarding messages and money order to Ruby or Oswald.
Do you have a copy of your report of December 2, referred to in
Exhibit 3003?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. May we mark that as Exhibit No. 3004 on the
deposition of Mr. Laurance R. Wilcox, Dallas, Tex., March 31, 1964?
Mr. Wilcox. [Hands document to attorney.]
Mr. Liebeler. I have initialed Exhibit No. 3004 and you have done
likewise, have you not?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. You had just indicated that you had talked to Mr.
Hamblen upon your return concerning the newspaper story. Would
you tell us what happened next?
Mr. Wilcox. I asked Mr. Hamblen if he had obtained the message
from the bookkeeping department as I had requested he do, and he
stated that he had not found it and did not have it and it wasn't
available.
On December 9, Mr. Semingsen, vice president of Western
Union, was at my office, and at this time we interviewed several
employees for the purpose of finding out if any of them could recall
having seen Oswald in our office. We also discussed the Ruby money
orders.](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-61-320.jpg)
![Mr. Wilcox. Yes. This statement was obtained at the time I
discussed the matter with Mr. Hamblen on December 2, Monday.
That was this statement.
Mr. Liebeler. That visit with Mr. Hamblen is described in Exhibit
No. 3003, your letter to Mr. Semingsen on December 3, 1963, is that
correct?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. And the statement of Mr. Hamblen, dated December
2, 1963, was obtained at that time? That time being the first time
that you talked to Mr. Hamblen after this matter appeared in the
newspapers; is that correct?
Mr. Wilcox. Yes.
Mr. Liebeler. Do you have the original of Mr. Hamblen's statement
of December 2, 1963?
Mr. Wilcox. I can't find it in my file. I am wondering if it was sent
to Mr. Semingsen's file, but we have gone through that.
Mr. Liebeler. I show you a thermofax copy that has been provided
to us, and ask you if that is an accurate copy of the original
statement?
Mr. Wilcox. Yes. That was made in our office on the thermofax
machine.
Mr. Liebeler. Let us mark the thermofax copy of the statement of
Mr. C. A. Hamblen to Mr. Wilcox dated December 2, 1963, as Exhibit
3005 on the deposition of Mr. Laurance R. Wilcox, at Dallas, Tex., on
March 31, 1964.
I have initialed Exhibit 3005, Mr. Wilcox. Will you do the same?
Mr. Wilcox. [Initials.]
Mr. Liebeler. Does the statement of Mr. Hamblen, Exhibit No.
3005, accurately reflect the conversation which you had with him on
December 2, 1963?](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-63-320.jpg)
![Mr. Wilcox. On December 4, 1963.
Mr. Liebeler. Do you have a copy of that statement?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. Could we mark that as Exhibit No. 3006 on the
deposition of Laurance R. Wilcox, at Dallas, Tex., March 31, 1964?
In this case you have actually provided me the original of Mr.
Lewis' statement, is that correct?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. I have initialed Exhibit No. 3006 and would like to
have you do the same, if you would.
Mr. Wilcox. [Initials.]
Mr. Liebeler. Mr. Lewis' statement refers to that portion of Mr.
Hamblen's statement, dated December 2, 1963, which we just read,
does it not?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. And Mr. Lewis says that he does recall the difficulty
he had paying a small money order to a gentleman who struck him
as being a feminine type of person, but says he cannot remember
the name of that person; is that correct?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. Do you remember discussing that with Mr. Lewis to
any extent other than as reflected on this statement?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. Could you tell us what you said and what Mr. Lewis
said about this?
Before you tell us what was said, this discussion that you are
referring to was held on December 4, 1963, or some other time?
Mr. Wilcox. Yes, sir; at that time.](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-65-320.jpg)
![Mr. Wilcox. That's right.
Mr. Liebeler. Mr. Semingsen's memorandum also indicates that a
search was made of the files in Fort Worth and in New Orleans, but
you have no direct personal knowledge of what happened at those
offices, do you?
Mr. Wilcox. No, sir.
Mr. Liebeler. We also have here a statement from Mr. Hamblen
dated December 5, 1963. Do you have the original of that
statement?
Mr Wilcox. Yes; I do have.
Mr. Liebeler. We will mark the original as Exhibit No. 3007 on the
deposition of Laurance R. Wilcox, at Dallas, Tex., March 31, 1963.
I have initialed Exhibit No. 3007, Mr. Wilcox, and I would like you
do the same.
Mr. Wilcox. [Initials.]
Mr. Liebeler. This statement was apparently taken on December
5, 1963. Do you recall the circumstances under which it was given?
Mr. Wilcox. This was following a meeting and discussion that we
held on December 4, at which time we discussed the money orders
and messages with Mr. Hamblen, Mrs. McClure, and Mr. Lewis.
Mr. Liebeler. Who was present at this discussion other than the
individuals you just mentioned and yourself?
(Mr. Wilcox phoned his office re: correct date of meeting.)
Mr. Wilcox. I did hold this meeting.
Mr. Liebeler. Let the record indicate that Mr. Wilcox has just
conferred telephonically with Mr. Semingsen and wishes now to
clarify the statements concerning the time on which certain meetings
were held with the employees.
Mr. Wilcox. I did hold the meetings with these people on
December 4, and did obtain these statements, including the](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-75-320.jpg)
![Mr. Liebeler. You picked that period because Mr. Hamblen said
that he thought the event occurred about 10 days prior to the
assassination; is that correct?
Mr. Wilcox. That's right. Now, I think you have the only
thermofax of Mrs. McClure's. Would you want the original?
Mr. Liebeler. Yes; I do want to mark the original statement of
Mrs. McClure, dated December 4, 1963, which has been previously
referred to by Mr. Wilcox. I will mark it as Exhibit 3008 on the
deposition of Mr. Laurance R. Wilcox at Dallas, Tex., on March 31,
1964.
I have initialed that exhibit, Mr. Wilcox, and would like to have
you do the same.
Mr. Wilcox. [Initials.]
Mr. Liebeler. Who prepared Mrs. McClure's statement?
Mr. Wilcox. Mrs. McClure wrote that.
Mr. Liebeler. She typed it on the typewriter herself?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. Were you present at the time she prepared it?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. Did you discuss these incidents with her prior to the
statement?
Mr. Wilcox. Only during the course of the meeting that she
attended in my office.
Mr. Liebeler. During the course of that time, Mrs. McClure
indicated to you, did she not, that she had no recollection of any of
the events described by Mr. Hamblen concerning the message
allegedly sent to Washington, D.C.?
Mr. Wilcox. That's right.](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-77-320.jpg)
![Mr. Liebeler. Let's mark these telegrams as Exhibits 3009 through
3014. I have initialed each one of them and would like to have you
do the same. I have not indicated on each one of them in detail. It is
on your deposition and on the dates, as I have on the others, but
that fact will appear from the record.
Mr. Wilcox. [Initials.]
Mr. Liebeler. You have now provided me with six telegrams,
cables which have been marked for identification as Exhibits 3009
through 3014, and it is my understanding that these telegrams and
cables were all shown to Mr. Hamblen and he was unable to identify
any of them as answering the description of the telegram or
cablegram with which Mrs. McClure had had difficulty, and which Mr.
Hamblen thought had been sent by a person resembling Lee Harvey
Oswald, is that correct?
Mr. Wilcox. That's right.
Mr. Liebeler. And a thorough search of the files along the lines
that you have previously indicated was unable to produce any other
telegrams even remotely falling into the category described by Mr.
Hamblen; i.e., a telegram to Washington, D.C. or in a peculiar hand
script such as described by Mr. Hamblen to any destination, whether
it be Washington or otherwise; is that correct?
Mr. Wilcox. That's right.
Mr. Liebeler. Now, you have provided me with a letter from
yourself to Mr. Semingsen, dated December 6, 1963, which we will
mark as Exhibit 3015, on deposition of Laurance R. Wilcox, at Dallas,
Tex., March 31, 1964. I have initialed that exhibit and ask you to do
the same, sir.
Mr. Wilcox. [Initials.]
Mr. Liebeler. And ask you if you prepared the original of that
letter on or about December 6, 1963?
Mr. Wilcox. That's right.](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-78-320.jpg)
![Mr. Liebeler. You sent the original of it to Mr. Semingsen, did you
not?
Mr. Wilcox. Yes, sir; that's right.
Mr. Liebeler. And the letter accurately sets forth the events
preceding that date which we have been discussing here, does it
not?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. You have also provided me with a copy of a letter
from yourself to Mr. Semingsen, dated December 9, 1963, to which
is attached the original of statement from Mrs. Betty Bedwell, dated
December 6, 1963, and A. I. English, dated December 6, 1963. I
notice that Mr. English's statement is not signed.
Mr. Wilcox. [Signs.]
Mr. Liebeler. And I ask you if the original of it was signed at the
time you received it?
Mr. Wilcox. No, sir; he just signed this on the typewriter to me.
Mr. Liebeler. No; he did not himself sign it?
Mr. Wilcox. No.
Mr. Liebeler. You have indicated below that it was in effect signed
by Mr. English when he delivered it to you?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. Also attached is a statement of Miss Bess Mildred
Francis, dated December 9, 1963. Also attached is a statement of
Doyle E. Lane, dated December 9, 1963; and one of Mr. E. T. Pirtle,
dated December 6, 1963; and one of Ward Townsley dated
December 6, 1963. Did you receive those statements from the
individuals thus described?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. We have marked the letter described above,
together with the attachments just described as Exhibit 3016 on the](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-79-320.jpg)
![deposition of Mr. Laurance R. Wilcox at Dallas, Tex., on March 31,
1964. I have initialed the first page of that exhibit and would like to
have you do the same.
Mr. Wilcox. [Initials.]
Mr. Liebeler. Did you send the original of your letter dated
December 9, 1963, to Mr. Semingsen on or about that date?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. The statements made in that letter are true and
correct to the best of your knowledge, are they not?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. You have also given me the original of a statement
dated December 6, 1963, of Mr. Doyle E. Lane. Was that also
attached to your letter of December 9, 1963, to Mr. Semingsen?
Mr. Wilcox. No, sir.
Mr. Liebeler. Did you receive yourself from Mr. Lane the
statement just described?
Mr. Wilcox. Yes, sir.
Mr. Liebeler. Let me mark it as Exhibit 3017 on this deposition.
Would you initial the statement of Mr. Lane which we have
marked 3017?
Mr. Wilcox. Yes, sir. [Initials.]
Mr. Liebeler. Thank you.
You previously testified, Mr. Wilcox, that you had a meeting on
or about December 4, 1963, with Mr. Hamblen, Mrs. McClure, and
Mr. Lewis in regard to the meeting we have been discussing, and
you also testified that you thought you had an additional meeting on
December 9, 1963, at which Mr. Semingsen was present. Would you
care to elaborate on that?](https://image.slidesharecdn.com/21578834-250610045303-b34e4142/85/Numerical-Methods-In-Engineering-With-Python-3-3rd-Edition-Jaan-Kiusalaas-80-320.jpg)
Numerical Methods In Engineering With Python 3 3rd Edition Jaan Kiusalaas
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- 5. more information -www.cambridge.org/9781107033856
- 7. Numerical Methods inEngineering with Python 3 This book is an introduction to numerical methods for students in engi- neering. It covers the usual topics found in an engineering course: solu- tion of equations, interpolation and data fitting, solution of differential equations, eigenvalue problems, and optimization. The algorithms are implemented in Python 3, a high-level programming language that ri- vals MATLAB R in readability and ease of use. All methods include pro- grams showing how the computer code is utilized in the solution of problems. The book is based on Numerical Methods in Engineering with Python, which used Python 2. Apart from the migration from Python 2 to Python 3, the major change in this new text is the introduction of the Python plotting package Matplotlib. Jaan Kiusalaas is a Professor Emeritus in the Department of Engi- neering Science and Mechanics at Pennsylvania State University. He has taught computer methods, including finite element and bound- ary element methods, for more than 30 years. He is also the co-author or author of four books – Engineering Mechanics: Statics; Engineering Mechanics: Dynamics; Mechanics of Materials; Numerical Methods in Engineering with MATLAB (2nd edition); and two previous editions of Numerical Methods in Engineering with Python.
- 9. NUMERICAL METHODS IN ENGINEERING WITH PYTHON3 Jaan Kiusalaas The Pennsylvania State University
- 10. cambridge university press Cambridge,New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9781107033856 C Jaan Kiusalaas 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Kiusalaas, Jaan. Numerical methods in engineering with Python 3 / Jaan Kiusalaas. pages cm Includes bibliographical references and index. ISBN 978-1-107-03385-6 1. Engineering mathematics – Data processing. 2. Python (Computer program language) I. Title. TA345.K58 2013 620.002855133–dc23 2012036775 ISBN 978-1-107-03385-6 Hardback Additional resources for this publication at www.cambridge.org/kiusalaaspython. Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. Contents Preface....................................ix 1 Introduction toPython ....................................................... 1 1.1 General Information ............................................................. 1 1.2 Core Python.......................................................................4 1.3 Functions and Modules.........................................................16 1.4 Mathematics Modules .......................................................... 18 1.5 numpy Module.................................................................. 20 1.6 Plotting with matplotlib.pyplot ........................................ 25 1.7 Scoping of Variables ............................................................ 28 1.8 Writing and Running Programs ............................................... 29 2 Systems of Linear Algebraic Equations....................................31 2.1 Introduction ..................................................................... 31 2.2 Gauss Elimination Method ..................................................... 37 2.3 LU Decomposition Methods ................................................... 44 Problem Set 2.1........................................................................55 2.4 Symmetric and Banded Coefficient Matrices.................................59 2.5 Pivoting .......................................................................... 69 Problem Set 2.2........................................................................78 ∗ 2.6 Matrix Inversion.................................................................84 ∗ 2.7 Iterative Methods ............................................................... 87 Problem Set 2.3........................................................................98 2.8 Other Methods ................................................................ 102 3 Interpolation and Curve Fitting...........................................104 3.1 Introduction....................................................................104 3.2 Polynomial Interpolation ..................................................... 105 3.3 Interpolation with Cubic Spline .............................................. 120 Problem Set 3.1 ...................................................................... 126 3.4 Least-Squares Fit...............................................................129 Problem Set 3.2 ...................................................................... 141 4 Roots of Equations ......................................................... 145 4.1 Introduction....................................................................145 4.2 Incremental Search Method .................................................. 146 v
- 12. vi Contents 4.3 Methodof Bisection...........................................................148 4.4 Methods Based on Linear Interpolation.....................................151 4.5 Newton-Raphson Method .................................................... 156 4.6 Systems of Equations..........................................................161 Problem Set 4.1 ...................................................................... 166 ∗ 4.7 Zeros of Polynomials .......................................................... 173 Problem Set 4.2 ...................................................................... 180 4.8 Other Methods ................................................................ 182 5 Numerical Differentiation ................................................. 183 5.1 Introduction....................................................................183 5.2 Finite Difference Approximations............................................183 5.3 Richardson Extrapolation ..................................................... 188 5.4 Derivatives by Interpolation..................................................191 Problem Set 5.1 ...................................................................... 195 6 Numerical Integration......................................................199 6.1 Introduction....................................................................199 6.2 Newton-Cotes Formulas ...................................................... 200 6.3 Romberg Integration..........................................................207 Problem Set 6.1 ...................................................................... 212 6.4 Gaussian Integration .......................................................... 216 Problem Set 6.2 ...................................................................... 230 ∗ 6.5 Multiple Integrals..............................................................232 Problem Set 6.3 ...................................................................... 243 7 Initial Value Problems......................................................246 7.1 Introduction....................................................................246 7.2 Euler’s Method.................................................................247 7.3 Runge-Kutta Methods ........................................................ 252 Problem Set 7.1 ...................................................................... 263 7.4 Stability and Stiffness ......................................................... 268 7.5 Adaptive Runge-Kutta Method .............................................. 271 7.6 Bulirsch-Stoer Method ........................................................ 280 Problem Set 7.2 ...................................................................... 287 7.7 Other Methods ................................................................ 292 8 Two-Point Boundary Value Problems.....................................293 8.1 Introduction....................................................................293 8.2 Shooting Method..............................................................294 Problem Set 8.1 ...................................................................... 304 8.3 Finite Difference Method.....................................................307 Problem Set 8.2 ...................................................................... 317 9 Symmetric Matrix Eigenvalue Problems ................................. 321 9.1 Introduction....................................................................321 9.2 Jacobi Method ................................................................. 324 9.3 Power and Inverse Power Methods..........................................336 Problem Set 9.1 ...................................................................... 345 9.4 Householder Reduction to Tridiagonal Form ............................... 351
- 13. vii Contents 9.5 Eigenvaluesof Symmetric Tridiagonal Matrices ............................ 359 Problem Set 9.2 ...................................................................... 368 9.6 Other Methods ................................................................ 373 10 Introduction to Optimization ............................................. 374 10.1 Introduction....................................................................374 10.2 Minimization Along a Line ................................................... 376 10.3 Powell’s Method ............................................................... 382 10.4 Downhill Simplex Method .................................................... 392 Problem Set 10.1.....................................................................399 Appendices..................................................................407 A1 Taylor Series....................................................................407 A2 Matrix Algebra.................................................................410 List of Program Modules (by Chapter) ....... 417 Index....................................421
- 15. Preface This book istargeted toward engineers and engineering students of advanced stand- ing (juniors, seniors, and graduate students). Familiarity with a computer language is required; knowledge of engineering mechanics (statics, dynamics, and mechanics of materials) is useful, but not essential. The primary purpose of the text is to teach numerical methods. It is not a primer on Python programming. We introduce just enough Python to implement the nu- merical algorithms. That leaves the vast majority of the language unexplored. Most engineers are not programmers, but problem solvers. They want to know what methods can be applied to a given problem, what their strengths and pitfalls are, and how to implement them. Engineers are not expected to write computer code for basic tasks from scratch; they are more likely to use functions and subroutines that have been already written and tested. Thus, programming by engineers is largely confined to assembling existing bits of code into a coherent package that solves the problem at hand. The “bit” of code is usually a function that implements a specific task. For the user the details of the code are unimportant. What matters are the interface (what goes in and what comes out) and an understanding of the method on which the al- gorithm is based. Because no numerical algorithm is infallible, the importance of understanding the underlying method cannot be overemphasized; it is, in fact, the rationale behind learning numerical methods. This book attempts to conform to the views outlined earlier. Each numerical method is explained in detail and its shortcomings are pointed out. The examples that follow individual topics fall into two categories: hand computations that illus- trate the inner workings of the method, and small programs that show how the com- puter code is utilized in solving a problem. Problems that require programming are marked with . The material consists of the usual topics covered in an engineering course on nu- merical methods: solution of equations, interpolation and data fitting, numerical dif- ferentiation and integration, solution of ordinary differential equations, and eigen- value problems. The choice of methods within each topic is tilted toward relevance to engineering problems. For example, there is an extensive discussion of symmetric, sparsely populated coefficient matrices in the solution of simultaneous equations. ix
- 16. x Preface In thesame vein, the solution of eigenvalue problems concentrates on methods that efficiently extract specific eigenvalues from banded matrices. An important criterion used in the selection of methods was clarity. Algorithms requiring overly complex bookkeeping were rejected regardless of their efficiency and robustness. This decision, which was taken with great reluctance, is in keeping with the intent to avoid emphasis on programming. The selection of algorithms was also influenced by current practice. This disqual- ified several well-known historical methods that have been overtaken by more recent developments. For example, the secant method for finding roots of equations was omitted as having no advantages over Ridder’s method. For the same reason, the mul- tistep methods used to solve differential equations (e.g., Milne and Adams methods) were left out in favor of the adaptive Runge-Kutta and Bulirsch-Stoer methods. Notably absent is a chapter on partial differential equations. It was felt that this topic is best treated by finite element or boundary element methods, which are out- side the scope of this book. The finite difference model, which is commonly intro- duced in numerical methods texts, is just too impractical in handling multidimen- sional boundary value problems. As usual, the book contains more material than can be covered in a three-credit course. The topics that can be skipped without loss of continuity are tagged with an asterisk (*). What Is New in This Edition This book succeeds Numerical Methods in Engineering with Python, which was based on Python 2. As the title implies, the new edition migrates to Python 3. Because the two versions are not entirely compatible, almost all computer routines required some code changes. We also took the opportunity to make a few changes in the material covered: • An introduction to the Python plotting package matplotlib.pyplot was added to Chapter 1. This package is used in numerous example problems, mak- ing the book more graphics oriented than before. • The function plotPoly, which plots data points and the corresponding polyno- mial interpolant, was added to Chapter 3. This program provides a convenient means of evaluating the fit of the interpolant. • At the suggestion of reviewers, the Taylor series method of solving initial value problems in Chapter 7 was dropped. It was replaced by Euler’s method. • The Jacobi method for solving eigenvalue problems in Chapter 9 now uses the threshold method in choosing the matrix elements marked for elimination. This change increases the speed of the algorithm. • The adaptive Runge-Kutta method in Chapter 7 was recoded, and the Cash-Karp coefficients replaced with the Dormand-Prince coefficients. The result is a more efficient algorithm with tighter error control.
- 17. xi Preface • Twenty-onenew problems were introduced, most of them replacing old prob- lems. • Some example problems in Chapters 4 and 7 were rearranged or replaced with new problems. The result of these changes is better coordination of examples with the text. The programs listed in the book were tested with Python 3.2 under Windows 7. The source codes are available at www.cambridge.org/kiusalaaspython.
- 19. 1 Introduction toPython 1.1 General Information Quick Overview This chapter is not a comprehensive manual of Python. Its sole aim is to provide suf- ficient information to give you a good start if you are unfamiliar with Python. If you know another computer language, and we assume that you do, it is not difficult to pick up the rest as you go. Python is an object-oriented language that was developed in the late 1980s as a scripting language (the name is derived from the British television series, Monty Python’s Flying Circus). Although Python is not as well known in engineering circles as are some other languages, it has a considerable following in the programming com- munity. Python may be viewed as an emerging language, because it is still being de- veloped and refined. In its current state, it is an excellent language for developing engineering applications. Python programs are not compiled into machine code, but are run by an interpreter.1 The great advantage of an interpreted language is that programs can be tested and debugged quickly, allowing the user to concentrate more on the principles behind the program and less on the programming itself. Because there is no need to compile, link, and execute after each correction, Python programs can be developed in much shorter time than equivalent Fortran or C programs. On the negative side, interpreted programs do not produce stand-alone applications. Thus a Python pro- gram can be run only on computers that have the Python interpreter installed. Python has other advantages over mainstream languages that are important in a learning environment: • Python is an open-source software, which means that it is free; it is included in most Linux distributions. • Python is available for all major operating systems (Linux, Unix, Windows, Mac OS, and so on). A program written on one system runs without modification on all systems. 1 The Python interpreter also compiles byte code, which helps speed up execution somewhat. 1
- 20. 2 Introduction toPython • Python is easier to learn and produces more readable code than most languages. • Python and its extensions are easy to install. Development of Python has been clearly influenced by Java and C++, but there is also a remarkable similarity to MATLABR (another interpreted language, very popular in scientific computing). Python implements the usual concepts of object-oriented languages such as classes, methods, inheritance etc. We do not use object-oriented programming in this text. The only object that we need is the N-dimensional array available in the module numpy (this module is discussed later in this chapter). To get an idea of the similarities and differences between MATLAB and Python, let us look at the codes written in the two languages for solution of simultaneous equations Ax = b by Gauss elimination. Do not worry about the algorithm itself (it is explained later in the text), but concentrate on the semantics. Here is the function written in MATLAB: function x = gaussElimin(a,b) n = length(b); for k = 1:n-1 for i= k+1:n if a(i,k) ˜= 0 lam = a(i,k)/a(k,k); a(i,k+1:n) = a(i,k+1:n) - lam*a(k,k+1:n); b(i)= b(i) - lam*b(k); end end end for k = n:-1:1 b(k) = (b(k) - a(k,k+1:n)*b(k+1:n))/a(k,k); end x = b; The equivalent Python function is from numpy import dot def gaussElimin(a,b): n = len(b) for k in range(0,n-1): for i in range(k+1,n): if a[i,k] != 0.0: lam = a [i,k]/a[k,k] a[i,k+1:n] = a[i,k+1:n] - lam*a[k,k+1:n] b[i] = b[i] - lam*b[k] for k in range(n-1,-1,-1): b[k] = (b[k] - dot(a[k,k+1:n],b[k+1:n]))/a[k,k] return b
- 21. 3 1.1 GeneralInformation The command from numpy import dot instructs the interpreter to load the function dot (which computes the dot product of two vectors) from the module numpy. The colon (:) operator, known as the slicing operator in Python, works the same way as it does in MATLAB and Fortran90—it defines a slice of an array. The statement for k = 1:n-1 in MATLAB creates a loop that is executed with k = 1, 2, . . . , n − 1. The same loop appears in Python as for k in range(n-1). Here the function range(n-1) creates the sequence [0, 1, . . . , n − 2]; k then loops over the elements of the sequence. The differences in the ranges of k reflect the native offsets used for arrays. In Python all sequences have zero offset, meaning that the index of the first element of the sequence is always 0. In contrast, the native offset in MATLAB is 1. Also note that Python has no end statements to terminate blocks of code (loops, subroutines, and so on). The body of a block is defined by its indentation; hence in- dentation is an integral part of Python syntax. Like MATLAB, Python is case sensitive. Thus the names n and N would represent different objects. Obtaining Python The Python interpreter can be downloaded from http : //www.python.org/getit It normally comes with a nice code editor called Idle that allows you to run programs directly from the editor. If you use Linux, it is very likely that Python is already in- stalled on your machine. The download includes two extension modules that we use in our programs: the numpy module that contains various tools for array operations, and the matplotlib graphics module utilized in plotting. The Python language is well documented in numerous publications. A com- mendable teaching guide is Python by Chris Fehly (Peachpit Press, CA, 2nd ed.). As a reference, Python Essential Reference by David M. Beazley (Addison-Wesley, 4th ed.) is highly recommended. Printed documentation of the extension modules is scant. However, tutorials and examples can be found on various websites. Our favorite ref- erence for numpy is http://www.scipy.org/Numpy Example List For matplotlib we rely on http://matplotlib.sourceforge.net/contents.html If you intend to become a serious Python programmer, you may want to acquire A Primer on Scientific Programming with Python by Hans P. Langtangen (Springer- Verlag, 2009).
- 22. 4 Introduction toPython 1.2 Core Python Variables In most computer languages the name of a variable represents a value of a given type stored in a fixed memory location. The value may be changed, but not the type. This is not so in Python, where variables are typed dynamically. The following interac- tive session with the Python interpreter illustrates this feature ( is the Python prompt): b = 2 # b is integer type print(b) 2 b = b*2.0 # Now b is float type print(b) 4.0 The assignment b = 2 creates an association between the name b and the in- teger value 2. The next statement evaluates the expression b*2.0 and associates the result with b; the original association with the integer 2 is destroyed. Now b refers to the floating point value 4.0. The pound sign (#) denotes the beginning of a comment—all characters between # and the end of the line are ignored by the interpreter. Strings A string is a sequence of characters enclosed in single or double quotes. Strings are concatenated with the plus (+) operator, whereas slicing (:) is used to extract a por- tion of the string. Here is an example: string1 = ’Press return to exit’ string2 = ’the program’ print(string1 + ’ ’ + string2) # Concatenation Press return to exit the program print(string1[0:12]) # Slicing Press return A string can be split into its component parts using the split command. The components appear as elements in a list. For example, s = ’3 9 81’ print(s.split()) # Delimiter is white space [’3’, ’9’, ’81’] A string is an immutable object—its individual characters cannot be modified with an assignment statement, and it has a fixed length. An attempt to violate im- mutability will result in TypeError, as follows:
- 23. 5 1.2 CorePython s = ’Press return to exit’ s[0] = ’p’ Traceback (most recent call last): File ’’pyshell#1’’, line 1, in ? s[0] = ’p’ TypeError: object doesn’t support item assignment Tuples A tuple is a sequence of arbitrary objects separated by commas and enclosed in parentheses. If the tuple contains a single object, a final comma is required; for example, x = (2,). Tuples support the same operations as strings; they are also immutable. Here is an example where the tuple rec contains another tuple (6,23,68): rec = (’Smith’,’John’,(6,23,68)) # This is a tuple lastName,firstName,birthdate = rec # Unpacking the tuple print(firstName) John birthYear = birthdate[2] print(birthYear) 68 name = rec[1] + ’ ’ + rec[0] print(name) John Smith print(rec[0:2]) (’Smith’, ’John’) Lists A list is similar to a tuple, but it is mutable, so that its elements and length can be changed. A list is identified by enclosing it in brackets. Here is a sampling of opera- tions that can be performed on lists: a = [1.0, 2.0, 3.0] # Create a list a.append(4.0) # Append 4.0 to list print(a) [1.0, 2.0, 3.0, 4.0] a.insert(0,0.0) # Insert 0.0 in position 0 print(a) [0.0, 1.0, 2.0, 3.0, 4.0] print(len(a)) # Determine length of list 5 a[2:4] = [1.0, 1.0, 1.0] # Modify selected elements print(a) [0.0, 1.0, 1.0, 1.0, 1.0, 4.0]
- 24. 6 Introduction toPython If a is a mutable object, such as a list, the assignment statement b = a does not result in a new object b, but simply creates a new reference to a. Thus any changes made to b will be reflected in a. To create an independent copy of a list a, use the statement c = a[:], as shown in the following example: a = [1.0, 2.0, 3.0] b = a # ’b’ is an alias of ’a’ b[0] = 5.0 # Change ’b’ print(a) [5.0, 2.0, 3.0] # The change is reflected in ’a’ c = a[:] # ’c’ is an independent copy of ’a’ c[0] = 1.0 # Change ’c’ print(a) [5.0, 2.0, 3.0] # ’a’ is not affected by the change Matrices can be represented as nested lists, with each row being an element of the list. Here is a 3 × 3 matrix a in the form of a list: a = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] print(a[1]) # Print second row (element 1) [4, 5, 6] print(a[1][2]) # Print third element of second row 6 The backslash () is Python’s continuation character. Recall that Python se- quences have zero offset, so that a[0] represents the first row, a[1] the second row, etc. With very few exceptions we do not use lists for numerical arrays. It is much more convenient to employ array objects provided by the numpy module. Array objects are discussed later. Arithmetic Operators Python supports the usual arithmetic operators: + Addition − Subtraction ∗ Multiplication / Division ∗∗ Exponentiation % Modular division Some of these operators are also defined for strings and sequences as follows: s = ’Hello ’ t = ’to you’
- 25. 7 1.2 CorePython a = [1, 2, 3] print(3*s) # Repetition Hello Hello Hello print(3*a) # Repetition [1, 2, 3, 1, 2, 3, 1, 2, 3] print(a + [4, 5]) # Append elements [1, 2, 3, 4, 5] print(s + t) # Concatenation Hello to you print(3 + s) # This addition makes no sense Traceback (most recent call last): File pyshell#13, line 1, in module print(3 + s) TypeError: unsupported operand type(s) for +: ’int’ and ’str’ Python also has augmented assignment operators, such asa+ = b, that are famil- iar to the users of C. The augmented operators and the equivalent arithmetic expres- sions are shown in following table. a += b a = a + b a -= b a = a - b a *= b a = a*b a /= b a = a/b a **= b a = a**b a %= b a = a%b Comparison Operators The comparison (relational) operators return True or False. These operators are Less than Greater than = Less than or equal to = Greater than or equal to == Equal to != Not equal to Numbers of different type (integer, floating point, and so on) are converted to a common type before the comparison is made. Otherwise, objects of different type are considered to be unequal. Here are a few examples: a = 2 # Integer b = 1.99 # Floating point c = ’2’ # String print(a b) True
- 26. 8 Introduction toPython print(a == c) False print((a b) and (a != c)) True print((a b) or (a == b)) True Conditionals The if construct if condition: block executes a block of statements (which must be indented) if the condition returns True. If the condition returns False, the block is skipped. The if conditional can be followed by any number of elif (short for “else if”) constructs elif condition: block that work in the same manner. The else clause else: block can be used to define the block of statements that are to be executed if none of the if-elif clauses are true. The function sign of a illustrates the use of the condi- tionals. def sign_of_a(a): if a 0.0: sign = ’negative’ elif a 0.0: sign = ’positive’ else: sign = ’zero’ return sign a = 1.5 print(’a is ’ + sign_of_a(a)) Running the program results in the output a is positive
- 27. 9 1.2 CorePython Loops The while construct while condition: block executes a block of (indented) statements if the condition is True. After execution of the block, the condition is evaluated again. If it is still True, the block is exe- cuted again. This process is continued until the condition becomes False. The else clause else: block can be used to define the block of statements that are to be executed if the condition is false. Here is an example that creates the list [1, 1/2, 1/3, . . .]: nMax = 5 n = 1 a = [] # Create empty list while n nMax: a.append(1.0/n) # Append element to list n = n + 1 print(a) The output of the program is [1.0, 0.5, 0.33333333333333331, 0.25] We met the for statement in Section 1.1. This statement requires a target and a sequence over which the target loops. The form of the construct is for target in sequence: block You may add an else clause that is executed after the for loop has finished. The previous program could be written with the for construct as nMax = 5 a = [] for n in range(1,nMax): a.append(1.0/n) print(a) Here n is the target, and the range object [1, 2, . . . , nMax − 1] (created by calling the range function) is the sequence. Any loop can be terminated by the break
- 28. 10 Introduction toPython statement. If there is an else cause associated with the loop, it is not executed. The following program, which searches for a name in a list, illustrates the use of break and else in conjunction with a for loop: list = [’Jack’, ’Jill’, ’Tim’, ’Dave’] name = eval(input(’Type a name: ’)) # Python input prompt for i in range(len(list)): if list[i] == name: print(name,’is number’,i + 1,’on the list’) break else: print(name,’is not on the list’) Here are the results of two searches: Type a name: ’Tim’ Tim is number 3 on the list Type a name: ’June’ June is not on the list The continue statement allows us to skip a portion of an iterative loop. If the interpreter encounters the continue statement, it immediately returns to the beginning of the loop without executing the statements that follow continue. The following example compiles a list of all numbers between 1 and 99 that are divisible by 7. x = [] # Create an empty list for i in range(1,100): if i%7 != 0: continue # If not divisible by 7, skip rest of loop x.append(i) # Append i to the list print(x) The printout from the program is [7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98] Type Conversion If an arithmetic operation involves numbers of mixed types, the numbers are automatically converted to a common type before the operation is carried out.
- 29. 11 1.2 CorePython Type conversions can also achieved by the following functions: int(a) Converts a to integer float(a) Converts a to floating point complex(a) Converts to complex a + 0j complex(a,b) Converts to complex a + bj These functions also work for converting strings to numbers as long as the lit- eral in the string represents a valid number. Conversion from a float to an integer is carried out by truncation, not by rounding off. Here are a few examples: a = 5 b = -3.6 d = ’4.0’ print(a + b) 1.4 print(int(b)) -3 print(complex(a,b)) (5-3.6j) print(float(d)) 4.0 print(int(d)) # This fails: d is a string Traceback (most recent call last): File pyshell#30, line 1, in module print(int(d)) ValueError: invalid literal for int() with base 10: ’4.0’ Mathematical Functions Core Python supports only the following mathematical functions: abs(a) Absolute value of a max(sequence) Largest element of sequence min(sequence) Smallest element of sequence round(a,n) Round a to n decimal places cmp(a,b) Returns ⎧ ⎪ ⎨ ⎪ ⎩ −1 if a b 0 if a = b 1 if a b The majority of mathematical functions are available in the math module. Reading Input The intrinsic function for accepting user input is input(prompt)
- 30. 12 Introduction toPython It displays the prompt and then reads a line of input that is converted to a string. To convert the string into a numerical value use the function eval(string) The following program illustrates the use of these functions: a = input(’Input a: ’) print(a, type(a)) # Print a and its type b = eval(a) print(b,type(b)) # Print b and its type The function type(a) returns the type of the object a; it is a very useful tool in debugging. The program was run twice with the following results: Input a: 10.0 10.0 class ’str’ 10.0 class ’float’ Input a: 11**2 11**2 class ’str’ 121 class ’int’ A convenient way to input a number and assign it to the variable a is a = eval(input(prompt)) Printing Output Output can be displayed with the print function print(object1, object2, . . .) that converts object1, object2, and so on, to strings and prints them on the same line, separated by spaces. The newline character ’n’ can be used to force a new line. For example, a = 1234.56789 b = [2, 4, 6, 8] print(a,b) 1234.56789 [2, 4, 6, 8] print(’a =’,a, ’nb =’,b) a = 1234.56789 b = [2, 4, 6, 8] The print function always appends the newline character to the end of a line. We can replace this character with something else by using the keyword argument end. For example, print(object1, object2, ...,end=’ ’) replaces n with a space.
- 31. 13 1.2 CorePython Output can be formatted with the format method. The simplest form of the con- version statement is ’{:fmt1}{:fmt2}. . .’.format(arg1,arg2,...) where fmt1, fmt2,. . . are the format specifications for arg1, arg2,. . ., respectively. Typ- ically used format specifications are wd Integer w.df Floating point notation w.de Exponential notation where w is the width of the field and d is the number of digits after the decimal point. The output is right justified in the specified field and padded with blank spaces (there are provisions for changing the justification and padding). Here are several examples: a = 1234.56789 n = 9876 print(’{:7.2f}’.format(a)) 1234.57 print(’n = {:6d}’.format(n)) # Pad with spaces n = 9876 print(’n = {:06d}’.format(n)) # Pad with zeros n =009876 print(’{:12.4e} {:6d}’.format(a,n)) 1.2346e+03 9876 Opening and Closing a File Before a data file on a storage device (e.g., a disk) can be accessed, you must create a file object with the command file object = open(filename, action) where filename is a string that specifies the file to be opened (including its path if necessary) and action is one of the following strings: ’r’ Read from an existing file. ’w’ Write to a file. If filename does not exist, it is created. ’a’ Append to the end of the file. ’r+’ Read to and write from an existing file. ’w+’ Same as ’r+’, but filename is created if it does not exist. ’a+’ Same as ’w+’, but data is appended to the end of the file. It is good programming practice to close a file when access to it is no longer re- quired. This can be done with the method file object.close()
- 32. 14 Introduction toPython Reading Data from a File There are three methods for reading data from a file. The method file object.read(n) reads n characters and returns them as a string. If n is omitted, all the characters in the file are read. If only the current line is to be read, use file object.readline(n) which reads n characters from the line. The characters are returned in a string that terminates in the newline character n. Omission of n causes the entire line to be read. All the lines in a file can be read using file object.readlines() This returns a list of strings, each string being a line from the file ending with the newline character. A convenient method of extracting all the lines one by one is to use the loop for line in file object: do something with line As an example, let us assume that we have a file named sunspots.txt in the working directory. This file contains daily data of sunspot intensity, each line having the format (year/month/date/intensity), as follows: 1896 05 26 40.94 1896 05 27 40.58 1896 05 28 40.20 etc. Our task is to read the file and create a list x that contains only the intensity. Since each line in the file is a string, we first split the line into its pieces using the split command. This produces a list of strings, such as[’1896’,’05’,’26’,’40.94’]. Then we extract the intensity (element [3] of the list), evaluate it, and append the result to x. Here is the algorithm: x = [] data = open(’sunspots.txt’,’r’) for line in data: x.append(eval(line.split()[3])) data.close()
- 33. 15 1.2 CorePython Writing Data to a File The method file object.write(string) writes a string to a file, whereas file object.writelines(list of strings) is used to write a list of strings. Neither method appends a newline character to the end of a line. As an example, let us write a formatted table of k and k2 from k = 101 to 110 to the file testfile. Here is the program that does the writing: f = open(’testfile’,’w’) for k in range(101,111): f.write(’{:4d} {:6d}’.format(k,k**2)) f.write(’n’) f.close() The contents of testfile are 101 10201 102 10404 103 10609 104 10816 105 11025 106 11236 107 11449 108 11664 109 11881 110 12100 The print function can also be used to write to a file by redirecting the output to a file object: print(object1, object2, . . . ,file = file object) Apart from the redirection, this works just like the regular print function. Error Control When an error occurs during execution of a program an exception is raised and the program stops. Exceptions can be caught with try and except statements: try: do something except error: do something else
- 34. 16 Introduction toPython where error is the name of a built-in Python exception. If the exception error is not raised, the try block is executed; otherwise the execution passes to the except block. All exceptions can be caught by omitting error from the except statement. The following statement raises the exception ZeroDivisionError: c = 12.0/0.0 Traceback (most recent call last): File pyshell#0, line 1, in module c=12.0/0.0 ZeroDivisionError: float division by zero This error can be caught by try: c = 12.0/0.0 except ZeroDivisionError: print(’Division by zero’) 1.3 Functions and Modules Functions The structure of a Python function is def func name(param1, param2,. . .): statements return return values where param1, param2,. . . are the parameters. A parameter can be any Python ob- ject, including a function. Parameters may be given default values, in which case the parameter in the function call is optional. If the return statement or return values are omitted, the function returns the null object. The following function computes the first two derivatives of f (x) by finite differences: def derivatives(f,x,h=0.0001): # h has a default value df =(f(x+h) - f(x-h))/(2.0*h) ddf =(f(x+h) - 2.0*f(x) + f(x-h))/h**2 return df,ddf Let us now use this function to determine the two derivatives of arctan(x) at x = 0.5: from math import atan df,ddf = derivatives(atan,0.5) # Uses default value of h print(’First derivative =’,df) print(’Second derivative =’,ddf)
- 35. 17 1.3 Functionsand Modules Note that atan is passed to derivatives as a parameter. The output from the program is First derivative = 0.799999999573 Second derivative = -0.639999991892 The number of input parameters in a function definition may be left arbitrary. For example, in the following function definition def func(x1,x2,*x3) x1 and x2 are the usual parameters, also called positional parameters, whereas x3 is a tuple of arbitrary length containing the excess parameters. Calling this function with func(a,b,c,d,e) results in the following correspondence between the parameters: a ←→ x1, b ←→ x2, (c,d,e) ←→ x3 The positional parameters must always be listed before the excess parameters. If a mutable object, such as a list, is passed to a function where it is modified, the changes will also appear in the calling program. An example follows: def squares(a): for i in range(len(a)): a[i] = a[i]**2 a = [1, 2, 3, 4] squares(a) print(a) # ’a’ now contains ’a**2’ The output is [1, 4, 9, 16] Lambda Statement If the function has the form of an expression, it can be defined with the lambda statement func name = lambda param1, param2,...: expression Multiple statements are not allowed. Here is an example: c = lambda x,y : x**2 + y**2 print(c(3,4)) 25
- 36. 18 Introduction toPython Modules It is sound practice to store useful functions in modules. A module is simply a file where the functions reside; the name of the module is the name of the file. A module can be loaded into a program by the statement from module name import * Python comes with a large number of modules containing functions and meth- ods for various tasks. Some of the modules are described briefly in the next two sec- tions. Additional modules, including graphics packages, are available for download- ing on the Web. 1.4 Mathematics Modules math Module Most mathematical functions are not built into core Python, but are available by load- ing the math module. There are three ways of accessing the functions in a module. The statement from math import * loads all the function definitions in the math module into the current function or module. The use of this method is discouraged because it is not only wasteful but can also lead to conflicts with definitions loaded from other modules. For example, there are three different definitions of the sine function in the Python modules math, cmath, and numpy. If you have loaded two or more of these modules, it is unclear which definition will be used in the function call sin(x)(it is the definition in the module that was loaded last). A safer but by no means foolproof method is to load selected definitions with the statement from math import func1, func2, . . . as illustrated as follows: from math import log,sin print(log(sin(0.5))) -0.735166686385 Conflicts can be avoided altogether by first making the module accessible with the statement import math and then accessing the definitions in the module by using the module name as a prefix. Here is an example: import math print(math.log(math.sin(0.5))) -0.735166686385
- 37. 19 1.4 MathematicsModules A module can also be made accessible under an alias. For example, the math module can be made available under the alias m with the command import math as m Now the prefix to be used is m rather than math: import math as m print(m.log(m.sin(0.5))) -0.735166686385 The contents of a module can be printed by calling dir(module). Here is how to obtain a list of the functions in the math module: import math dir(math) [’__doc__’, ’__name__’, ’acos’, ’asin’, ’atan’, ’atan2’, ’ceil’, ’cos’, ’cosh’, ’e’, ’exp’, ’fabs’, ’floor’, ’fmod’, ’frexp’, ’hypot’, ’ldexp’, ’log’, ’log10’, ’modf’, ’pi’, ’pow’, sign’, sin’, ’sinh’, ’sqrt’, ’tan’, ’tanh’] Most of these functions are familiar to programmers. Note that the module in- cludes two constants: π and e. cmath Module The cmath module provides many of the functions found in the math module, but these functions accept complex numbers. The functions in the module are [’__doc__’, ’__name__’, ’acos’, ’acosh’, ’asin’, ’asinh’, ’atan’, ’atanh’, ’cos’, ’cosh’, ’e’, ’exp’, ’log’, ’log10’, ’pi’, ’sin’, ’sinh’, ’sqrt’, ’tan’, ’tanh’] Here are examples of complex arithmetic: from cmath import sin x = 3.0 -4.5j y = 1.2 + 0.8j z = 0.8 print(x/y) (-2.56205313375e-016-3.75j) print(sin(x)) (6.35239299817+44.5526433649j) print(sin(z)) (0.7173560909+0j)
- 38. 20 Introduction toPython 1.5 numpy Module General Information The numpy module2 is not a part of the standard Python release. As pointed out ear- lier, it must be installed separately (the installation is very easy). The module intro- duces array objects that are similar to lists, but can be manipulated by numerous functions contained in the module. The size of an array is immutable, and no empty elements are allowed. The complete set of functions in numpy is far too long to be printed in its entirety. The following list is limited to the most commonly used functions. [’complex’, ’float’, ’abs’, ’append’, arccos’, ’arccosh’, ’arcsin’, ’arcsinh’, ’arctan’, ’arctan2’, ’arctanh’, ’argmax’, ’argmin’, ’cos’, ’cosh’, ’diag’, ’diagonal’, ’dot’, ’e’, ’exp’, ’floor’, ’identity’, ’inner, ’inv’, ’log’, ’log10’, ’max’, ’min’, ’ones’, ’outer’, ’pi’, ’prod’ ’sin’, ’sinh’, ’size’, ’solve’, ’sqrt’, ’sum’, ’tan’, ’tanh’, ’trace’, ’transpose’, ’vectorize’,’zeros’] Creating an Array Arrays can be created in several ways. One of them is to use the array function to turn a list into an array: array(list,type) Following are two examples of creating a 2 × 2 array with floating-point elements: from numpy import array a = array([[2.0, -1.0],[-1.0, 3.0]]) print(a) [[ 2. -1.] [-1. 3.]] b = array([[2, -1],[-1, 3]],float) print(b) [[ 2. -1.] [-1. 3.]] Other available functions are zeros((dim1,dim2),type) which creates a dim1 × dim2 array and fills it with zeroes, and ones((dim1,dim2),type) which fills the array with ones. The default type in both cases is float. 2 NumPy is the successor of older Python modules called Numeric and NumArray. Their interfaces and capabilities are very similar. Although Numeric and NumArray are still available, they are no longer supported.
- 39. 21 1.5 numpyModule Finally, there is the function arange(from,to,increment) which works just like the range function, but returns an array rather than a se- quence. Here are examples of creating arrays: from numpy import * print(arange(2,10,2)) [2 4 6 8] print(arange(2.0,10.0,2.0)) [ 2. 4. 6. 8.] print(zeros(3)) [ 0. 0. 0.] print(zeros((3),int)) [0 0 0] print(ones((2,2))) [[ 1. 1.] [ 1. 1.]] Accessing and Changing Array Elements If a is a rank-2 array, then a[i,j] accesses the element in row i and column j, whereas a[i] refers to row i. The elements of an array can be changed by assign- ment as follows: from numpy import * a = zeros((3,3),int) print(a) [[0 0 0] [0 0 0] [0 0 0]] a[0] = [2,3,2] # Change a row a[1,1] = 5 # Change an element a[2,0:2] = [8,-3] # Change part of a row print(a) [[ 2 3 2] [ 0 5 0] [ 8 -3 0]] Operations on Arrays Arithmetic operators work differently on arrays than they do on tuples and lists—the operation is broadcast to all the elements of the array; that is, the operation is applied to each element in the array. Here are examples: from numpy import array a = array([0.0, 4.0, 9.0, 16.0])
- 40. 22 Introduction toPython print(a/16.0) [ 0. 0.25 0.5625 1. ] print(a - 4.0) [ -4. 0. 5. 12.] The mathematical functions available in numpy are also broadcast, as follows: from numpy import array,sqrt,sin a = array([1.0, 4.0, 9.0, 16.0]) print(sqrt(a)) [ 1. 2. 3. 4.] print(sin(a)) [ 0.84147098 -0.7568025 0.41211849 -0.28790332] Functions imported from the math module will work on the individual elements, of course, but not on the array itself. An example follows: from numpy import array from math import sqrt a = array([1.0, 4.0, 9.0, 16.0]) print(sqrt(a[1])) 2.0 print(sqrt(a)) Traceback (most recent call last): . . . TypeError: only length-1 arrays can be converted to Python scalars Array Functions There are numerous functions in numpy that perform array operations and other use- ful tasks. Here are a few examples: from numpy import * A = array([[4,-2,1],[-2,4,-2],[1,-2,3]],float) b = array([1,4,3],float) print(diagonal(A)) # Principal diagonal [ 4. 4. 3.] print(diagonal(A,1)) # First subdiagonal [-2. -2.] print(trace(A)) # Sum of diagonal elements 11.0 print(argmax(b)) # Index of largest element 1 print(argmin(A,axis=0)) # Indices of smallest col. elements [1 0 1]
- 41. 23 1.5 numpyModule print(identity(3)) # Identity matrix [[ 1. 0. 0.] [ 0. 1. 0.] [ 0. 0. 1.]] There are three functions in numpy that compute array products. They are illus- trated by the following program. For more details, see Appendix A2. from numpy import * x = array([7,3]) y = array([2,1]) A = array([[1,2],[3,2]]) B = array([[1,1],[2,2]]) # Dot product print(dot(x,y) =n,dot(x,y)) # {x}.{y} print(dot(A,x) =n,dot(A,x)) # [A]{x} print(dot(A,B) =n,dot(A,B)) # [A][B] # Inner product print(inner(x,y) =n,inner(x,y)) # {x}.{y} print(inner(A,x) =n,inner(A,x)) # [A]{x} print(inner(A,B) =n,inner(A,B)) # [A][B_transpose] # Outer product print(outer(x,y) =n,outer(x,y)) print(outer(A,x) =n,outer(A,x)) print(outer(A,B) =n,outer(A,B)) The output of the program is dot(x,y) = 17 dot(A,x) = [13 27] dot(A,B) = [[5 5] [7 7]] inner(x,y) = 17 inner(A,x) = [13 27] inner(A,B) = [[ 3 6] [ 5 10]] outer(x,y) =
- 42. 24 Introduction toPython [[14 7] [ 6 3]] outer(A,x) = [[ 7 3] [14 6] [21 9] [14 6]] Outer(A,B) = [[1 1 2 2] [2 2 4 4] [3 3 6 6] [2 2 4 4]] Linear Algebra Module The numpy module comes with a linear algebra module called linalg that contains routine tasks such as matrix inversion and solution of simultaneous equations. For example, from numpy import array from numpy.linalg import inv,solve A = array([[ 4.0, -2.0, 1.0], [-2.0, 4.0, -2.0], [ 1.0, -2.0, 3.0]]) b = array([1.0, 4.0, 2.0]) print(inv(A)) # Matrix inverse [[ 0.33333333 0.16666667 0. ] [ 0.16666667 0.45833333 0.25 ] [ 0. 0.25 0.5 ]] print(solve(A,b)) # Solve [A]{x} = {b} [ 1. , 2.5, 2. ] Copying Arrays We explained earlier that if a is a mutable object, such as a list, the assignment state- ment b = a does not result in a new object b, but simply creates a new reference to a, called a deep copy. This also applies to arrays. To make an independent copy of an array a, use the copy method in the numpy module: b = a.copy() Vectorizing Algorithms Sometimes the broadcasting properties of the mathematical functions in the numpy module can be used to replace loops in the code. This procedure is known as
- 43. 25 1.6 Plottingwith matplotlib.pyplot vectorization. Consider, for example, the expression s = 100 i=0 iπ 100 sin iπ 100 The direct approach is to evaluate the sum in a loop, resulting in the following ”scalar” code: from math import sqrt,sin,pi x = 0.0; s = 0.0 for i in range(101): s = s + sqrt(x)*sin(x) x = x + 0.01*pi print(s) The vectorized version of the algorithm is from numpy import sqrt,sin,arange from math import pi x = arange(0.0, 1.001*pi, 0.01*pi) print(sum(sqrt(x)*sin(x))) Note that the first algorithm uses the scalar versions of sqrt and sin functions in the math module, whereas the second algorithm imports these functions from numpy. The vectorized algorithm executes much faster, but uses more memory. 1.6 Plotting with matplotlib.pyplot The module matplotlib.pyplot is a collection of 2D plotting functions that pro- vide Python with MATLAB-style functionality. Not being a part of core Python, it requires separate installation. The following program, which plots sine and cosine functions, illustrates the application of the module to simple xy plots. import matplotlib.pyplot as plt from numpy import arange,sin,cos x = arange(0.0,6.2,0.2) plt.plot(x,sin(x),’o-’,x,cos(x),’ˆ-’) # Plot with specified # line and marker style plt.xlabel(’x’) # Add label to x-axis plt.legend((’sine’,’cosine’),loc = 0) # Add legend in loc. 3 plt.grid(True) # Add coordinate grid plt.savefig(’testplot.png’,format=’png’) # Save plot in png # format for future use plt.show() # Show plot on screen input(nPress return to exit)
- 44. 26 Introduction toPython The line and marker styles are specified by the string characters shown in the following table (only some of the available characters are shown). ’-’ Solid line ’--’ Dashed line ’-.’ Dash-dot line ’:’ Dotted line ’o’ Circle marker ’ˆ’ Triangle marker ’s’ Square marker ’h’ Hexagon marker ’x’ x marker Some of the location (loc) codes for placement of the legend are 0 ”Best” location 1 Upper right 2 Upper left 3 Lower left 4 Lower right Running the program produces the following screen:
- 45. 27 1.6 Plottingwith matplotlib.pyplot It is possible to have more than one plot in a figure, as demonstrated by the fol- lowing code: import matplotlib.pyplot as plt from numpy import arange,sin,cos x = arange(0.0,6.2,0.2) plt.subplot(2,1,1) plt.plot(x,sin(x),’o-’) plt.xlabel(’x’);plt.ylabel(’sin(x)’) plt.grid(True) plt.subplot(2,1,2) plt.plot(x,cos(x),’ˆ-’) plt.xlabel(’x’);plt.ylabel(’cos(x)’) plt.grid(True) plt.show() input(nPress return to exit) The command subplot(rows,cols,plot number)establishes a subplot window within the current figure. The parameters row and col divide the figure into row × col grid of subplots (in this case, two rows and one column). The commas between the parameters may be omitted. The output from this above program is
- 46. 28 Introduction toPython 1.7 Scoping of Variables Namespace is a dictionary that contains the names of the variables and their values. Namespaces are automatically created and updated as a program runs. There are three levels of namespaces in Python: 1. Local namespace is created when a function is called. It contains the variables passed to the function as arguments and the variables created within the func- tion. The namespace is deleted when the function terminates. If a variable is cre- ated inside a function, its scope is the function’s local namespace. It is not visible outside the function. 2. A global namespace is created when a module is loaded. Each module has its own namespace. Variables assigned in a global namespace are visible to any function within the module. 3. A built-in namespace is created when the interpreter starts. It contains the func- tions that come with the Python interpreter. These functions can be accessed by any program unit. When a name is encountered during execution of a function, the interpreter tries to resolve it by searching the following in the order shown: (1) local namespace, (2) global namespace, and (3) built-in namespace. If the name cannot be resolved, Python raises a NameError exception. Because the variables residing in a global namespace are visible to functions within the module, it is not necessary to pass them to the functions as arguments (although it is good programming practice to do so), as the following program illustrates: def divide(): c = a/b print(’a/b =’,c) a = 100.0 b = 5.0 divide() a/b = 20.0 Note that the variable c is created inside the function divide and is thus not accessible to statements outside the function. Hence an attempt to move the print statement out of the function fails: def divide(): c = a/b a = 100.0 b = 5.0 divide() print(’a/b =’,c)
- 47. 29 1.8 Writingand Running Programs Traceback (most recent call last): File C:Python32test.py, line 6, in module print(’a/b =’,c) NameError: name ’c’ is not defined 1.8 Writing and Running Programs When the Python editor Idle is opened, the user is faced with the prompt , in- dicating that the editor is in interactive mode. Any statement typed into the editor is immediately processed on pressing the enter key. The interactive mode is a good way both to learn the language by experimentation and to try out new programming ideas. Opening a new window places Idle in the batch mode, which allows typing and saving of programs. One can also use a text editor to enter program lines, but Idle has Python-specific features, such as color coding of keywords and automatic inden- tation, which make work easier. Before a program can be run, it must be saved as a Python file with the .py extension (e.g., myprog.py). The program can then be exe- cuted by typing python myprog.py; in Windows; double-clicking on the program icon will also work. But beware: The program window closes immediately after exe- cution, before you get a chance to read the output. To prevent this from happening, conclude the program with the line input(’press return’) Double-clicking the program icon also works in Unix and Linux if the first line of the program specifies the path to the Python interpreter (or a shell script that provides a link to Python). The path name must be preceded by the sym- bols #!. On my computer the path is /usr/bin/python, so that all my programs start with the line #!/usr/bin/python. On multi-user systems the path is usually /usr/local/bin/python. When a module is loaded into a program for the first time with the import state- ment, it is compiled into bytecode and written in a file with the extension .pyc. The next time the program is run, the interpreter loads the bytecode rather than the origi- nal Python file. If in the meantime changes have been made to the module, the mod- ule is automatically recompiled. A program can also be run from Idle using Run/Run Module menu. It is a good idea to document your modules by adding a docstring at the begin- ning of each module. The docstring, which is enclosed in triple quotes, should ex- plain what the module does. Here is an example that documents the module error (we use this module in several of our programs): ## module error ’’’ err(string). Prints ’string’ and terminates program.
- 48. 30 Introduction toPython ’’’ import sys def err(string): print(string) input(’Press return to exit’) sys.exit() The docstring of a module can be printed with the statement print(module name. doc ) For example, the docstring of error is displayed by import error print(error.__doc__) err(string). Prints ’string’ and terminates program. Avoid backslashes in the docstring because they confuse the Python 3 inter- preter.
- 49. 2 Systems ofLinear Algebraic Equations Solve the simultaneous equations Ax = b. 2.1 Introduction In this chapter we look at the solution of n linear, algebraic equations in n unknowns. It is by far the longest and arguably the most important topic in the book. There is a good reason for its importance—it is almost impossible to carry out numerical anal- ysis of any sort without encountering simultaneous equations. Moreover, equation sets arising from physical problems are often very large, consuming a lot of com- putational resources. It usually possible to reduce the storage requirements and the run time by exploiting special properties of the coefficient matrix, such as sparseness (most elements of a sparse matrix are zero). Hence there are many algorithms dedi- cated to the solution of large sets of equations, each one being tailored to a particular form of the coefficient matrix (symmetric, banded, sparse, and so on). A well-known collection of these routines is LAPACK—Linear Algebra PACKage, originally written in Fortran77.1 We cannot possibly discuss all the special algorithms in the limited space avail- able. The best we can do is to present the basic methods of solution, supplemented by a few useful algorithms for banded coefficient matrices. Notation A system of algebraic equations has the form A11x1 + A12x2 + · · · + A1nxn = b1 A21x1 + A22x2 + · · · + A2nxn = b2 (2.1) . . . An1x1 + An2x2 + · · · + Annxn = bn 1 LAPACK is the successor of LINPACK, a 1970s and 80s collection of Fortran subroutines. 31
- 50. 32 Systems ofLinear Algebraic Equations where the coefficients Aij and the constants bj are known, and xi represents the un- knowns. In matrix notation the equations are written as ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ A11 A12 · · · A1n A21 A22 · · · A2n . . . . . . ... . . . An1 An2 · · · Ann ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ x1 x2 . . . xn ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ b1 b2 . . . bn ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (2.2) or simply Ax = b. (2.3) A particularly useful representation of the equations for computational purposes is the augmented coefficient matrix obtained by adjoining the constant vector b to the coefficient matrix A in the following fashion: A b = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ A11 A12 · · · A1n b1 A21 A22 · · · A2n b2 . . . . . . ... . . . . . . An1 An2 · · · An3 bn ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (2.4) Uniqueness of Solution A system of n linear equations in n unknowns has a unique solution, provided that the determinant of the coefficient matrix is nonsingular; that is, |A| = 0. The rows and columns of a nonsingular matrix are linearly independent in the sense that no row (or column) is a linear combination of other rows (or columns). If the coefficient matrix is singular, the equations may have an infinite number of solutions or no solutions at all, depending on the constant vector. As an illustration, take the equations 2x + y = 3 4x + 2y = 6 Since the second equation can be obtained by multiplying the first equation by two, any combination of x and y that satisfies the first equation is also a solution of the second equation. The number of such combinations is infinite. In contrast, the equa- tions 2x + y = 3 4x + 2y = 0 have no solution because the second equation, being equivalent to 2x + y = 0, contradicts the first one. Therefore, any solution that satisfies one equation cannot satisfy the other one.
- 51. 33 2.1 Introduction IllConditioning The obvious question is, What happens when the coefficient matrix is almost singu- lar (i.e., if |A| is very small). To determine whether the determinant of the coefficient matrix is “small,” we need a reference against which the determinant can be mea- sured. This reference is called the norm of the matrix and is denoted by A. We can then say that the determinant is small if |A| A Several norms of a matrix have been defined in existing literature, such as the Euclidan norm Ae = n i=1 n j=1 A2 ij (2.5a) and the row-sum norm, also called the infinity norm A∞ = max 1≤i≤n n j=1 Aij (2.5b) A formal measure of conditioning is the matrix condition number, defined as cond(A) = A A−1 (2.5c) If this number is close to unity, the matrix is well conditioned. The condition number increases with the degree of ill conditioning, reaching infinity for a singular matrix. Note that the condition number is not unique, but depends on the choice of the ma- trix norm. Unfortunately, the condition number is expensive to compute for large matrices. In most cases it is sufficient to gauge conditioning by comparing the deter- minant with the magnitudes of the elements in the matrix. If the equations are ill conditioned, small changes in the coefficient matrix result in large changes in the solution. As an illustration, take the equations 2x + y = 3 2x + 1.001y = 0 that have the solution x = 1501.5, y = −3000. Since |A| = 2(1.001) − 2(1) = 0.002 is much smaller than the coefficients, the equations are ill conditioned. The effect of ill conditioning can verified by changing the second equation to 2x + 1.002y = 0 and re-solving the equations. The result is x = 751.5, y = −1500. Note that a 0.1% change in the coefficient of y produced a 100% change in the solution! Numerical solutions of ill-conditioned equations are not to be trusted. The rea- son is that the inevitable roundoff errors during the solution process are equivalent to introducing small changes into the coefficient matrix. This in turn introduces large errors into the solution, the magnitude of which depends on the severity of ill con- ditioning. In suspect cases the determinant of the coefficient matrix should be com- puted so that the degree of ill conditioning can be estimated. This can be done during or after the solution with only a small computational effort.
- 52. 34 Systems ofLinear Algebraic Equations Linear Systems Linear, algebraic equations occur in almost all branches of numerical analysis. But their most visible application in engineering is in the analysis of linear systems (any system whose response is proportional to the input is deemed to be linear). Linear systems include structures, elastic solids, heat flow, seepage of fluids, electromag- netic fields, and electric circuits (i.e., most topics taught in an engineering curri- culum). If the system is discrete, such as a truss or an electric circuit, then its analysis leads directly to linear algebraic equations. In the case of a statically determinate truss, for example, the equations arise when the equilibrium conditions of the joints are written down. The unknowns x1, x2, . . . , xn represent the forces in the members and the support reactions, and the constantsb1, b2, . . . , bn are the prescribed external loads. The behavior of continuous systems is described by differential equations, rather than algebraic equations. However, because numerical analysis can deal only with discrete variables, it is first necessary to approximate a differential equation with a system of algebraic equations. The well-known finite difference, finite element, and boundary element methods of analysis work in this manner. They use different ap- proximations to achieve the “discretization,” but in each case the final task is the same: solve a system (often a very large system) of linear, algebraic equations. In summary, the modeling of linear systems invariably gives rise to equations of the form Ax = b, where b is the input and x represents the response of the sys- tem. The coefficient matrix A, which reflects the characteristics of the system, is in- dependent of the input. In other words, if the input is changed, the equations have to be solved again with a different b, but the same A. Therefore, it is desirable to have an equation-solving algorithm that can handle any number of constant vectors with minimal computational effort. Methods of Solution There are two classes of methods for solving systems of linear, algebraic equations: direct and iterative methods. The common characteristic of direct methods is that they transform the original equations into equivalent equations (equations that have the same solution) that can be solved more easily. The transformation is carried out by applying the following three operations. These so-called elementary operations do not change the solution, but they may affect the determinant of the coefficient matrix as indicated in parenthesis. 1. Exchanging two equations (changes sign of |A|) 2. Multiplying an equation by a nonzero constant (multiplies |A| by the same constant) 3. Multiplying an equation by a nonzero constant and then subtracting it from an- other equation (leaves |A| unchanged)
- 53. 35 2.1 Introduction Iterative,or indirect methods, start with a guess of the solution x and then re- peatedly refine the solution until a certain convergence criterion is reached. Itera- tive methods are generally less efficient than their direct counterparts because of the large number of iterations required. Yet they do have significant computational ad- vantages if the coefficient matrix is very large and sparsely populated (most coeffi- cients are zero). Overview of Direct Methods Table 2.1 lists three popular direct methods, each of which uses elementary opera- tions to produce its own final form of easy-to-solve equations. Method Initial form Final form Gauss elimination Ax = b Ux = c LU decomposition Ax = b LUx = b Gauss-Jordan elimination Ax = b Ix = c Table 2.1. Three Popular Direct Methods In Table 2.1 U represents an upper triangular matrix, L is a lower triangular ma- trix, and I denotes the identity matrix. A square matrix is called triangular if it con- tains only zero elements on one side of the leading diagonal. Thus a 3 × 3 upper tri- angular matrix has the form U = ⎡ ⎢ ⎣ U11 U12 U13 0 U22 U23 0 0 U33 ⎤ ⎥ ⎦ and a 3 × 3 lower triangular matrix appears as L = ⎡ ⎢ ⎣ L11 0 0 L21 L22 0 L31 L32 L33 ⎤ ⎥ ⎦ Triangular matrices play an important role in linear algebra, because they sim- plify many computations. For example, consider the equations Lx = c, or L11x1 = c1 L21x1 + L22x2 = c2 L31x1 + L32x2 + L33x3 = c3 If we solve the equations forward, starting with the first equation, the compu- tations are very easy, because each equation contains only one unknown at a time.
- 54. 36 Systems ofLinear Algebraic Equations The solution would thus proceed as follows: x1 = c1/L11 x2 = (c2 − L21x1)/L22 x3 = (c3 − L31x1 − L32x2)/L33 This procedure is known as forward substitution. In a similar way, Ux = c, en- countered in Gauss elimination, can easily be solved by back substitution, which starts with the last equation and proceeds backward through the equations. The equations LUx = b, which are associated with LU decomposition, can also be solved quickly if we replace them with two sets of equivalent equations: Ly = b and Ux = y. Now Ly = b can be solved for y by forward substitution, followed by the solution of Ux = y by means of back substitution. The equations Ix = c, which are produced by Gauss-Jordan elimination, are equivalent to x = c (recall the identity Ix = x), so that c is already the solution. EXAMPLE 2.1 Determine whether the following matrix is singular: A = ⎡ ⎢ ⎣ 2.1 −0.6 1.1 3.2 4.7 −0.8 3.1 −6.5 4.1 ⎤ ⎥ ⎦ Solution. Laplace’s development of the determinant (see Appendix A2) about the first row of A yields |A| = 2.1 4.7 −0.8 −6.5 4.1 − (−0.6) 3.2 −0.8 3.1 4.1 + 1.1 3.2 4.7 3.1 −6.5 = 2.1(14.07) + 0.6(15.60) + 1.1(35.37) = 0 Since the determinant is zero, the matrix is singular. It can be verified that the singu- larity is due to the following row dependency: (row 3) = (3 × row 1) − (row 2). EXAMPLE 2.2 Solve the equations Ax = b, where A = ⎡ ⎢ ⎣ 8 −6 2 −4 11 −7 4 −7 6 ⎤ ⎥ ⎦ b = ⎡ ⎢ ⎣ 28 −40 33 ⎤ ⎥ ⎦ knowing that the LU decomposition of the coefficient matrix is (you should verify this) A = LU = ⎡ ⎢ ⎣ 2 0 0 −1 2 0 1 −1 1 ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ 4 −3 1 0 4 −3 0 0 2 ⎤ ⎥ ⎦
- 55. Another Random Documenton Scribd Without Any Related Topics
- 56. TESTIMONY OF LAURANCER. WILCOX The testimony of Laurance R. Wilcox was taken at 2 p. m., on March 31, 1964, in the office of the U.S. attorney, 301 Post Office Building, Bryan and Ervay Streets, Dallas, Tex., by Mr. Wesley J. Liebeler, assistant counsel of the President's Commission. Mr. Liebeler. Before I start, I want to swear you in as a witness. Do you solemnly swear that the testimony you are about to give will be the truth, the whole truth, and nothing but the truth, so help you God? Mr. Wilcox. I do. Mr. Liebeler. Mr. Wilcox, my name is Wesley J. Liebeler. I am a member of the legal staff of the Commission appointed by President Johnson to investigate the assassination of President Kennedy. Staff counsel have been authorized to take the testimony of witnesses by the Commission pursuant to authority granted to the Commission under Executive Order 11130, dated November 29, 1963, and Joint Resolution of Congress No. 137. The Commission rules of procedure require that copies of that Executive order and the congressional resolution, as well as copies of the Commission's rules and procedures relating to the taking of testimony be provided to each witness prior to the time his testimony is taken. I now provide you with copies of those documents. The general nature of the testimony that we wish to get
- 57. from you todayrelates to investigations made by Western Union Telegraph Co. concerning the possibility that Lee Harvey Oswald received money orders through the offices of your company, either in Dallas or the surrounding area, and the possibility that he may have sent telegrams to other persons through the facilities of your company. Mr. Liebeler. Before we get into the details of your testimony, would you please state your full name for the record? Mr. Wilcox. Laurance R. Wilcox. Mr. Liebeler. By whom are you employed? Mr. Wilcox. Western Union Telegraph Co. Mr. Liebeler. In what capacity? Mr. Wilcox. District manager. Mr. Liebeler. What are your duties as district manager? Mr. Wilcox. Administrative; in charge of the operation for Western Union in the city of Dallas. Mr. Liebeler. You are general manager then of the area which includes just the city of Dallas; is that correct? Mr. Wilcox. District manager. Mr. Liebeler. Of just the city of Dallas; is that correct? Mr. Wilcox. Yes; and immediate surrounding towns such as Garland, Mesquite, Grand Prairie, Irving, and Lancaster. That is all I can think of right now. Mr. Liebeler. You are generally in charge of the operations of the company within that particular area; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. Did there come a time, Mr. Wilcox, when you caused a search to be made of the records of the Western Union Telegraph Co. to determine whether or not Lee Harvey Oswald had ever
- 58. received or sentany telegrams through the offices under your jurisdiction? Mr. Wilcox. Yes, sir. Mr. Liebeler. Would you tell us about that? Mr. Wilcox. Can I refer to these papers? Mr. Liebeler. You may refer to any papers that you wish. Mr. Wilcox. I want to so that I will have this exactly right as to what took place. My first knowledge of the message that was supposed to have been sent by Oswald was when Mr. Hamblen, early night manager at my office, visited with me telling me—— Mr. Liebeler. Is that Mr. C. A. Hamblen? Mr. Wilcox. Yes, sir. Mr. Liebeler. Go ahead. Mr. Wilcox. Telling me that he had been watching the television, and when he saw the picture of Oswald, he recognized that as being a man that had been in our office and had filed a message. Mr. Liebeler. Do you remember when Mr. Hamblen told you this? Mr. Wilcox. This was on a Wednesday following the assassination —following the shooting of Oswald. He told me that he was positive that he had seen Mr. Oswald in our office. Mr. Liebeler. He told you this on Wednesday following the Sunday on which Mr. Oswald was shot by Ruby; is that correct? Mr. Wilcox. That's right. Mr. Liebeler. That would have been November 27, 1963? Thanksgiving was on November the 28th. Mr. Wilcox. Yes. It was on a Wednesday preceding Thanksgiving. I asked Mr. Hamblen to tell me exactly what had taken place. He stated that Oswald had filed a message going to Washington —a cash message, and it was written in a peculiar script, as Mr.
- 59. Hamblen put it.It was typed in pencil. I think he meant is printed in pencil. I asked him to immediately set about to obtain copies of that, as I was leaving town, and to get a copy of the message, and to do that it would be necessary for him to see Mr. Pirtle, our bookkeeping manager, and to get the message, put it in an envelope, and address it to my personal attention while I was in Kansas City. Mr. Liebeler. Did Mr. Hamblen tell you anything about what the message said or to whom it was addressed at this first conversation? Mr. Wilcox. No, he didn't. We left for Kansas City and was there over the Thanksgiving holidays. Mr. Liebeler. You and your family? Mr. Wilcox. Yes. And while there, this article appeared in the Kansas City Star, their evening paper; very much the same story as appeared in the Dallas Times Herald. Mr. Liebeler. I show you a copy of a clipping which purports to have appeared in the Dallas Times Herald on November 30, 1963. I ask you if this tells a story similar to the report you saw in the Kansas City Star? Mr. Wilcox. Yes, same story. Mr. Liebeler. We will mark this clipping as Exhibit 3002 on the deposition of Laurance R. Wilcox, at Dallas, Tex., March 31, 1964. I have initialed the clipping in question, Mr. Wilcox, and ask you to do the same. Mr. Wilcox. [Initials.] Immediately when I read this story in the Kansas City paper I recognized it as being the same story that Mr. Hamblen had told me just a couple of days before in my office. We returned to Dallas Sunday, and immediately on my return to the office Monday—— Mr. Liebeler. Monday, I suggest would have been December 2, 1963?
- 60. Mr. Wilcox. Yes. Mr.Liebeler. And you fix that by recalling that the assassination was on the 22d. The following Friday would have been the 29th. Saturday, the 30th. Sunday the 1st; and Monday would have been December 2? Mr. Wilcox. Yes, December 2. Now, this was on my return to the office, and we indicated that was December 2. I went over the information that was contained in the newspaper article with Mr. Hamblen, reminding him that this was in effect the same story as he had told me just 2 days before, a few days before. Particularly, its comment about the strange printing on the telegram which he had seen presented by Oswald. Mr. Hamblen admitted that he had discussed several matters with different reporters, but insisted that he hadn't given out such detailed information as appeared in the newspaper article. However, I was constrained to feel that he had because it was exactly the same story as he had given me initially in the past week. There was no doubt in my mind but that the newspaper article stemmed from Mr. Hamblen's visit with a newspaper reporter. Mr. Liebeler. In this connection, Mr. Wilcox, you have referred to a copy of a letter dated December 3, 1963, which appears to be a letter from you to Mr. Semingsen; is that correct? Mr. Wilcox. Yes, sir; would you like to have a copy? Mr. Liebeler. Yes. Did you prepare that letter on or about December 3? Mr. Wilcox. Yes, I did. Mr. Liebeler. Does it accurately reflect the events that occurred prior to that time? Mr. Wilcox. Yes.
- 61. Mr. Liebeler. Imark that letter as Exhibit 3003, on the deposition of Mr. Laurance R. Wilcox, at Dallas, Tex., March 31, 1964. I have initialed the copy in question, Mr. Wilcox, and would like to have you do the same thing if you would. Mr. Wilcox. [Initials.] Mr. Liebeler. Before we go on, I call your attention to the fact that this letter of December 3, 1963, starts out by saying This is to supplement my report of December 2 in connection with newspaper article regarding messages and money order to Ruby or Oswald. Do you have a copy of your report of December 2, referred to in Exhibit 3003? Mr. Wilcox. Yes, sir. Mr. Liebeler. May we mark that as Exhibit No. 3004 on the deposition of Mr. Laurance R. Wilcox, Dallas, Tex., March 31, 1964? Mr. Wilcox. [Hands document to attorney.] Mr. Liebeler. I have initialed Exhibit No. 3004 and you have done likewise, have you not? Mr. Wilcox. Yes, sir. Mr. Liebeler. You had just indicated that you had talked to Mr. Hamblen upon your return concerning the newspaper story. Would you tell us what happened next? Mr. Wilcox. I asked Mr. Hamblen if he had obtained the message from the bookkeeping department as I had requested he do, and he stated that he had not found it and did not have it and it wasn't available. On December 9, Mr. Semingsen, vice president of Western Union, was at my office, and at this time we interviewed several employees for the purpose of finding out if any of them could recall having seen Oswald in our office. We also discussed the Ruby money orders.
- 62. Mr. Liebeler. Canyou tell us the names of the employees who were interviewed on December 9, 1963, in connection with the possibility that Oswald might have sent or received telegrams through your company? Do you have a memorandum reflecting what happened? Mr. Wilcox. No, because what we did, we asked for statements from these people, and I have those statements and that was the result of that meeting. Mr. Liebeler. The meeting you referred to now is the meeting held in your office on December 9? Mr. Wilcox. The meeting held in my office on December 9. Mr. Liebeler. I call your attention to a message that was sent by Mr. Semingsen to Mr. J. H. Waters in New York City, which has been attached as Attachment No. 5 to Exhibit 3001 on Mr. Semingsen's deposition. I show you that message and ask you if that accurately reflects what occurred at the meeting in your office on December 9, 1963? Mr. Wilcox. This doesn't. Mr. Liebeler. You say it does not? Mr. Wilcox. This was pertaining to Mr. Ruby. This did not have anything to do with that. Mr. Liebeler. Let me further point out to you, Mr. Wilcox, that we have statements of Mr. Hamblen dated December 2 and December 5, 1963, which are apparently addressed to you. I show you copies, first, of the statement dated December 2, 1963, and ask you if you can remember the circumstances under which that statement was obtained from Mr. Hamblen? Mr. Wilcox. This statement was obtained—— Mr. Liebeler. The statement you are referring to is the statement dated December 2, 1963; is that correct?
- 63. Mr. Wilcox. Yes.This statement was obtained at the time I discussed the matter with Mr. Hamblen on December 2, Monday. That was this statement. Mr. Liebeler. That visit with Mr. Hamblen is described in Exhibit No. 3003, your letter to Mr. Semingsen on December 3, 1963, is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. And the statement of Mr. Hamblen, dated December 2, 1963, was obtained at that time? That time being the first time that you talked to Mr. Hamblen after this matter appeared in the newspapers; is that correct? Mr. Wilcox. Yes. Mr. Liebeler. Do you have the original of Mr. Hamblen's statement of December 2, 1963? Mr. Wilcox. I can't find it in my file. I am wondering if it was sent to Mr. Semingsen's file, but we have gone through that. Mr. Liebeler. I show you a thermofax copy that has been provided to us, and ask you if that is an accurate copy of the original statement? Mr. Wilcox. Yes. That was made in our office on the thermofax machine. Mr. Liebeler. Let us mark the thermofax copy of the statement of Mr. C. A. Hamblen to Mr. Wilcox dated December 2, 1963, as Exhibit 3005 on the deposition of Mr. Laurance R. Wilcox, at Dallas, Tex., on March 31, 1964. I have initialed Exhibit 3005, Mr. Wilcox. Will you do the same? Mr. Wilcox. [Initials.] Mr. Liebeler. Does the statement of Mr. Hamblen, Exhibit No. 3005, accurately reflect the conversation which you had with him on December 2, 1963?
- 64. Mr. Wilcox. Yes. Mr.Liebeler. Did you prepare that statement yourself? Mr. Wilcox. No, sir; Mr. Hamblen prepared that. Mr. Liebeler. Did Mr. Hamblen dictate it himself, or write it out? Mr. Wilcox. Wrote it on the typewriter himself. Mr. Liebeler. Now in the December 2, 1963, statement, Mr. Hamblen says: As I recall, several weeks ago Mr. Aubrey Lewis had trouble paying this party a money order. He expected to get same without proper ID. He was informed to get some ID and come back and get paid, and as Mr. Lewis recalled he returned about 1 hour later with Navy ID card and a library card and was paid a small amount with this ID and after payment made, party left the office. Mr. Lewis remarked to me, I would like to pinch the heads off people of his character. Do you recall discussing this with Mr. Hamblen on December 2, 1963? Mr. Wilcox. Yes, sir. Mr. Liebeler. Can you remember anything else that Mr. Hamblen might have said about this incident other than that which is reflected in the statement? Mr. Wilcox. No. Mr. Liebeler. Do you recall discussing this statement with Mr. Lewis at any time? Mr. Wilcox. Yes, sir. Mr. Liebeler. When did you first discuss it with Mr. Lewis? Mr. Wilcox. That is what I am trying to find right here. Must have been on December 4 that I talked to Mr. Lewis about it, and his statement addressed to me was written by himself. Mr. Liebeler. On December 4, 1963?
- 65. Mr. Wilcox. OnDecember 4, 1963. Mr. Liebeler. Do you have a copy of that statement? Mr. Wilcox. Yes, sir. Mr. Liebeler. Could we mark that as Exhibit No. 3006 on the deposition of Laurance R. Wilcox, at Dallas, Tex., March 31, 1964? In this case you have actually provided me the original of Mr. Lewis' statement, is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. I have initialed Exhibit No. 3006 and would like to have you do the same, if you would. Mr. Wilcox. [Initials.] Mr. Liebeler. Mr. Lewis' statement refers to that portion of Mr. Hamblen's statement, dated December 2, 1963, which we just read, does it not? Mr. Wilcox. Yes, sir. Mr. Liebeler. And Mr. Lewis says that he does recall the difficulty he had paying a small money order to a gentleman who struck him as being a feminine type of person, but says he cannot remember the name of that person; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. Do you remember discussing that with Mr. Lewis to any extent other than as reflected on this statement? Mr. Wilcox. Yes, sir. Mr. Liebeler. Could you tell us what you said and what Mr. Lewis said about this? Before you tell us what was said, this discussion that you are referring to was held on December 4, 1963, or some other time? Mr. Wilcox. Yes, sir; at that time.
- 66. Mr. Liebeler. Willyou tell us what that conversation was? Mr. Wilcox. The conversation with Mr. Lewis had to do with trying to identify the person that he was supposed to have paid this money order to, and the one with which he had some difficulty in establishing identification. Mr. Liebeler. I want you to tell me exactly what happened in as great detail as you can remember it. Now, at this conversation with Mr. Lewis, the possibility that this money order was to be delivered to someone at the YMCA on Ervay Street in Dallas was discussed, was it not? Mr. Wilcox. Yes, sir. Mr. Liebeler. Will you tell us all that you can remember about that aspect of the discussion that you had with Mr. Lewis? Mr. Wilcox. I have discussed the matter with Mr. Lewis regarding the money order which he was having difficulty in paying. He could not pay the money order because the recipient could not produce suitable identification. Mr. Lewis was a little hazy on the details about it, but was under the impression that it was a money order or a message in the nearby area, possibly on Ervay Street or at the YMCA. We instituted a search for all money orders during this period, that might be the message to which Mr. Hamblen was making reference. Mr. Liebeler. That was the one with which Mr. Lewis had difficulty in paying; is that correct? Mr. Wilcox. Yes. Now, we found some messages. One in the amount of $65. One in the amount of $35. But neither of these were the money orders that Mr. Hamblen had reference to. Mr. Liebeler. How did you establish the fact that they were not the money orders?
- 67. Mr. Wilcox. Helooked at them and decided it wasn't the one, because we couldn't find anything that had reference to this particular identification that Hamblen was talking about. Mr. Liebeler. Mr. Lewis looked at these telegrams; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. Does the telegram indicate what sort of identification was produced by the person to whom the money order was paid? Mr. Wilcox. Well, there are several others in here. Mr. Liebeler. You are looking through a number of telegrams covering money orders; is that correct? Mr. Wilcox. Going to different people at the YMCA, but none of them was the money order message. None of them was a money order message that Lewis could identify as being a money order of the type Hamblen was describing. In other words, he couldn't find anything that fitted into that pattern at all. Mr. Liebeler. You discussed these money orders with Mr. Lewis after you obtained them from the files; is that correct? Mr. Wilcox. That's right. Mr. Lewis later told me that it might have been a money order draft that could have been paid to some individual in our town, and that the party holding the draft had presented it at our office for payment, but was having difficulty in producing suitable identification for us to cash the draft. In that event, we would not have any message or record of message in our file locally. Mr. Liebeler. If I understand you correctly, Mr. Wilcox, the situation which you just described would occur when a money order telegram had been delivered to some party at some other office, some office other than the Dallas central office? Mr. Wilcox. That's right. Mr. Liebeler. And he would have the draft in his hand that would have been delivered to him by the telegraph company in some other
- 68. office, and thenhe would come to the Dallas central office and attempt to cash that draft; is that correct? Mr. Wilcox. Yes. Mr. Liebeler. And Mr. Lewis said it is possible that the transaction with which he had such difficulty could have been a transaction such as the one we have described; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. In that event, you would not have any record of it in the Dallas office? Mr. Wilcox. No, sir. Mr. Liebeler. Is it correct that you would not have any record of it in the Dallas office even when the draft had been successfully cashed, as it apparently subsequently was in this case? Mr. Wilcox. No, sir. Mr. Liebeler. You would still not have any record of it? Mr. Wilcox. No, sir. Mr. Liebeler. Now, you have produced 11 money orders in the form of telegrams, transmitting money to individuals living at the YMCA here in Dallas. Is it correct that these are the only 11 telegrams which you were able to find addressed to the YMCA or to anyone at the YMCA during some period beginning on or about October 1963, and running up to approximately the end of November 1963? Mr. Wilcox. Yes, sir. Mr. Liebeler. Did you, yourself, personally authorize or instruct that the search be conducted which produced these telegrams? Mr. Wilcox. Yes, sir. Mr. Liebeler. That would have been a search through every single money order delivered through the Dallas office through the month of October or November, 1963, is that correct?
- 69. Mr. Wilcox. Yes,sir. Mr. Liebeler. These 11 telegrams are the only 11 that were addressed to the YMCA? Mr. Wilcox. Yes. Mr. Liebeler. Now, the dates covered by the search would be October 1 through the 17th of November 1963? Or what was it, as best you can recall? Mr. Wilcox. The end of November, as I remember. Mr. Liebeler. Beginning when? Mr. Wilcox. Beginning the first part of October. Mr. Liebeler. It is a fact, is it not, that none of these 11 telegrams are addressed to Lee Harvey Oswald? Mr. Wilcox. No, sir; none of them are. Mr. Liebeler. Or to anybody using any of Mr. Oswald's known aliases? Mr. Wilcox. No, sir. Mr. Liebeler. I think that we had better, for the record, indicate the names of the people to whom these telegrams are addressed. There is a telegram dated October 4, 1963, addressed to George McMurray, transmitting $15, is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. There is also a telegram dated October 10, 1963, addressed to Michael C. Robinson, transmitting $100 to Mr. Robinson at the YMCA, is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. And there is a telegram dated October 11, 1963, transmitting $150 to James McGinley, is that right? Mr. Wilcox. Yes, sir.
- 70. Mr. Liebeler. Alsofor the record we better indicate who sent these telegrams. The telegram to Mr. McMurray was sent by Mrs. Mildred McMurray in Plainsfield, N.J.; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. The telegram to Mr. Robinson was sent by Mrs. S. R. Robinson, of Charleston, S.C.; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. The telegram to Mr. McGinley was sent by Cornelius McGinley of Chicago; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. There is a telegram dated October 15, 1963, to Mr. Jack Burge from Rosalie A. Burge, Tulsa. Okla., transmitting $10; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. There is also a telegram dated October 19, 1963, to John A. Casber from John Casber of Midland. Tex., transmitting $35? Mr. Wilcox. Yes, sir. Mr. Liebeler. A telegram dated October 28, 1963, to Gary Aue from Mrs. G. C. O'Quinn, Ft. Morgan, Colo., transmitting $65. Mr. Wilcox. Yes, sir. Mr. Liebeler. There is a telegram dated October 31, 1963, to Stanley S. Qupiec from Stayea Houston of Ware, Mass., transmitting $50; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. There is also a telegram dated November 22, 1963, to Welton Hayes from Louis W. Hayes, of Rome, N.Y., transmitting $25; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. And a telegram dated November either 27 or 28.
- 71. Mr. Wilcox. Itwould be November 7. This is a transmission, November 7. Mr. Liebeler. 1963 to John M. Brandes, Jr., from Helen Tuttle, San Antonio, transmitting $20; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. And one dated November 22, 1963, to Frank or Grace Fitzell, from the Akron Dime Bank in Akron, Ohio, transmitting $200; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. And one dated November 28, 1963, to George McMurray from Mildred McMurray, Plainfield. N.J., transmitting $6; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. And the addresses in each case were located then at the YMCA on North Ervay in Dallas; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. Those were the only money order telegrams to individuals at the YMCA that you were able to find in the search of your records, and you are satisfied that those are the only money order telegrams in existence addressed to people in the YMCA during that period; is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. Now, Mr. Lewis' statement of December 4, 1963, mentions that the individual with whom he had difficulty in paying the money order was a man of Spanish descent, is that correct? Mr. Wilcox. Yes, sir. Mr. Liebeler. Did you have any discussion with Mr. Lewis about that? Mr. Wilcox. We may have had it at the time we read this statement, but I don't recall anything specific.
- 72. Mr. Liebeler. Didyou have any discussion with Mr. Lewis as to whether he recognized this individual with whom he had difficulty in paying the money order, as Lee Harvey Oswald, from pictures of Oswald which appeared in the newspaper? Mr. Wilcox. I asked him about that and he said he couldn't describe or associate the party to the extent that he could associate it with Oswald's picture or anything like that. Mr. Liebeler. He just had no recognition? Mr. Wilcox. Other than that he had some difficulty in paying a money order. Mr. Liebeler. Now, as a result of these events that followed Mr. Hamblen's statement to the newspaper reporter and the subsequent investigation that was requested or instigated by the FBI and the Secret Service and other investigatory agencies, you conducted certain searches of certain files in your office in an attempt to locate any money orders that Oswald might have received, or any telegrams that Oswald might have sent during certain periods; is that correct? Mr. Wilcox. Yes; I can give you those dates. Mr. Liebeler. Let me show you Exhibit 3001, which is a memorandum from Mr. Semingsen, and ask you if that memorandum accurately sets forth the dates and the checks that were made of the files in the Dallas and Irving, Tex., offices of your company to determine whether or not Oswald had received or sent any telegrams? Mr. Wilcox. Yes, sir; that's right. Mr. Liebeler. Mr. Semingsen testified this morning that the paying office—that is, the office through which a money order is delivered, maintains a chronological record or file of all money orders passing through that office; is that correct? Mr. Wilcox. That's right.
- 73. Mr. Liebeler. Itis my understanding that you caused the files for both Dallas and Irving to be searched for the periods indicated in Exhibit 3001? Mr. Wilcox. That's right. Mr. Liebeler. You were unable to find any money order payable to Lee Harvey Oswald or O. H. Lee or Alex James Hidell; is that correct? Mr. Wilcox. That's correct. Mr. Liebeler. Is it possible that money orders could be sent to someone just by using initials or some shorthand name? Mr. Wilcox. No; you never see anything because you would have no way to identify the man or associate it that you are paying it to them correctly, associate it with any identification that he might have. Mr. Liebeler. Since each individual must produce identification at the time the money order is paid, the money order is naturally in the full name? Mr. Wilcox. Correct. They could be accepting it for someone—it might not be their correct name, but they could have some identification that, if it was just a small money order, but we can't change that identification. If it meets what shows on the money order, then we pay it. Mr. Liebeler. But you have never heard of a situation where a money order is paid to somebody just addressed to him by initials or something like that? Mr. Wilcox. No. Mr. Liebeler. How many offices, approximately, does the Western Union office have in Dallas? Mr. Wilcox. There are eight branch offices. Mr. Liebeler. Plus a central office?
- 74. Mr. Wilcox. Plusa central office. Mr. Liebeler. Is it possible to receive money orders through the branch office? Mr. Wilcox. Yes, sir. Mr. Liebeler. Are the files of money orders received through branch offices kept in the branch offices, or the Dallas central office? Mr. Wilcox. Kept at our main office. Mr. Liebeler. So, that the search that you made of the records of the central office would include money orders that were received through any branch office located within the City of Dallas, is that correct? Mr. Wilcox. That's right. Mr. Liebeler. You said before that your district covered not only the City of Dallas, but surrounding communities including Irving? Mr. Wilcox. Yes, sir. Mr. Liebeler. Are the records relating to money orders received through these other offices of suburban communities located or kept in the suburban offices or the Dallas central office? Mr. Wilcox. In this case, this was an agency office and it would be—the records were kept at the Irving agency office. We did search those records at the Irving agency office. Mr. Liebeler. That fact is indicated in Exhibit 3001, is it not? Mr. Wilcox. Yes, sir. Mr. Liebeler. The records of telegrams or money orders received through other suburban offices such as Garland, for example, would also be kept in the Garland office, would they not? Mr. Wilcox. Yes, sir. Mr. Liebeler. But no search was made of any suburban office other than Irving; is that correct?
- 75. Mr. Wilcox. That'sright. Mr. Liebeler. Mr. Semingsen's memorandum also indicates that a search was made of the files in Fort Worth and in New Orleans, but you have no direct personal knowledge of what happened at those offices, do you? Mr. Wilcox. No, sir. Mr. Liebeler. We also have here a statement from Mr. Hamblen dated December 5, 1963. Do you have the original of that statement? Mr Wilcox. Yes; I do have. Mr. Liebeler. We will mark the original as Exhibit No. 3007 on the deposition of Laurance R. Wilcox, at Dallas, Tex., March 31, 1963. I have initialed Exhibit No. 3007, Mr. Wilcox, and I would like you do the same. Mr. Wilcox. [Initials.] Mr. Liebeler. This statement was apparently taken on December 5, 1963. Do you recall the circumstances under which it was given? Mr. Wilcox. This was following a meeting and discussion that we held on December 4, at which time we discussed the money orders and messages with Mr. Hamblen, Mrs. McClure, and Mr. Lewis. Mr. Liebeler. Who was present at this discussion other than the individuals you just mentioned and yourself? (Mr. Wilcox phoned his office re: correct date of meeting.) Mr. Wilcox. I did hold this meeting. Mr. Liebeler. Let the record indicate that Mr. Wilcox has just conferred telephonically with Mr. Semingsen and wishes now to clarify the statements concerning the time on which certain meetings were held with the employees. Mr. Wilcox. I did hold the meetings with these people on December 4, and did obtain these statements, including the
- 76. statement from Mrs.McClure, Mr. Lewis, both indicated as December 4, and the statement from Mr. Hamblen which is dated December 5. Following this meeting I endeavored to find the message or messages that Mr. Hamblen was referring to, which he insisted Mrs. McClure had accepted from Mr. Oswald. I did extract from our files all messages matching the message numbers on the cash sheet prepared by Mrs. McClure. Mr. Liebeler. Did the FBI ask you to do this because Mr. Hamblen said that a message with which Mrs. McClure had difficulty was given to her by a man who Hamblen thought was Oswald, and that the message was one to Washington, D.C., specifically to the Secretary of the Navy—— Mr. Wilcox. Yes, sir. Mr. Liebeler. And that it was accounted for on Mrs. McClure's sheet as a night letter that was set forth on her cash sheet; is that correct? Mr. Wilcox. That's right. Mr. Liebeler. So you then got all of the telegrams that were listed on Mrs. McClure's cash sheets; is that correct? Mr. Wilcox. From the 1st of November through the November 22. We could not find any such messages. However, we did extract all messages going to Washington, D.C., regardless of the names to whom they were sent, or signed, including some messages going to other points, because of their peculiar type of printing. Now, would you like to see those messages? Mr. Liebeler. Yes, sir; I would. Let me ask you specifically if the period covered in terms of this extraction was from October? Mr. Wilcox. From November 1. Mr. Liebeler. From November 1? Mr. Wilcox. Yes, sir.
- 77. Mr. Liebeler. Youpicked that period because Mr. Hamblen said that he thought the event occurred about 10 days prior to the assassination; is that correct? Mr. Wilcox. That's right. Now, I think you have the only thermofax of Mrs. McClure's. Would you want the original? Mr. Liebeler. Yes; I do want to mark the original statement of Mrs. McClure, dated December 4, 1963, which has been previously referred to by Mr. Wilcox. I will mark it as Exhibit 3008 on the deposition of Mr. Laurance R. Wilcox at Dallas, Tex., on March 31, 1964. I have initialed that exhibit, Mr. Wilcox, and would like to have you do the same. Mr. Wilcox. [Initials.] Mr. Liebeler. Who prepared Mrs. McClure's statement? Mr. Wilcox. Mrs. McClure wrote that. Mr. Liebeler. She typed it on the typewriter herself? Mr. Wilcox. Yes, sir. Mr. Liebeler. Were you present at the time she prepared it? Mr. Wilcox. Yes, sir. Mr. Liebeler. Did you discuss these incidents with her prior to the statement? Mr. Wilcox. Only during the course of the meeting that she attended in my office. Mr. Liebeler. During the course of that time, Mrs. McClure indicated to you, did she not, that she had no recollection of any of the events described by Mr. Hamblen concerning the message allegedly sent to Washington, D.C.? Mr. Wilcox. That's right.
- 78. Mr. Liebeler. Let'smark these telegrams as Exhibits 3009 through 3014. I have initialed each one of them and would like to have you do the same. I have not indicated on each one of them in detail. It is on your deposition and on the dates, as I have on the others, but that fact will appear from the record. Mr. Wilcox. [Initials.] Mr. Liebeler. You have now provided me with six telegrams, cables which have been marked for identification as Exhibits 3009 through 3014, and it is my understanding that these telegrams and cables were all shown to Mr. Hamblen and he was unable to identify any of them as answering the description of the telegram or cablegram with which Mrs. McClure had had difficulty, and which Mr. Hamblen thought had been sent by a person resembling Lee Harvey Oswald, is that correct? Mr. Wilcox. That's right. Mr. Liebeler. And a thorough search of the files along the lines that you have previously indicated was unable to produce any other telegrams even remotely falling into the category described by Mr. Hamblen; i.e., a telegram to Washington, D.C. or in a peculiar hand script such as described by Mr. Hamblen to any destination, whether it be Washington or otherwise; is that correct? Mr. Wilcox. That's right. Mr. Liebeler. Now, you have provided me with a letter from yourself to Mr. Semingsen, dated December 6, 1963, which we will mark as Exhibit 3015, on deposition of Laurance R. Wilcox, at Dallas, Tex., March 31, 1964. I have initialed that exhibit and ask you to do the same, sir. Mr. Wilcox. [Initials.] Mr. Liebeler. And ask you if you prepared the original of that letter on or about December 6, 1963? Mr. Wilcox. That's right.
- 79. Mr. Liebeler. Yousent the original of it to Mr. Semingsen, did you not? Mr. Wilcox. Yes, sir; that's right. Mr. Liebeler. And the letter accurately sets forth the events preceding that date which we have been discussing here, does it not? Mr. Wilcox. Yes, sir. Mr. Liebeler. You have also provided me with a copy of a letter from yourself to Mr. Semingsen, dated December 9, 1963, to which is attached the original of statement from Mrs. Betty Bedwell, dated December 6, 1963, and A. I. English, dated December 6, 1963. I notice that Mr. English's statement is not signed. Mr. Wilcox. [Signs.] Mr. Liebeler. And I ask you if the original of it was signed at the time you received it? Mr. Wilcox. No, sir; he just signed this on the typewriter to me. Mr. Liebeler. No; he did not himself sign it? Mr. Wilcox. No. Mr. Liebeler. You have indicated below that it was in effect signed by Mr. English when he delivered it to you? Mr. Wilcox. Yes, sir. Mr. Liebeler. Also attached is a statement of Miss Bess Mildred Francis, dated December 9, 1963. Also attached is a statement of Doyle E. Lane, dated December 9, 1963; and one of Mr. E. T. Pirtle, dated December 6, 1963; and one of Ward Townsley dated December 6, 1963. Did you receive those statements from the individuals thus described? Mr. Wilcox. Yes, sir. Mr. Liebeler. We have marked the letter described above, together with the attachments just described as Exhibit 3016 on the
- 80. deposition of Mr.Laurance R. Wilcox at Dallas, Tex., on March 31, 1964. I have initialed the first page of that exhibit and would like to have you do the same. Mr. Wilcox. [Initials.] Mr. Liebeler. Did you send the original of your letter dated December 9, 1963, to Mr. Semingsen on or about that date? Mr. Wilcox. Yes, sir. Mr. Liebeler. The statements made in that letter are true and correct to the best of your knowledge, are they not? Mr. Wilcox. Yes, sir. Mr. Liebeler. You have also given me the original of a statement dated December 6, 1963, of Mr. Doyle E. Lane. Was that also attached to your letter of December 9, 1963, to Mr. Semingsen? Mr. Wilcox. No, sir. Mr. Liebeler. Did you receive yourself from Mr. Lane the statement just described? Mr. Wilcox. Yes, sir. Mr. Liebeler. Let me mark it as Exhibit 3017 on this deposition. Would you initial the statement of Mr. Lane which we have marked 3017? Mr. Wilcox. Yes, sir. [Initials.] Mr. Liebeler. Thank you. You previously testified, Mr. Wilcox, that you had a meeting on or about December 4, 1963, with Mr. Hamblen, Mrs. McClure, and Mr. Lewis in regard to the meeting we have been discussing, and you also testified that you thought you had an additional meeting on December 9, 1963, at which Mr. Semingsen was present. Would you care to elaborate on that?
- 81. Mr. Wilcox. Themeeting as you have outlined—the meetings on the dates you have outlined are correct. The meeting on December 9, at which Mr. Semingsen attended was for the purpose of confronting Mr. Hamblen with the messages that we had extracted that were going to Washington, or those which were in peculiar print, that had been accepted by Mrs. McClure during the period of 1st of November until about November 22. Mr. Hamblen, of course, could not identify any of these telegrams as having been the message he described in his statement of December 2 and December 5. Mr. Liebeler. Did you form a conclusion at that time as to the accuracy of Mr. Hamblen's recollection concerning the events which he had related to you and to the press? Mr. Wilcox. Yes. Mr. Liebeler. What was that conclusion? Mr. Wilcox. That this whole thing was a figment of Mr. Hamblen's imagination. I am fearful that he was just emotionally upset over the events as they transpired, and this may have been the factor causing him to say what he had said about the acceptance of the messages and the money order, and possibly have something to do with his statements to the press. Mr. Liebeler. As I understand you, you were perfectly satisfied at that time that Oswald had never in fact been in the office either to receive money orders or to send any telegram of the type described by Mr. Hamblen, or as far as you have been able to determine, any other telegram; is that correct? Mr. Wilcox. That's right. Mr. Liebeler. At this point I can't think of any further questions, Mr. Wilcox. If you can think of anything that you would like to say that in your opinion would be of assistance to the Commission in its work, please indicate that at this time? Can you think of anything else?
- 82. Mr. Wilcox. Ican't think of anything else now, but if I do, I will be happy to pass it along to you. Mr. Liebeler. If you do think of anything subsequently, call it to my attention and I would appreciate it very much. Thank you very much, Mr. Wilcox. You have been very helpful and very cooperative. The Commission appreciates the cooperation you and Western Union have shown. Thank you very much.
- 83. Transcriber's Notes Punctuation andspelling were made consistent when a predominant preference was found in this book; otherwise they were not changed. Misspellings in quoted evidence not changed; misspellings that could be due to mispronunciations were not changed. Some simple typographical errors were corrected. Inconsistent hyphenation of compound words retained. Ambiguous end-of-line hyphens retained. Occasional uses of Mr. for Mrs. and of Mrs. for Mr. corrected. Dubious repeated words, (e.g., What took place by way of of conversation?) retained. Several unbalanced quotation marks not remedied. Occasional periods that should be question marks, and question marks that probably should be periods, not changed. Occasional periods that should be commas, and commas that should be periods, were changed only when they clearly had been misprinted (at the end of a paragraph or following a speaker's name in small-caps at the beginning of a line). Some commas and semi-colons were printed so faintly that they appear to be periods or colons: some were found and corrected, but some almost certainly remain.
- 84. The Index andillustrated Exhibits volumes of this series may not be available at Project Gutenberg. Page 1: had an acquaintance with the Oswald's was printed that way. Page 24: thinking she said this latter probably should be later. Page 54: October 18, 1938 is the date printed in Testimony; Oswald's actual date of birth was October 18, 1939. Page 110: ever been a subscribed to The Militant should be either subscriber or ever subscribed. Page 112: examine Watts' No. 8 probably should be Dobbs' No. 8. Page 117: Sentence beginning If my voice is low, young lady is unattributed, but in context, probably was spoken by Mr. Jenner. Page 130: or anyone of a number of was printed that way. Page 204: through his hurrying be made perhaps should be he made. Page 208: NVD is either misprint or mispronunciation for MVD. Page 273: you know, what night or probably should be that night. Page 277: Rubenstein is spelled Rubinstein elsewhere in this and other volumes of the Hearings. Page 282: To be best of your recollection was printed that way. Page 351: Bogard was bound to own probably should be found.
- 85. Page 352: Vendingmachines. What kind of vending machines? was missing the period; changed here.
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