Probability formula sheet
Set theory, sample space, events, concepts of randomness and uncertainty, basic principles of probability, axioms and properties of probability, conditional probability, independent events, Baye’s formula, Bernoulli trails, sequential experiments, discrete and continuous random variable, distribution and density functions, one and two dimensional random variables, marginal and joint distributions and density functions. Expectations, probability distribution families (binomial, poisson, hyper geometric, geometric distribution, normal, uniform and exponential), mean, variance, standard deviations, moments and moment generating functions, law of large numbers, limits theorems
for more visit http://tricntip.blogspot.com/
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez where ez is the exponential function and z is any complex number. In other words
{\displaystyle z=f^{-1}(ze^{z})=W(ze^{z})} z=f^{-1}(ze^{z})=W(ze^{z})
By substituting {\displaystyle z'=ze^{z}} z'=ze^{z} in the above equation, we get the defining equation for the W function (and for the W relation in general):
{\displaystyle z'=W(z')e^{W(z')}} z'=W(z')e^{W(z')}
for any complex number z'.
Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W, the complex variable z is then replaced by the real variable x, and the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0). The additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W−1(x). It decreases from W−1(−1/e) = −1 to W−1(0−) = −∞.
The Lambert W relation cannot be expressed in terms of elementary functions.[1] It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
Probability formula sheet
Set theory, sample space, events, concepts of randomness and uncertainty, basic principles of probability, axioms and properties of probability, conditional probability, independent events, Baye’s formula, Bernoulli trails, sequential experiments, discrete and continuous random variable, distribution and density functions, one and two dimensional random variables, marginal and joint distributions and density functions. Expectations, probability distribution families (binomial, poisson, hyper geometric, geometric distribution, normal, uniform and exponential), mean, variance, standard deviations, moments and moment generating functions, law of large numbers, limits theorems
for more visit http://tricntip.blogspot.com/
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez where ez is the exponential function and z is any complex number. In other words
{\displaystyle z=f^{-1}(ze^{z})=W(ze^{z})} z=f^{-1}(ze^{z})=W(ze^{z})
By substituting {\displaystyle z'=ze^{z}} z'=ze^{z} in the above equation, we get the defining equation for the W function (and for the W relation in general):
{\displaystyle z'=W(z')e^{W(z')}} z'=W(z')e^{W(z')}
for any complex number z'.
Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W, the complex variable z is then replaced by the real variable x, and the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0). The additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W−1(x). It decreases from W−1(−1/e) = −1 to W−1(0−) = −∞.
The Lambert W relation cannot be expressed in terms of elementary functions.[1] It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docxanhlodge
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
More instructions for the lab write-up:
1) You are not obligated to use the 'diary' function. It was presented only for you convenience. You
should be copying and pasting your code, plots, and results into some sort of "Word" type editor that
will allow you to import graphs and such. Make sure you always include the commands to generate
what is been asked and include the outputs (from command window and plots), unless the pr.
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
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Engineering Research Publication
Best International Journals, High Impact Journals,
International Journal of Engineering & Technical Research
ISSN : 2321-0869 (O) 2454-4698 (P)
www.erpublication.org
ER Publication,
IJETR, IJMCTR,
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International Journals,
High Impact Journals,
Monthly Journal,
Good quality Journals,
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Quantum algorithm for solving linear systems of equationsXequeMateShannon
Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.
A study on number theory and its applicationsItishree Dash
A STUDY ON NUMBER THEORY AND ITS APPLICATIONS
Applications
Modular Arithmetic
Congruence and Pseudorandom Number
Congruence and CRT(Chinese Remainder Theorem)
Congruence and Cryptography
I am Leonard K. I am a Differential Equations Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From California, USA. I have been helping students with their homework for the past 8 years. I solve homework related to Differential Equations.
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Métodos directos para solución de sistemas ecuaciones lineales
Metodos jacobi y gauss seidel
1. Special Matrices. A band matrix is a square matrix in which all elements are zero, except for a band on the main diagonal. A tridiagonal system (ie with a bandwidth of 3) can be expressed generally as: <br /> <br />f0g0 x0 b0e1f1g1 x1=b1 e2f2g2 x2 b2 e3f3 x3 b3<br /> <br />Based on LU decomposition we can see that the Thomas algorithm is: <br /> <br /> <br /> <br />The forward substitution is <br /> <br /> <br />and back is: <br /> <br /> <br /> Example: Solve the following tridiagonal system using the Thomas algorithm. <br />2.04-1 x0 40.8-12.04-1 x1=0.8 -12.04-1 x2 0.8 -12.04 x3 200.8<br /> <br /> The solution of the triangular decomposition is: <br /> <br />2.04-1 -0.491.55-1 -0.6451.395-1 -0.7171.323<br /> <br /> <br />The solution of the system is: <br /> <br />X = [65.970, 93.778, 124.538, 159.480]T<br /> <br />Descomposición de Cholesky.<br /> This algorithm is based on the fact that a symmetric matrix can be decomposed into [A] = [L] [L] T, since the matrix [A] is a symmetric matrix. In this case we will apply the Crout elimination of the lower and upper matrix, simply have the same values. So taking the equations for the LU factorization can be adapted as: We can see that any element under the diagonal is calculated as:<br />para todo i=0,...,n-1 y j = 0,...,i-1.<br /> <br /> For the terms above the diagonal, in this case only the diagonal<br />para todo i=0,...,n-1.<br /> <br /> The Java implementation is: <br /> static public void Cholesky(double A[][]) {<br /> int i, j, k, n, s;<br /> double fact, suma = 0;<br /> <br /> n = A.length;<br /> <br /> for (i = 0; i < n; i++) { //k = i<br /> for (j = 0; j <= i - 1; j++) { //i = j<br /> suma = 0;<br /> for (k = 0; k <= j - 1; k++) // j = k<br /> suma += A[i][k] * A[j][k];<br /> <br /> A[i][j] = (A[i][j] - suma) / A[j][j];<br /> }<br /> <br /> suma = 0;<br /> for (k = 0; k <= i - 1; k++)<br /> suma += A[i][k] * A[i][k];<br /> A[i][i] = Math.sqrt(A[i][i] - suma);<br /> }<br /> }<br /> Jacobi Method<br /> In numerical analysis method of Jacobi is an iterative method, used for solving systems of linear equations Ax = b. Type The algorithm is named after the German mathematician Carl Gustav Jakob Jacobi. • <br />Description <br />The basis of the method is to construct a convergent sequence defined iteratively. The limit of this sequence is precisely the solution of the system. For practical purposes if the algorithm stops after a finite number of steps leads to an approximation of the value of x in the solution of the system. The sequence is constructed by decomposing the system matrix as follows: <br />where<br />, Is a diagonal matrix<br />, Is a lower triangular matrix.<br />, Is an upper triangular matrix. <br />Starting from,, we can rewrite this equation as: <br />Then <br /> If aii ≠ 0 for each i. For the iterative rule, the definition of the Jacobi method can be expressed as: <br />where k is the iteration counter, finally we have: <br />Note that the calculation of xi (k +1) requires all elements in x (k), except the one with the same i. Therefore, unlike in the Gauss-Seidel method, you can not overwrite xi (k) xi (k +1), since its value will be for the remainder of the calculations. This is the most significant difference between the methods of Jacobi and Gauss-Seidel. The minimum amount of storage is two vectors of dimension n, and will need to make an explicit copy. <br />Convergence <br />Jacobi's method always converges if the matrix A is strictly diagonally dominant and can converge even if this condition is not satisfied. It is necessary, however, that the diagonal elements in the matrix are greater (in magnitude) than the other elements.<br /> Algorithm<br /> The Jacobi method can be written in the form of an algorithm as follows: <br />Algoritmo Método de Jacobifunción Jacobi (A, x0)//x0 es una aproximación inicial a la solución//para hasta convergencia hacer para hasta hacer para hasta hacer si entonces fin parafin paracomprobar si se alcanza convergenciafin para<br />algorithm in java <br />public class Jacobi {<br />double [][]matriz={{4,-2,1},{1,-5,3},{2,1,4}};<br />double []vector={2,1,3};<br />double []vectorR={1,2,3};<br />double []x2=vectorR;<br />double sumatoria=1;<br />int max=50;<br /> public void SolJacobi(){<br /> int tam = matriz.length;<br />for (int y = 0; y < 10; y++) {<br /> system.outtt.println(quot;
vector quot;
+ y + quot;
quot;
);<br /> for(int t=0;t>max;t++){<br /> x2=vectorR.clone();<br /> for (int i = 0; i < tam; i++) {<br /> sumatoria=0;<br /> for (int s = 0; s < tam; s++) {<br /> if(s!=i)sumatoria += matriz[i][s]*x2[s];<br />}<br />vectorR[i]=(vector[i]-sumatoria)/matriz[i][i];<br /> System.out.print(quot;
quot;
+ vectorR[i]);<br />}<br />}<br /> <br />}<br />} <br />public static void main(String[] args) {<br /> jacobi obj=new Jacobi();<br /> obj.SolJacobi();<br />}<br />}<br />Gauss-Seidel Method <br />In numerical analysis method is a Gauss-Seidel iterative method used to solve systems of linear equations. The method is named in honor of the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel and is similar to the method of Jacobi. <br />Description <br />It is an iterative method, which means that part of an initial approximation and the process is repeated until a solution with a margin of error as small as desired. We seek the solution to a system of linear equations, in matrix notation: <br />The method of Gauss-Seidel iteration<br />where<br />para i=j, o para .<br />and<br />This is alsoque :<br />If<br />define<br />and<br />.<br />Whereas the system Ax = b, with the proviso that, i = 1, ..., n. Then we can write the iteration formula of the method <br />, i=1,...,n(*)<br />The difference between this method and Jacobi is that, in the latter, improvements to conventional approaches are not used to complete the iterations. <br />Convergence Theorem: Suppose a matrix is a nonsingular matrix that satisfies the condition or. ó .Then the Gauss-Seidel method converges to a solution of the system of equations Ax = b, and the convergence is at least as fast as the convergence of Jacobi method. <br />For cases where the method converges show first that can be written as follows: <br />(**)<br />(The term is the approximation obtained after the k-th iteration) this way of writing the iteration is the general form of a stationary iterative method. <br />First we must show that we want to solve the linear problem can be represented in the form (**), for this reason we must try to write the matrix as the sum of a lower triangular matrix, a diagonal and upper triangular A = D ( L + I + U), D = diag (). Making the necessary clearances write the method this way <br />hence B =- (L + I) -1 U. Now we can see that the relation between errors, which can be calculated to subtract x = Bx + c (**) <br />Now suppose that , i= 1, ..., n, are the eigenvalues corresponding to eigenvectors ui, i = 1 ,..., n, which are linearly independent, then we can write the initial error <br />(***)<br />Therefore, the iteration converges if and only if | λi | <1, i = 1, ..., n. From this it follows the following theorem:<br />Theorem: A necessary and sufficient condition for a stationary iterative method converges for an arbitrary approximation x ^ ((0)) is that where ρ (B) is the spectral radius of B.<br />Explanation <br />We choose an initial approximation. M matrices are calculated and the vector c with the formulas mentioned. The process is repeated until xk is sufficiently close to xk - 1, where k represents the number of steps in the iteration. <br />Algorithm<br />The Gauss-Seidel can be written algorithm as follows:<br />Algoritmo Método de Gauss-Seidelfunción Gauss-Seidel (A, x0)//x0 es una aproximación inicial a la solución//para hasta convergencia hacer para hasta hacer para hasta hacer si entonces σ = σ + aijxjfin parafin paracomprobar si se alcanza convergenciafin para<br />EXAMPLE JACOBI AND GAUSS SEIDEL METHOD <br />Two numerical methods, which allows us to find solutions to systems with the same number of equations than unknowns. <br />In both methods the following process is done with a little variation on Gauss-Seidel <br />We have these equations: <br />5x-2y+z=3-x-7y+3z=-22x-y+8z=1<br />1. Solve each of the unknowns in terms of the others<br />x=(3+2y-z)/5y=(x-3z-2)/-7z=(1-2x+y)/8<br />2. Give initial values to the unknowns<br />x1=0y1=0z1=0<br />By Jacobi: Replace in each equation the initial values, this will give new values to be used in the next iteration <br />x=(3+2*0-0)/5=0,60y=(0-3*0-2)/-7=0,28z=(1-2x+y)/8=0,12<br />Gauss-Seidel Replace the values in each equation but found next. <br />x=(3+2*0-0)/5=0,6y=(0,6-3*0-2)/-7=0,2z=(1-2*0,6+0,2)/8=0<br />It performs many iterations you want, using as initial values the new values found. You can stop the execution of the algorithm to calculate the error of calculation, which we can find with this formula: sqr ((x1-x0) ^ 2 + (y1-y0) ^ 2 + (z1-z0) ^ 2)<br /> jacobi<br /> Gauss-Seidel<br />The main difference is that the method of gauss_seidel uses the values found immediately, then makes the whole process faster, and consequently makes this a more effective method. <br />The formulas used in the excel sheet for the method of Jacobi is <br />=(3+2*D5-E5)/5=(C5-3*E5-2)/-7=(1-2*C5+D5)/8=RAIZ((C6-C5)^2 + (D6-D5)^2 + (E6-E5)^2)<br />Corresponding to the variable X, Y, Z and failure respectively<br />To the Gauss-Seidel:<br />=(3+2*J5-K5)/5=(I6-3*K5-2)/-7=(1-2*I6+J6)/8=RAIZ((I6-I5)^2 + (J6-J5)^2 + (K6-K5)^2)<br />