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- 1. Chapter 2: Error Analysis – – – – – – – Significant Figures Accuracy & Precision Error Definitions Round-off Errors Truncation Errors Error Propagation Formulation Errors & Data uncertainty
- 2. Significant Figures The significant figures are the digits that carry meaning to the precision of the measurement. Consider three measurements for the length of a table: L1=3.2 m L2: 3.27 m L3: 3.270 m Number of significant figures for L1 is two, for L2 is three, and for L3 is four. First digit is the most significant figure, and the last digit is the least significant digit in the measurement. We can assign error associated with each measurement: L1=3.2 +- 0.2 m L2: 3.27 m +- 0.01 m L3: 3.270 +- 0.003 m Any digit beyond the error carrying digits is meaningless. Leading zeros are not significant. They are only used to show the location of the decimal point. e. g. 0.00052 has only two significant digits . To avoid confusion, scientists prefer scientific notation (e.g., 5.2x10-4).
- 3. Accuracy & Precision Accuracy refers to how closely a computed or measured value agrees with the true value. Inaccuracy (also called bias) is a systematic deviation from the truth. Precision refers to how closely individual computed or measured values agree with each other. Imprecision (also called uncertainty) refers to the magnitude of the scatter. accuracy and precision are independent from each other.
- 4. Error Definitions In numerical methods both accuracy and precision is required for a particular problem. We will use the collective term error to represent both inaccuracy and imprecision in our predictions. Numerical errors arise from the use of approximation to represent exact mathematic operations or quantities. Consider the approximation we did in the problem of falling object in air. We observed some error between the exact (true) and numerical solutions (approximation). The relationship between them: True value = approximation + error or Et = true value – approximation
- 5. Note that in this equation, we included all factors contributing to the error. So, we used the subscript t to designate that this is the true error). To take into account different magnitudes in different measurements, we prefer to normalize the error. Then, we define the fractional relative error: Fractional relative error = (true value-approximation)/(true value) or the percent relative error: t = (true value-approximation)/(true value) x 100 Most of the times, we just say “error” to mean percent relative error.
- 6. So, we define true error as : (true value approximated value) t 100 % (true value) In most cases we don’t have the knowledge of the “true value”, so we define approximate error as (approximate error ) a 100 % (approximate value) Approximate error can be defined in different ways depending on the problem. For example, in iterative methods, error is defined with respect to the previous calculation. (current approx. previous approx.) a 100 % (current approx.)
- 7. Round-off Errors Round-off errors result from the omission of the significant figures. Base-10 (decimal) versus Base-2 (binary) system: Base-10 Base-2 103 102 101 100 23 22 21 20 a b c d a b a b = ax103 + bx102 + cx101 + dx100 = ax23 + bx22 + ax21 + bx20 Positional Notation Computers knows only two numbers (on/off states). So computers can only store numbers in binary (base-2) system. e.g. 100101 a bit (binary digit). 1 byte= 8 bits computer uses 6 bits to store this number.
- 8. Integer Representation: First bit is used to store the sign (0 for “+” and 1 for “-”); remaining bits are used to store the number. In integer representation, numbers can be defined exactly but only a limited range of numbers are allowed in a limited memory. Also fractional quantities can not be represented. Ex: How -3 is stored in a computer in integer representation? Ex: Find the range of numbers that you can store in a 16-bit computer in integer representation. (-32767 to 32767).
- 9. Floating Point Representation: FPR allows a much wider range of numbers than integer representation. It allows storing fractional quantities. It is similar to the scientific notation. Mantissa (significand) mb e e.g. exponent base 0.015678 1.5678 10 2 (base-10) Ex: Assume you have a hypotetical base-10 computer with a 5-digit word size (one digit for sign, two for exponent with sign, two for mantissa). a) Find the range of values that can be represented. b) Calculate the error of representing 2-5 using this representation.
- 10. IEEE floating point representation standards: 32-bit (single precision) word format: 64-bit (double precision) word format: Mantissa takes only a limited number of significant digits round-off error Increasing the number of digits (32-bit versus 64-bit) decreases the roundoff error.
- 11. Range: In FPR there is still a limit for the representation numbers but the range is much bigger. In 64-bit representiaton in IEEE format: 52 digits 11 digits Max value= +1.111…1111 x 2 +(1111111111) = 1.7977 x 10+308 Min value= 1.000…0000 x 2 -(1111111111) = 2.2251 x 10 -308 Numbers larger than the max. value cannot be represented by the computer overflow error. >> realmax ans= 1.7976931e+308 Any value bigger than this is set to infinity Numbers smaller than the min. value cannot be represented. There is a “hole” at zero. underflow error. >> realmin ans= 2.22500738e-308 Any value smalller than this is set to zero
- 12. Precision: 52 bits used for the mantissa correspond to about 15-16 base-10 significant units. >> pi ans= 3.142857 >> formatlong >> pi ans= 3.1428571428571 32-bit representation (single precision) 64-bit representation (double precision) Ex: Find the smallest possible value floating point number for a hypothetical base-2 machine that stores information using 7-bits words (first bit for the sign of the number, next three for the sign and magnitude of the exponent , and the last three for the magnitude of the mantissa). (1x2-3)
- 13. Chopping versus Rounding: Assume a computer that can store 7 significant digits: Rounding Chopping error error 4.2428576428...... 4.2428576428...... 4.242857 4.242858 Rounding is a better choice since the sign of error can be either positive and negative leading to smaller total numerical error. Whereas error in chopping is always positive and adds up. Rounding costs an extra processing to the computer, so most computers just chops off the number. Error associated with rounding/chopping Quantization error
- 14. Machine epsilon: As a result of quantization of numbers, there is a finite length of interval between two numbers in floating point representation. x Machine epsilon (or machine precision) is the upper bound on the relative error due to chopping/rounding in floating point arithmetic. x For a 64-bit x representation, b=2, t=53 =2-52 The machine epsilon can be computed as =2.22044.. x 10-16 b 1t b=number base t= number of digits in mantissa >> eps ans= 2.2204460e-16
- 15. Arithmetic operations: Besides the limitations of the computer for storage of numbers, arithmetic operations of these numbers also contribute to the round-off error. Consider a hypotetical base-10 computer with 4-digit mantissa and 1-digit exponent: 1.345 + 0.03406 = 0.1345 x 101 + 0.003406 x 101 = 0.137906 x 101 in arithmetic operations numbers are converted as with same exponents chopped-off Ex: a) Evaluate the polynomial y x 3 5 x 2 6 x 0.55 at x=1.73. Use 3-digit arithmetic with chopping. Evaluate the error. b) If the function is expressed as y x( x 5) 6x 0.55 What is the percent relative error? Compare with part a.
- 16. Subtractive cancellation: Subtructive cancellation occurs when subtracting two nearly equal number. 0.7549 x 103 - 0.7548 x 103 = 0.0001 x 103 4 S.D. 4 S.D. Also called loss of significance 1 S.D. Many problems in numerical analysis are prone to subtractive cencallation error. They can be mitigated by manipulations in the formulation of the problem or by increasing the precision. Consider finding the roots of a 2nd order polynomial: b b 2 4ac x2 2a x1 b 4ac 2 Subtractive cancellation - Can use double precision, or - Can use an alternative formulation: x1 x2 2c b b 2 4ac
- 17. Truncation Errors Truncation errors result from using an approximations in place of exact mathematical representations. Remember the approximation in the falling object in air problem: dv v v(ti 1 ) v(ti ) dt t ti 1 ti Taylor theorem: Taylor’s theorem give us insight for estimating the truncation error in the numerical approximation. Taylor’s theorem states that if the function f and its n+1 drivatives are continous on an interval containing a and x, then the values of the function at x is given by f ( 2) (a) f ( n ) (a) 2 f ( x) f (a) f (a)( x a) ( x a) ... ( x a) n Rn 2! n! '
- 18. exact solution In other words, any smooth function can be approximated as a polynomial of order n within a given interval. The error gets smaller as n increases. base point(a)=1 Ex: Use second order Taylor series expansion to approximate the function f ( x) 0.1x 4 0.15 x 3 0.5 x 2 0.25 x 1.2 at x=1 from a=0. Calculate the truncation error from this approximation.
- 19. Suppose you have f(xi ) and want to evaulate f(xi+1): f 2 ( xi ) 2 f n ( xi ) n f ( xi 1 ) f ( xi ) f ' ( xi )h h ... h Rn 2! n! h ( xi 1 xi ) Step size where f n 1 ( ) n 1 Rn h ( n 1)! xi , xi 1 Here Rn represents the remainder (or the error) from the n-th order approximation of the function. It provides and exact determination of the error. We can estimate the order of the magnitude of the error in terms of step size (h): Rn O (h n 1 ) we can change ‘h’ to control the magnitude of the error in the calculation!
- 20. Falling object in air problem: We can evalate the truncation error for the “falling object in air” problem. Express v(ti+1 ) in Taylor series: v '' (ti ) 2 v n (ti ) n v(ti 1 ) v(ti ) v (ti )h h ... h Rn 2! n! ' h (ti 1 ti ) Taylor series to n=1: v(ti 1 ) v(ti ) v ' (ti )h R1 R1 O(h 2 ) or v ' (ti ) v(ti 1 ) v(ti ) R1 h h Finite difference appr. Truncation error R1 O(h 2 ) O ( h) h h Error Then the error associated with finite difference approximation is in the order of h.
- 21. Error Propogation Error propagation concerns how an error in x is propagated to the function f(x). x=true v. xo= approx. v. x x x f ( x ) f ( x) f ( x ) Propagation of error Taylor expansion can be used to estimate the error propagation. Lets evaluate f(x) near f(xo): f ( x) f ( x ) f ' ( x)x ... f ( x) f ( x) f ( x ) f ' ( x)x dropping 2nd and higher order terms f ( x ) f ' ( xo ) x Ex 2.3: Given a measured value of x0= 2.50.01, estimate the resulting error in the function f(x)=x3.
- 22. Functions of more than one variable: x x x y y y x z z f ( x , y o , z o ,..) Propagation of error ... error in error out Error propagation for functions of more than one variable can be understood as the generalization of the case of functions with a single variable: f f f f ( x , y , z ,..) x y z ... x y z o o Ex 2.3: Open channel flow formula for a rectangular channel is given by : 1 (bh) 5 / 3 Q n (b 2h) 2 / 3 s (Q=flow rate , n=roughness coff.,) (b=width, h=depth, s=slope) Assume that b=20 m and h=0.3 m for the channel. If you know that n=0.030±0.002 and s=0.05±0.01, what is the resulting error in calculation of Q?
- 23. Condition and stability: Condition of a mathematical computation is the sensitivity to the input values. It is a measure of how much an uncertainity is magnified by the computation. Condition Number = Error in the output Error in the input f ( x) f ( x o ) f ' ( x o )( x x o ) Condition xo f ' ( xo ) f ( xo ) f ( xo ) o o Number (x x ) (x x ) f ( xo ) xo xo If the uncertainty in the input results in gross changes in the output, we say that the problem is unstable or ill-conditioned. C.N.1 C.N.>>1 (ill-conditioned)
- 24. Total numerical error: Total Numerical Error = Round-off Error + Truncation Error Round-off errors can be minimized by increasing the number of significant digits. Subtractive cancellations and number of computations increases the roundoff-error. Truncation errors can be reduced by decreasing the step size (h). But this may result in subtractive cancellation error too. So, there is a trade-off between truncation error and round-off error in terms of step size (h). Note that there is no systematic and general approach to evaluating numerical errors for all problems.
- 25. Formulation Errors & Data Uncertainity These errors are totally independant from numerical errors, and are not directly connected to most numerical methods. Blunders: In other words: stupid mistakes. They can only be mitigated by experience, or by consulting to experienced persons. Formulation (model) errors: Formulation (or model) errors are causes by incomplete formulation of the mathematical model. (e.g., in “the falling object in air problem”, not taking the effect of air fricton into account). Data uncertainity: If your data contain large inaccuracies or imprecisions (may be due to problems with measurement device), this will directly affect the quality of the results. Statistical analyses on the data helps to minimize these errors.

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