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EP208 Computational Methods in Physics – Notes on Lecture 2 Numerical Differentiation
NUMERICAL DIFFERENTIATION
For a function f(x), first and second derivatives are given as follows:
Method
FDA
Truncation
Error order
First Derivative
f ' ( x)
f ( x h)
h
f ( x)
h
CDA
f ' ( x)
f ( x h) f ( x h)
2h
REA
f ' ( x)
f ( x 2h) 8 f ( x h) 8 f ( x h)
12h
Method
h2
f ( x 2h)
h4
Error
order
Second Derivative
FDA
f ' ' ( x)
CDA
f ' ' ( x)
REA
f ( x h) 2 f ( x )
h2
f ( x h)
f ( x 2h) 16 f ( x h) 30 f ( x) 16 f ( x h)
12h 2
h2
f ( x 2h)
Partial Derivatives for a function f(x, y, z) [CDA only]
f
x
f ( x h, y , z ) f ( x h, y , z )
2h
2
f
x
2
2
f
x y
f ( x h, y, z ) 2 f ( x, y, z )
h2
f ( x h, y h, z )
f ( x h, y, z )
f ( x h, y h, z ) f ( x h, y h, z )
4h 2
here h is small but not zero
1
f ( x h, y h, z )
h4
EP208 Computational Methods in Physics – Notes on Lecture 2 Numerical Differentiation
EXAMPLE
Comparison of the numerical (first) derivative methods for f(x) = ex – 2x at x = 3 with h = 0.01
Method
True
FDA (single precision)
CDA (single precision)
REA (single precision)
f’(x)
18.085537
18.186188
18.085766
18.085377
Error
0.000000
0.100651 (0.5565%)
0.000229 (0.0012%)
0.000160 (0.0008%)
C Pseudo code to compute the first derivative of a function f(x).
C The FDA, CDA and REA methods are implemented for comparison.
function definition f(x) = ...
input "input x ", x
input "input h ", h
fda
cda
rea
= (f(x+h)-f(x))/h
= (f(x+h)-f(x-h))/(2*h)
= (f(x-2*h)-8*f(x-h)+8*f(x+h)-f(x+2*h))/(12*h)
output "FDA = ", fda
output "CDA = ", cda
output "REA = ", rea
Fortran
C++
PROGRAM FirstDerivative
IMPLICIT NONE
#include <iostream>
#include <cmath>
using namespace std;
REAL :: x, h, fda, cda, rea
PRINT
READ
PRINT
READ
fda
cda
rea
float f(float x){
float y = exp(x)-2*x;
return y;
}
*,"Input x"
*,x
*,"Input h"
*,h
int main(){
float x, h, fda, cda, rea;
= (f(x+h)-f(x))/h
= (f(x+h)-f(x-h))/(2*h)
= (f(x-2*h)-8*f(x-h) + &
8*f(x+h)-f(x+2*h))/(12*h)
PRINT *,"FDA = ", fda
PRINT *,"CDA = ", cda
PRINT *,"REA = ", rea
CONTAINS
REAL FUNCTION F(x)
REAL, INTENT(IN) :: x
F = EXP(x)-2*x
END FUNCTION
<<
>>
<<
>>
"Input x ";
x;
"Input h ";
h;
fda
cda
rea
= (f(x+h)-f(x))/h;
= (f(x+h)-f(x-h))/(2*h);
= (f(x-2*h)-8*f(x-h) +
8*f(x+h)-f(x+2*h))/(12*h);
cout << "FDA = " << fda << endl;
cout << "CDA = " << cda << endl;
cout << "REA = " << rea << endl;
END PROGRAM FirstDeivative
Input x
Input h
FDA =
CDA =
REA =
cout
cin
cout
cin
return 0;
}
Input x 3
Input h 0.01
FDA = 18.1862
CDA = 18.0858
REA = 18.0854
3
0.01
18.186188
18.085766
18.085377
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EP208 Computational Methods in Physics – Notes on Lecture 2 Numerical Differentiation
EXAMPLE
2
Comparison of the numerical (second) derivative methods for f(x) = 3sin(x) + x at x = π/4 with h= 0.01
Method
True
CDA2 (double precision)
REA2 (double precision)
f’(x)
-0.121320343
-0.121302666
-0.121320343
Error
0.000000000
0.000017677 (0.02%)
0.000000000
C Pseudo code to compute the second derivative of a function f(x).
C The FDA, CDA and REA methods are implemented for comparison.
function definition f(x) = ...
input "input x ", x
input "input h ", h
cda2
rea2
= ( f(x-h) - 2f(x) + f(x+h) ) / h2
= (-f(x-2h)+16f(x-h)-30f(x)+16f(x+h)-f(x+2h) ) / (12h2)
output "CDA2 = ", cda2
output "REA2 = ", rea2
Fortran
C++
PROGRAM SecondDerivative
IMPLICIT NONE
INTEGER, PARAMETER :: K = 8
REAL(KIND=K) :: x, h, cda2, rea2
#include <iostream>
#include <cmath>
using namespace std;
#define Real double
x = 3.14159265358979323846_K/4.0_K
h = 0.01_K
Real f(Real x){
Real y = 3.0*sin(x) + x*x;
return y;
}
cda2 =(f(x-h)-2*f(x)+f(x+h))/h**2
rea2 =(-f(x-2*h)+ 16*f(x-h) - &
30*f(x) + 16*f(x+h) - &
f(x+2*h))/(12*h**2)
int main(){
Real x, h, cda2, rea2;
PRINT *,"CDA2 = ", cda2
PRINT *,"REA2 = ", rea2
x = M_PI/4.0;
h = 0.01;
CONTAINS
cda2 = (f(x-h)-2*f(x)+f(x+h))/(h*h);
rea2 = (-f(x-2*h) + 16*f(x-h) 30*f(x) + 16*f(x+h) f(x+2*h))/(12*h*h);
REAL (KIND=K) FUNCTION F(x)
REAL(KIND=K), INTENT(IN) :: x
F = 3*sin(x) + x**2
END FUNCTION
cout << "CDA2 = " << cda2 << endl;
cout << "REA2 = " << rea2 << endl;
END PROGRAM SecondDerivative
return 0;
}
if KIND
CDA2
REA2
if KIND
CDA2
REA2
= 4
= -0.12397766
= -0.12477239
= 8
= -0.12130266594301276
= -0.12132034331665172
if Real
CDA2
REA2
if Real
CDA2
REA2
3
is float
= -0.123978
= -0.124772
is double
= -0.121303
= -0.12132
EP208 Computational Methods in Physics – Notes on Lecture 2 Numerical Differentiation
Errors in Numerical Derivative
The approximation methods FDA, CDA, and REA can be used to demonstrate the
effect of truncation errors and round-off errors. The error, for example (h2/6).f```(x)
inherent to the CDA method, is an example of a truncation error, i.e. by truncating
higher order terms in the Taylor expansion the method becomes only approximate.
Another source of error exists when the FDA, CDA or REA are computed; this is the
round-off error due to limited precision in numerical arithmetic (numerical values are
stored in the computer with a limited number of binary bits). Round-off errors are
compounded in arithmetic operations.
The total error is therefore a combination of the two error sources:
Total Error = Truncation Error + Round-off Error
due to limited
precision in numerical
arithmetic
due to truncating higher
order terms in Taylor
expansion
The important parameter here is the value of h;
the truncation error increases with increasing h,
the round-off error decreases with increasing h
Given a particular method, for example the CDA, the most accurate computed
derivative is obtained by minimizing the total error, this corresponds to finding the
optimal value of h.
This optimal value will differ depending on
i.
ii.
iii.
The numerical method (FDA, CDA, REA, etc).
The function being differentiated, and the value of x.
The precision of the arithmetic (single-, double-, quad-precision).
To arrive at the optimal value some study of the output of your program is needed.
The total error in the CDA is given by:
Error = CDA(x) – f`(x)
We can form a plot of |Error| against log(h) to indicate the effect of h on error See Fig 2.1.
A minimum error exists at some intermediate value of h corresponding to a minimum
in the plot. If f`(x) is unknown we can only plot CDA versus h, but, as f'(x) is a
constant the plot will have the same shape (only shift up or down). In this case again
a minimum (or stationary) value in the plot will be observed corresponding to a
minimum error.
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EP208 Computational Methods in Physics – Notes on Lecture 2 Numerical Differentiation
Fig 2.1: Error = CDA(x)- f ’(x) vs h plots for the function f(x) = ex – 2x at x = 3 for
single- and double-precision arithmetic. Optimum values of h corresponding to
minimum error are about 10–2 and 10–5 respectively. The rise on the left (as h
increases) is due to truncation error which has the form of h2 and the rise on
the right (as h decreases) is due to round-off errors.
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EP208 Computational Methods in Physics – Notes on Lecture 2 Numerical Differentiation
Exercise
1. Consider the position of a particle in meters is given by function x(t) = −1/t.
[ v = x’(t) = 1/t2 and a = x’’(t)= -2/t3 ]
1st derivative of the function at t=3 s is f’(3) = 0.111 111 111 111 111 … m/s
2nd derivative of the function at t=3 is f’(3) = -0.074 074 074 074 074 … m/s2
(a) For the first derivative, compare the accuracy of FDA, CDA and REA.
For this, use h = 0.01 s, and double precision data type.
(b) For the second derivative, compare the accuracy of FDA, CDA and REA.
For this, use h = 0.01 s, and double precision data type.
2. Write a program to find the partial derivatives ∂f/∂x and ∂f/∂y at x = y = 1 for the
function f(x,y) = xy + sin(x)/y.
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