HP 71B: Simpson’s Rule Approximation for f(x,y)

HP 71B: Simpson’s Rule Approximation for f(x,y)

Happy New Year everyone!  Hope everyone had a great New Years Day!  J

For the TI-84 Plus version, please visit here:  http://edspi31415.blogspot.com/2016/02/ti-84-plus-simpsons-rule-for-fxy.html

Today, I will present apply the Simpson’s Rule to functions of two variables like f(x,y) for the HP 71B. Here is the general algorithm:

Start with the function f(x,y) and your integration limits A, B, C, and D.

Then determine Δx and Δy (labeled E and F in the programs below) by:

Δx = (B – A)/(N – 1)

Δy = (D – C)/(N – 1)

Where N is the number of partitions.  Unlike the Simpson’s Rule for one variable, in this case N must be odd.  Generally, the higher N is, the more accurate the approximation is, with the expense of additional computational time.

Next, build a matrix, let’s say [ I ].   This is your Simpson’s Matrix.  The Simpson’s Matrix is built by the expression

[ I ] = [1, 4, 2, 4, 2, 4, 2, 4, 2, …, 4, 1]^T * [1, 4, 2, 4, 2, 4, 2, 4, 2, …, 4, 1]

The length of the vector used to determine [ I ] depends on N.  A way to build it by the routine:

Store 1 into the element 1 of [ I ]  (first element)

Store 1 into the element N of [ I ]  (last element)

For J from 2 to N – 1

If J is divisible by 2, then store 4 in the jth element of [ I ],

Else store 2 in the jth element of [ I ]

For N = 5, the vector would be built is [1, 4, 2, 4, 1]

And

[ I ] = [1,4,2,4,1]^T * [1,4,2,4,1] =

[[1, 4, 2, 4, 1]

[4, 16, 8, 16, 4]

[2, 8, 4, 8, 2]

[4, 16, 8, 16, 4]

[1, 4, 2, 4, 1]]

Build another matrix [ J ].  The elements are determined by the following formulas:

For row j and column k, the element is f(A + Δx*(j – 1),  C + Δy*(k – 1))

Once finished, multiply every element of [ I ] by [ J ].  This is NOT matrix multiplication.   Then sum all of the elements of the results.  In essence:

S = ∑ (j = 1 to N) ∑ (k = 1 to N)   [I](j,k)* [J](j,k)

Determine the final integral approximation as:

Integral = Δx * Δy * 1/9 * S

The program DBLSIMP uses N = 5.  This program works best for f(x,y) where they are polynomials.  On the HP 71B, matrices cannot be typed directly, elements have to be stored and recalled on element at a time.  The program presented does not use modules. 

HP 71B Program DBLSIMP

At least 580 Bytes

Edit f(x,y) at line 10. Use variables X and Y.

5 DESTROY I,J,A,B,C,D
8 DESTROY E,F,K,S,X,Y

10 DEF FNF(X,Y)= [ enter f(X,Y) here ]

12 RADIANS

14 DIM I(5,5)

20 I(1,1) = 1

21 I(1,2) = 4

22 I(1,3) = 2

23 I(1,4) = 4

24 I(1,5) = 1

25 I(2,1) = 4

26 I(2,2) = 16

27 I(2,3) = 8

28 I(2,4) = 16

29 I(2,5) = 4

30 I(3,1) = 2

31 I(3,2) = 8

32 I(3,3) = 4

33 I(3,4) = 8

34 I(3,5) = 2

35 I(4,1) = 4

36 I(4,2) = 16

37 I(4,3) = 8

38 I(4,4) = 16

39 I(4,5) = 4

40 I(5,1) = 1

41 I(5,2) = 4

42 I(5,3) = 2

43 I(5,4) = 4

44 I(5,5) = 1

50 DISP “X: from a to b” @ WAIT 1

52 INPUT “a = “; A

54 INPUT “b = “; B

56 DISP “Y: from c to d” @ WAIT 1

58 INPUT “c = “; C

60 INPUT “d = “; D

62 E = .25 * (B – A)

64 F = .25 * (D – C)

66 S = 0

70 FOR J = 1 TO 5

72 FOR K = 1 TO 5

74 X = A + E * (K – 1)

76 Y = C + F * (J – 1)

78 S = S + FNF(X,Y) * I(J,K)

80 NEXT K

82 NEXT J

90 S = S * E * F/9

95 DISP “INTEGRAL = “ @ WAIT 1

97 DISP S

Examples:

F(X,Y) = 2*Y – 3*X

A = 1, B = 2, C = 2, D = 5

Result:  Integral = 7.5

(Actual answer: 7.5)

F(X,Y) = X^2/Y^2

A = 1, B = 2, C = 2, D = 5

Result:  Integral ≈ 0.70212579101

(Actual answer:  0.7)

F(X,Y) = 0.5*X*EXP(Y)

A = 1, B = 2, C = 2, D = 5

Result:  Integral ≈ 105.942243008

(Actual answer is about 105.768077253)

Source:

Cooper, Ian.  “Doing Physics With Matlab:  Mathematical Routines”  School of Physics, University of Sydney

http://www.physics.usyd.edu.au/teach_res/mp/doc/math_integration_2D.pdf

Retrieved January 30, 2016

Thank you, and wishing you a happy, healthy, and successful 2017!

Eddie

This blog is property of Edward Shore, 2017