(71B) Simpson’s Rule Approximation for f(x,y)
01-04-2017, 04:26 AM
Post: #1 Eddie W. Shore Senior Member Posts: 1,000 Joined: Dec 2013
(71B) Simpson’s Rule Approximation for f(x,y)
integral( integral( f(x,y) dx from A to B) dy from C to 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.

Code:
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

F(X,Y) = X^2/Y^2
A = 1, B = 2, C = 2, D = 5
Result: Integral ≈ 0.70212579101

F(X,Y) = 0.5*X*EXP(Y)
A = 1, B = 2, C = 2, D = 5
Result: Integral ≈ 105.942243008