Swiss Micros DM32: Solving Integral Equations
Introduction
The programs presented today solves the following equation for X:
X
∫ F(T) dT = C
0
Since we are not able to use an integration command in solving a program on the DM32 (and the HP 32S and 32SII), we will have to use a manual method, mainly Newton's Method:
X_n+1 = X_n - [ ∫( F(T) dT from T = 0 to T = x_n) - C ] ÷ f(x_n)
with in tolerance D.
To see a similar program for the HP Prime, posted on April 3, 2020, please click here:
https://edspi31415.blogspot.com/2020/04/hp-prime-solving-integral-equations.html
DM32 Code: Integral Equations
LBL H: Help File
LBL H
SF 10
EQN: L B L _ F - F ( T )
EQN: L B L _ S - S O L V E
EQN: S _ H A S _ R C L _ T
EQN: C = C O N S T
EQN: D = T O L E R
EQN: G U E S S _ X E Q _ S
CF 10
RTN
Note: Underscore is the space key. Press R/S after each message.
LBL F
RCL T
enter f(T), the integrand, here
RTN
Note: The variable used is T. If you want to test out the function, store a value in the variable T first.
LBL S
RAD
STO X
LBL A
FN= F
0
RCL X
∫ FN d T
RCL- C
STO Y
RCL X
STO T
XEQ F
STO÷ Y
RCL X
RCL- Y
STO Y
RCL Y
RCL- X
ABS
RCL D
x<y?
GTO B
RCL Y
RTN
LBL B
RCL Y
STO X
GTO A
Note: This is the main program. Enter a guess, and then key in XEQ S.
Variables Used:
T = independent variable
C = constant
D = tolerance (i.e. 10^-4, 10^-5, 10^-6, etc)
X = x_n
Y = x_n+1, final approximation
Download the state file here: ntegralequ.dm32
The state file includes a sample integrand:
e^(-T ÷ 4):
LBL F
RCL T
x^2
+/-
4
÷
e^x
RTN
Examples
In the following examples, the tolerance is 10^-5 (5 +/- 10^x STO D) and FIX 5 mode is set.
1. ∫( e^(-T^2 ÷ 4) dT for T = 0 to X) = 1
C = 1
Guess = 2, Result: 1.10208
Guess = 1, Result: 1.10208
Guess = 3, Result: Division by 0 error
Note that initial guesses are important.
2. ∫( e^-sin(T + 1) dT for T = 0 to X) = 10
C = 10
LBL F
RCL T
1
+
SIN
+/-
e^x
RTN
Guess = 5; Result: 9.10014
3. ∫( T^3 - 2 × T dT for T = 0 to X) = 40
C = 40
LBL F
RCL T
3
y^x
RCL T
2
×
-
RTN
Guess = 5; Result: 3.84789
4. ∫( sin^2 T dT for T = 0 to X) = 1.4897
C = 1.4897
LBL F
RCL T
SIN
x^2
RTN
Guess = 2; Result: 2.49991
Eddie
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