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←Hints & Tips

DM32 Simultaneous Equations

** From the Owners Manual **

Statefile

Provided by Swiss Micros https://technical.swissmicros.com/dm32/examples/STATE/SIMEQ.d32

Local copy simeq.d32

Overview

This set of programs allows the solution of 3×3 and 2×2 systems of equations, as well as the inverse and determinate of the matrix representing the coefficients.

The set of equations is represented by a 3 x 4 array of numbers, with the first 3 columns representing the coefficients of the equations and the 4th column the “value” of the equations.

For example

'' Ax + Dy + Gz = J ''

'' Bx + Ey + Hz = K ''

'' Cx + Fy + Iz = L ''

These equations can be represented in matrix form

''

A D G x J
B E H y = K
C F I z L

''

This is effectively

'' A x X = B ''

and therefore

'' X = B / A ''

or

''X = B x A-1 ''

The unknown matrix is obtained by multiplying B by the inverse of A

Use

Enter the values of the 3×3 matrix and the 3×1 column vector by using prog. A

  • XEQ A
  • Enter each value at the prompt followed by R/S
  • Once all values A → L are entered find the inverse of the coefficient matrix A with prog. I
    • XEQ I
  • Multiply the column vector by this inverse matrix using prog. M
    • XEQ M
  • This gives the resulting X, Y and Z by pressing R/S to step through them

Example

23x + 15y + 17z = 31

8x + 11y - 6z = 17

4x + 15y + 12z = 14

Beware!

You enter each value in sequence going down the columns in sequence

Which is the opposite sense to the way matrices are entered in DM15L and DM41X

  • XEQ A
    • 23 R/S
    • 8 R/S
    • 4 R/S
    • 15 R/S
    • 11 R/S
    • 15 R/S
    • 17 R/S
    • 6 +/- R/S
    • 12 R/S
    • 31 R/S
    • 17 R/S
    • 14 R/S
  • Find the inverse XEQ I
  • Multiply XEQ M
  • Display shows solutions
    • X = 0.930622 R/S
    • Y = 0.794258 R/S
    • Z = -0.136364
  • You can inspect the inverted matrix by running A again
    • XEQ A
    • A ? 0.048 R/S
    • B ? -0.026 R/S
    • etc….etc…
  • You can re-invert A back to the original state by running prog I again
  • You can see the determinant of A by running prog D

2x2

The programs expect a 3×3 matrix. If the system is 2×2 you have to use dummy values to pad out to 3×3

Set C, F, H, G, L = 0 and I = 1

e.g.

'' 2x + 3y = 6 ''

'' 8x - 4y = 9 ''

This becomes

'' 2x + 3y + 0z = 6 ''

'' 8x - 4y + 0z = 9 ''

'' 0x + 0y + 1z = 0 ''

and is entered

  • XEQ A
    • 2 R/S
    • 8 R/S
    • 0 R/S
    • 3 R/S
    • 4 +/- R/S
    • 0 R/S
    • 0 R/S
    • 0 R/S
    • 1 R/S
    • 6 R/S
    • 9 R/S
    • 0 R/S
  • Find the inverse XEQ I
  • Multiply XEQ M to find the unknowns
    • X = 1.59375 R/S
    • Y = 0.93750 R/S
    • Z = 0.00000 C

Further Information

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Page created : 17/04/26 18:10 BST

Page updated : 17/04/26 19:03 BST