DM32 Simultaneous Equations

** From the Owners Manual **

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

Local copy simeq.d32

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

''

''

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

They are entered sequentially in the order

ABCDEFGHIJKL

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 you can
  • find the determinant of the coefficient matrix with prog D
    • this is not required for a simple solution, but might be useful for other purposes?
  • Invert 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

A system of equations with 3 unknowns x, y and z

23x + 15y + 17z = 31

'' 8x + 11y - 6z = 17''

'' 4x + 15y + 12z = 14''

• 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

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

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

Page updated : 19/04/26 21:07 BST