<-[[.:start]] ====== DM32 Simultaneous Equations ====== ** From the Owners Manual ** ===== Statefile ===== Provided by Swiss Micros [[https://technical.swissmicros.com/dm32/examples/STATE/SIMEQ.d32]] Local copy {{ :public:calculator:guides:simeq.d32 |}} ===== Overview ===== This set of programs allows the solution of 3x3 and 2x2 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'' They are entered sequentially in the order ''ABCDEFGHIJKL'' ===== Use ===== Enter the values of the 3x3 matrix and the 3x1 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? * **I**nvert the coefficient matrix A with prog. ''I'' * XEQ 'I' * **M**ultiply 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 ===== Examples ===== ==== 3 x 3 ==== A system of equations with 3 unknowns ''x'', ''y'' and ''z'' ''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 3x3 matrix. If the system is 2x2 you have to use dummy values to pad out to 3x3 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 ===== {{tag>calculator dm32}} Page created : 17/04/26 18:10 BST Page updated : ~~LASTMOD~~