The function DET returns the determinant of a square matrix, meaning that the array (or the part passed) must have two dimensions of equal size, otherwise DET breaks with error -15 (bad parameter).
The array needs to be a floating point array, any other type (including integer arrays) will also produce error -15.
If no parameter is given, DET will use the array that has been supplied to the previously executed MATINV command as its source. If however, this command has not yet been used, DET without a parameter will stop with the error -7 (not found).
You may ask what a determinant is? Briefly speaking, it represents a square matrix by a single number so that important characteristics of the matrix can be deduced from it, eg. the matrix cannot be inverted if the determinant is zero.
We will try to approach the eigenvalues of a matrix and list them all (the so-called “spectrum” of a matrix). Due to approximation errors and the simple algorithm employed, there can be more output values than there should be. This can be improved by increasing estep in line 130, but at the cost of speed.
The range of expected eigenvalues (eval1 to eval2) is adapted to the chosen matrix whose random elements only range between 0 and 1. There is no limit for the positive size n of the matrix, n=0 is allowed but does not make sense because CHARPOLY becomes constant:
100 CLEAR: RANDOMISE 10: PRINT "Eigenvalues:" 110 n=2: DIM matrix(n,n), one(n,n) 120 MATRND matrix: MATIDN one 130 : 140 eval1=-1: eval2=1: esteps=200 150 eprec<(eval2-eval1)/estep) 160 c1=CHARPOLY(matrix,eval1): count%=0 170 FOR eval=eval1+eprec TO eval2 STEP eprec 180 c2=CHARPOLY(matrix,eval) 190 IF SGN(c1)<>SGN(c2) THEN PRINT eval 200 c1=c2: count%=count%+1 210 AT#0,0,0: PRINT#0,INT(100*count%/esteps);"%" 220 END FOR eval 230 PRINT "absolute fault:"!eprec 240 : 250 DEFine FuNction CHARPOLY(matrix,lambda) 260 LOCal diff(n,n),i 270 FOR i=1 TO n: one(i,i)=lambda 280 MATSUB diff,matrix,one 290 RETurn DET(diff) 300 END DEFine CHARPOLY
In practice, a Newton iteration algorithm (or better) would be used.
MATINV co-operates closely with DET, so that for each of them a matrix parameter can be omitted if the other function has been called before; MATINV calls DET internally. In the example, we used the MATRND, MATIDN, SGN and MATSUB keywords which are all part of the same Toolkit.