Page 122 - Compact Numerical Methods For Computers

The algebraic eigenvalue problem 111
will be completely unrecognisable after multiplication by
if
c = re = r cos f + ir sin f (9.29)
that is, we obtain
x' + iy' = (xr cos f – yr sin f) + i(xr sin f + yr cos f). (9.30)

Therefore, it is useful to standardise every computed eigenvector so that the
largest. component is
1 + i0. (9.31)
Furthermore it is worthwhile to compute residuals for each vector. While this may
be a trivial programming tack for those familiar with complex numbers and linear
algebra, algorithms for both standardisation and residual calculation follow as an
aid to those less familiar with the eigenproblem of general square matrices.
We now present three algorithms which are intended to be used together:

algorithm 11, to standardise a complex (eigen)vector;
algorithm 12, to compute residuals for a purported complex eigenvector;
and
algorithm 26, Eberlein’s Jacobi-like method for eigensolutions of a complex
square matrix.

The driver program DR26.PAS on the software diskette is designed to use these three
algorithms in computing and presenting eigensolutions of a general square matrix,
that is, a square matrix which may have real or complex elements and need not have
any symmetries in its elements.

Algorithm 11. Standardisation of a complex vector

procedure stdceigv(n: integer; {number of elements in vector}
var T, U: x-matrix); {eigenvector k is given
(column k of T) + sqrt(-1) * (column k of U)
= T[.,k] + sqrt(-1) * U[.,k] }
{algll.pas == Standardisation of complex eigensolutions.
Copyright 1988 J.C.Nash
}
var
i, k, m : integer;
b, e, g, s : real;
begin
writeln(‘algll.pas -- standardized eigensolutions’);
for i := 1 to n do {loop over the eigensolutions}
begin {the standardization of the solution so that the largest
element of each solution is set to 1 + 0 sqrt(-1)}
g := T[1,i]*T[1,i]+U[1,i]*U[1,i]; {STEP 1}
{the magnitude of element 1 of eigensolution i}
k := 1; {to set index for the largest element so far}
if n>1 then
begin {STEP 2}
for m := 2 to n do {loop over the other elements of the solution}