SLALN2 - solve a system of the form (ca A - w D ) X = s B
or (ca A' - w D) X = s B with possible scaling ("s") and
perturbation of A
SYNOPSIS
SUBROUTINE SLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1,
D2, B, LDB, WR, WI, X, LDX, SCALE,
XNORM, INFO )
LOGICAL LTRANS
INTEGER INFO, LDA, LDB, LDX, NA, NW
REAL CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
REAL A( LDA, * ), B( LDB, * ), X( LDX, * )
PURPOSE
SLALN2 solves a system of the form (ca A - w D ) X = s B
or (ca A' - w D) X = s B with possible scaling ("s") and
perturbation of A. (A' means A-transpose.) A is an NA x
NA real matrix, ca is a real scalar, D is an NA x NA real
diagonal matrix, w is a real or complex value, and X and B
are NA x 1 matrices -- real if w is real, complex if w is
complex. NA may be 1 or 2.
If w is complex, X and B are represented as NA x 2 matri
ces, the first column of each being the real part and the
second being the imaginary part.
"s" is a scaling factor (.LE. 1), computed by SLALN2,
which is so chosen that X can be computed without over
flow. X is further scaled if necessary to assure that
norm(ca A - w D)*norm(X) is less than overflow.
If both singular values of (ca A - w D) are less than
SMIN, SMIN*identity will be used instead of (ca A - w D).
If only one singular value is less than SMIN, one element
of (ca A - w D) will be perturbed enough to make the
smallest singular value roughly SMIN. If both singular
values are at least SMIN, (ca A - w D) will not be per
turbed. In any case, the perturbation will be at most
some small multiple of max( SMIN, ulp*norm(ca A - w D) ).
The singular values are computed by infinity-norm approxi
mations, and thus will only be correct to a factor of 2 or
so.
Note: all input quantities are assumed to be smaller than
overflow by a reasonable factor. (See BIGNUM.)
ARGUMENTS
=.TRUE.: A-transpose will be used.
=.FALSE.: A will be used (not transposed.)
NA (input) INTEGER
The size of the matrix A. It may (only) be 1 or
2.
NW (input) INTEGER
1 if "w" is real, 2 if "w" is complex. It may
only be 1 or 2.
SMIN (input) REAL
The desired lower bound on the singular values of
A. This should be a safe distance away from
underflow or overflow, say, between (under
flow/machine precision) and (machine precision *
overflow ). (See BIGNUM and ULP.)
CA (input) REAL
The coefficient c, which A is multiplied by.
A (input) REAL array, dimension (LDA,NA)
The NA x NA matrix A.
LDA (input) INTEGER
The leading dimension of A. It must be at least
NA.
D1 (input) REAL
The 1,1 element in the diagonal matrix D.
D2 (input) REAL
The 2,2 element in the diagonal matrix D. Not
used if NW=1.
B (input) REAL array, dimension (LDB,NW)
The NA x NW matrix B (right-hand side). If NW=2
("w" is complex), column 1 contains the real part
of B and column 2 contains the imaginary part.
LDB (input) INTEGER
The leading dimension of B. It must be at least
NA.
WR (input) REAL
The real part of the scalar "w".
WI (input) REAL
The imaginary part of the scalar "w". Not used if
NW=1.
X (output) REAL array, dimension (LDX,NW)
The NA x NW matrix X (unknowns), as computed by
1 will contain the real part of X and column 2
will contain the imaginary part.
LDX (input) INTEGER
The leading dimension of X. It must be at least
NA.
SCALE (output) REAL
The scale factor that B must be multiplied by to
insure that overflow does not occur when computing
X. Thus, (ca A - w D) X will be SCALE*B, not B
(ignoring perturbations of A.) It will be at most
1.
XNORM (output) REAL
The infinity-norm of X, when X is regarded as an
NA x NW real matrix.
INFO (output) INTEGER
An error flag. It will be set to zero if no error
occurs, a negative number if an argument is in
error, or a positive number if ca A - w D had to
be perturbed. The possible values are:
= 0: No error occurred, and (ca A - w D) did not
have to be perturbed. = 1: (ca A - w D) had to be
perturbed to make its smallest (or only) singular
value greater than SMIN. NOTE: In the interests
of speed, this routine does not check the inputs
for errors.
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