SLALN2(3) LAPACK routine of NEC Numeric Library Collection SLALN2(3)
NAME
SLALN2 - 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 )
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 com-
plex. NA may be 1 or 2.
If w is complex, X and B are represented as NA x 2 matrices, 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 overflow. 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 perturbed. 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
approximations, and thus will only be correct to a factor of 2
or so.
Note: all input quantities are assumed to be smaller than over-
flow by a reasonable factor. (See BIGNUM.)
ARGUMENTS
LTRANS (input) LOGICAL
=.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 (underflow/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 com-
plex), column 1 contains the real part of B and column 2 con-
tains 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 SLALN2. If
NW=2 ("w" is complex), on exit, column 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 num-
ber 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.
LAPACK routine 31 October 2017 SLALN2(3)