DLALN2
November 2006
Purpose
DLALN2 solves a system of the form (ca A - w D ) X = s B
or (ca A**T - w D) X = s B with possible scaling ("s") and
perturbation of A. (A**T 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 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 DLALN2, 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 overflow
by a reasonable factor. (See BIGNUM.)
or (ca A**T - w D) X = s B with possible scaling ("s") and
perturbation of A. (A**T 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 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 DLALN2, 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 overflow
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) DOUBLE PRECISION
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) DOUBLE PRECISION
The coefficient c, which A is multiplied by.
|
| A |
(input) DOUBLE PRECISION 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) DOUBLE PRECISION
The 1,1 element in the diagonal matrix D.
|
| D2 |
(input) DOUBLE PRECISION
The 2,2 element in the diagonal matrix D. Not used if NW=1.
|
| B |
(input) DOUBLE PRECISION 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) DOUBLE PRECISION
The real part of the scalar "w".
|
| WI |
(input) DOUBLE PRECISION
The imaginary part of the scalar "w". Not used if NW=1.
|
| X |
(output) DOUBLE PRECISION array, dimension (LDX,NW)
The NA x NW matrix X (unknowns), as computed by DLALN2.
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) DOUBLE PRECISION
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) DOUBLE PRECISION
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. |