Provided by: liblapack-doc_3.12.0-3build1.1_all 
NAME
laln2 - laln2: 1x1 or 2x2 solve, step in trevc
SYNOPSIS
Functions
subroutine dlaln2 (ltrans, na, nw, smin, ca, a, lda, d1, d2, b, ldb, wr, wi, x, ldx,
scale, xnorm, info)
DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
subroutine slaln2 (ltrans, na, nw, smin, ca, a, lda, d1, d2, b, ldb, wr, wi, x, ldx,
scale, xnorm, info)
SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
Detailed Description
Function Documentation
subroutine dlaln2 (logical ltrans, integer na, integer nw, double precision smin, double
precision ca, double precision, dimension( lda, * ) a, integer lda, double precision d1,
double precision d2, double precision, dimension( ldb, * ) b, integer ldb, double
precision wr, double precision wi, double precision, dimension( ldx, * ) x, integer ldx,
double precision scale, double precision xnorm, integer info)
DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
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 (<= 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.)
Parameters
LTRANS
LTRANS is LOGICAL
=.TRUE.: A-transpose will be used.
=.FALSE.: A will be used (not transposed.)
NA
NA is INTEGER
The size of the matrix A. It may (only) be 1 or 2.
NW
NW is INTEGER
1 if 'w' is real, 2 if 'w' is complex. It may only be 1
or 2.
SMIN
SMIN is 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
CA is DOUBLE PRECISION
The coefficient c, which A is multiplied by.
A
A is DOUBLE PRECISION array, dimension (LDA,NA)
The NA x NA matrix A.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least NA.
D1
D1 is DOUBLE PRECISION
The 1,1 element in the diagonal matrix D.
D2
D2 is DOUBLE PRECISION
The 2,2 element in the diagonal matrix D. Not used if NA=1.
B
B is 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
LDB is INTEGER
The leading dimension of B. It must be at least NA.
WR
WR is DOUBLE PRECISION
The real part of the scalar 'w'.
WI
WI is DOUBLE PRECISION
The imaginary part of the scalar 'w'. Not used if NW=1.
X
X is 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
LDX is INTEGER
The leading dimension of X. It must be at least NA.
SCALE
SCALE is 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
XNORM is DOUBLE PRECISION
The infinity-norm of X, when X is regarded as an NA x NW
real matrix.
INFO
INFO is 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine slaln2 (logical ltrans, integer na, integer nw, real smin, real ca, real,
dimension( lda, * ) a, integer lda, real d1, real d2, real, dimension( ldb, * ) b, integer
ldb, real wr, real wi, real, dimension( ldx, * ) x, integer ldx, real scale, real xnorm,
integer info)
SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
Purpose:
SLALN2 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 (<= 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 overflow
by a reasonable factor. (See BIGNUM.)
Parameters
LTRANS
LTRANS is LOGICAL
=.TRUE.: A-transpose will be used.
=.FALSE.: A will be used (not transposed.)
NA
NA is INTEGER
The size of the matrix A. It may (only) be 1 or 2.
NW
NW is INTEGER
1 if 'w' is real, 2 if 'w' is complex. It may only be 1
or 2.
SMIN
SMIN is 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
CA is REAL
The coefficient c, which A is multiplied by.
A
A is REAL array, dimension (LDA,NA)
The NA x NA matrix A.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least NA.
D1
D1 is REAL
The 1,1 element in the diagonal matrix D.
D2
D2 is REAL
The 2,2 element in the diagonal matrix D. Not used if NA=1.
B
B is 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
LDB is INTEGER
The leading dimension of B. It must be at least NA.
WR
WR is REAL
The real part of the scalar 'w'.
WI
WI is REAL
The imaginary part of the scalar 'w'. Not used if NW=1.
X
X is 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
LDX is INTEGER
The leading dimension of X. It must be at least NA.
SCALE
SCALE is 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
XNORM is REAL
The infinity-norm of X, when X is regarded as an NA x NW
real matrix.
INFO
INFO is 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Author
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