slaln2()

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.

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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 () 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.
!>
!>  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

[in]	LTRANS	!> LTRANS is LOGICAL !> =.TRUE.: A-transpose will be used. !> =.FALSE.: A will be used (not transposed.) !>
[in]	NA	!> NA is INTEGER !> The size of the matrix A. It may (only) be 1 or 2. !>
[in]	NW	!> NW is INTEGER !> 1 if is real, 2 if is complex. It may only be 1 !> or 2. !>
[in]	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.) !>
[in]	CA	!> CA is REAL !> The coefficient c, which A is multiplied by. !>
[in]	A	!> A is REAL array, dimension (LDA,NA) !> The NA x NA matrix A. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of A. It must be at least NA. !>
[in]	D1	!> D1 is REAL !> The 1,1 element in the diagonal matrix D. !>
[in]	D2	!> D2 is REAL !> The 2,2 element in the diagonal matrix D. Not used if NA=1. !>
[in]	B	!> B is REAL array, dimension (LDB,NW) !> The NA x NW matrix B (right-hand side). If NW=2 ( is !> complex), column 1 contains the real part of B and column 2 !> contains the imaginary part. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of B. It must be at least NA. !>
[in]	WR	!> WR is REAL !> The real part of the scalar . !>
[in]	WI	!> WI is REAL !> The imaginary part of the scalar . Not used if NW=1. !>
[out]	X	!> X is REAL array, dimension (LDX,NW) !> The NA x NW matrix X (unknowns), as computed by SLALN2. !> If NW=2 ( is complex), on exit, column 1 will contain !> the real part of X and column 2 will contain the imaginary !> part. !>
[in]	LDX	!> LDX is INTEGER !> The leading dimension of X. It must be at least NA. !>
[out]	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. !>
[out]	XNORM	!> XNORM is REAL !> The infinity-norm of X, when X is regarded as an NA x NW !> real matrix. !>
[out]	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.