laln2 - Man Page

laln2: 1x1 or 2x2 solve, step in trevc

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 is INTEGER
          The size of the matrix A.  It may (only) be 1 or 2.

          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 is DOUBLE PRECISION
          The coefficient c, which A is multiplied by.

          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 is DOUBLE PRECISION
          The 1,1 element in the diagonal matrix D.

          D2 is DOUBLE PRECISION
          The 2,2 element in the diagonal matrix D.  Not used if NA=1.

          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 is DOUBLE PRECISION
          The real part of the scalar 'w'.

          WI is DOUBLE PRECISION
          The imaginary part of the scalar 'w'.  Not used if NW=1.

          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.

Definition at line 216 of file dlaln2.f.

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.