laln2(3)

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.

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.

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.

Definition at line 216 of file slaln2.f.

Author

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