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 () 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 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  is real, 2 if  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 ( 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 .
    !>

    WI

    !>          WI is DOUBLE PRECISION
    !>          The imaginary part of the scalar .  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 ( 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 () 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
    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  is real, 2 if  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 ( 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 .
    !>

    WI

    !>          WI is REAL
    !>          The imaginary part of the scalar .  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 ( 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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