latrs(3)

NAME
latrs - latrs: triangular solve with robust scaling

SYNOPSIS

  Functions
subroutine clatrs (uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
subroutine dlatrs (uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
DLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
subroutine slatrs (uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
SLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
subroutine zlatrs (uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
ZLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Detailed Description

Function Documentation

  subroutine clatrs (character uplo, character trans, character diag, character normin, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)

CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Purpose:

     CLATRS solves one of the triangular systems
        A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,
     with scaling to prevent overflow.  Here A is an upper or lower
     triangular matrix, A**T denotes the transpose of A, A**H denotes the
     conjugate transpose of A, x and b are n-element vectors, and s is a
     scaling factor, usually less than or equal to 1, chosen so that the
     components of x will be less than the overflow threshold.  If the
     unscaled problem will not cause overflow, the Level 2 BLAS routine
     CTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
     then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    TRANS

              TRANS is CHARACTER*1
              Specifies the operation applied to A.
              = 'N':  Solve A * x = s*b     (No transpose)
              = 'T':  Solve A**T * x = s*b  (Transpose)
              = 'C':  Solve A**H * x = s*b  (Conjugate transpose)

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    NORMIN

              NORMIN is CHARACTER*1
              Specifies whether CNORM has been set or not.
              = 'Y':  CNORM contains the column norms on entry
              = 'N':  CNORM is not set on entry.  On exit, the norms will
                      be computed and stored in CNORM.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is COMPLEX array, dimension (LDA,N)
              The triangular matrix A.  If UPLO = 'U', the leading n by n
              upper triangular part of the array A contains the upper
              triangular matrix, and the strictly lower triangular part of
              A is not referenced.  If UPLO = 'L', the leading n by n lower
              triangular part of the array A contains the lower triangular
              matrix, and the strictly upper triangular part of A is not
              referenced.  If DIAG = 'U', the diagonal elements of A are
              also not referenced and are assumed to be 1.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max (1,N).

    X

              X is COMPLEX array, dimension (N)
              On entry, the right hand side b of the triangular system.
              On exit, X is overwritten by the solution vector x.

    SCALE

              SCALE is REAL
              The scaling factor s for the triangular system
                 A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
              If SCALE = 0, the matrix A is singular or badly scaled, and
              the vector x is an exact or approximate solution to A*x = 0.

    CNORM

              CNORM is REAL array, dimension (N)
              If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
              contains the norm of the off-diagonal part of the j-th column
              of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
              to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
              must be greater than or equal to the 1-norm.
              If NORMIN = 'N', CNORM is an output argument and CNORM(j)
              returns the 1-norm of the offdiagonal part of the j-th column
              of A.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      A rough bound on x is computed; if that is less than overflow, CTRSV
      is called, otherwise, specific code is used which checks for possible
      overflow or divide-by-zero at every operation.
      A columnwise scheme is used for solving A*x = b.  The basic algorithm
      if A is lower triangular is
           x[1:n] := b[1:n]
           for j = 1, ..., n
                x(j) := x(j) / A(j,j)
                x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
           end
      Define bounds on the components of x after j iterations of the loop:
         M(j) = bound on x[1:j]
         G(j) = bound on x[j+1:n]
      Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
      Then for iteration j+1 we have
         M(j+1) <= G(j) / | A(j+1,j+1) |
         G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
      where CNORM(j+1) is greater than or equal to the infinity-norm of
      column j+1 of A, not counting the diagonal.  Hence
         G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                      1<=i<=j
      and
         |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                       1<=i< j
      Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTRSV if the
      reciprocal of the largest M(j), j=1,..,n, is larger than
      max(underflow, 1/overflow).
      The bound on x(j) is also used to determine when a step in the
      columnwise method can be performed without fear of overflow.  If
      the computed bound is greater than a large constant, x is scaled to
      prevent overflow, but if the bound overflows, x is set to 0, x(j) to
      1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
      Similarly, a row-wise scheme is used to solve A**T *x = b  or
      A**H *x = b.  The basic algorithm for A upper triangular is
           for j = 1, ..., n
                x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
           end
      We simultaneously compute two bounds
           G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
           M(j) = bound on x(i), 1<=i<=j
      The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
      add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
      Then the bound on x(j) is
           M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
                <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                          1<=i<=j
      and we can safely call CTRSV if 1/M(n) and 1/G(n) are both greater
      than max(underflow, 1/overflow).

Definition at line 237 of file clatrs.f.

  subroutine dlatrs (character uplo, character trans, character diag, character normin, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) x, double precision scale, double precision, dimension( * ) cnorm, integer info)

DLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Purpose:

     DLATRS solves one of the triangular systems
        A *x = s*b  or  A**T *x = s*b
     with scaling to prevent overflow.  Here A is an upper or lower
     triangular matrix, A**T denotes the transpose of A, x and b are
     n-element vectors, and s is a scaling factor, usually less than
     or equal to 1, chosen so that the components of x will be less than
     the overflow threshold.  If the unscaled problem will not cause
     overflow, the Level 2 BLAS routine DTRSV is called.  If the matrix A
     is singular (A(j,j) = 0 for some j), then s is set to 0 and a
     non-trivial solution to A*x = 0 is returned.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    TRANS

              TRANS is CHARACTER*1
              Specifies the operation applied to A.
              = 'N':  Solve A * x = s*b  (No transpose)
              = 'T':  Solve A**T* x = s*b  (Transpose)
              = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    NORMIN

              NORMIN is CHARACTER*1
              Specifies whether CNORM has been set or not.
              = 'Y':  CNORM contains the column norms on entry
              = 'N':  CNORM is not set on entry.  On exit, the norms will
                      be computed and stored in CNORM.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              The triangular matrix A.  If UPLO = 'U', the leading n by n
              upper triangular part of the array A contains the upper
              triangular matrix, and the strictly lower triangular part of
              A is not referenced.  If UPLO = 'L', the leading n by n lower
              triangular part of the array A contains the lower triangular
              matrix, and the strictly upper triangular part of A is not
              referenced.  If DIAG = 'U', the diagonal elements of A are
              also not referenced and are assumed to be 1.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max (1,N).

    X

              X is DOUBLE PRECISION array, dimension (N)
              On entry, the right hand side b of the triangular system.
              On exit, X is overwritten by the solution vector x.

    SCALE

              SCALE is DOUBLE PRECISION
              The scaling factor s for the triangular system
                 A * x = s*b  or  A**T* x = s*b.
              If SCALE = 0, the matrix A is singular or badly scaled, and
              the vector x is an exact or approximate solution to A*x = 0.

    CNORM

              CNORM is DOUBLE PRECISION array, dimension (N)
              If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
              contains the norm of the off-diagonal part of the j-th column
              of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
              to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
              must be greater than or equal to the 1-norm.
              If NORMIN = 'N', CNORM is an output argument and CNORM(j)
              returns the 1-norm of the offdiagonal part of the j-th column
              of A.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      A rough bound on x is computed; if that is less than overflow, DTRSV
      is called, otherwise, specific code is used which checks for possible
      overflow or divide-by-zero at every operation.
      A columnwise scheme is used for solving A*x = b.  The basic algorithm
      if A is lower triangular is
           x[1:n] := b[1:n]
           for j = 1, ..., n
                x(j) := x(j) / A(j,j)
                x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
           end
      Define bounds on the components of x after j iterations of the loop:
         M(j) = bound on x[1:j]
         G(j) = bound on x[j+1:n]
      Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
      Then for iteration j+1 we have
         M(j+1) <= G(j) / | A(j+1,j+1) |
         G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
      where CNORM(j+1) is greater than or equal to the infinity-norm of
      column j+1 of A, not counting the diagonal.  Hence
         G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                      1<=i<=j
      and
         |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                       1<=i< j
      Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
      reciprocal of the largest M(j), j=1,..,n, is larger than
      max(underflow, 1/overflow).
      The bound on x(j) is also used to determine when a step in the
      columnwise method can be performed without fear of overflow.  If
      the computed bound is greater than a large constant, x is scaled to
      prevent overflow, but if the bound overflows, x is set to 0, x(j) to
      1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
      Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
      algorithm for A upper triangular is
           for j = 1, ..., n
                x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
           end
      We simultaneously compute two bounds
           G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
           M(j) = bound on x(i), 1<=i<=j
      The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
      add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
      Then the bound on x(j) is
           M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
                <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                          1<=i<=j
      and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater
      than max(underflow, 1/overflow).

Definition at line 236 of file dlatrs.f.

  subroutine slatrs (character uplo, character trans, character diag, character normin, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)

SLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Purpose:

     SLATRS solves one of the triangular systems
        A *x = s*b  or  A**T*x = s*b
     with scaling to prevent overflow.  Here A is an upper or lower
     triangular matrix, A**T denotes the transpose of A, x and b are
     n-element vectors, and s is a scaling factor, usually less than
     or equal to 1, chosen so that the components of x will be less than
     the overflow threshold.  If the unscaled problem will not cause
     overflow, the Level 2 BLAS routine STRSV is called.  If the matrix A
     is singular (A(j,j) = 0 for some j), then s is set to 0 and a
     non-trivial solution to A*x = 0 is returned.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    TRANS

              TRANS is CHARACTER*1
              Specifies the operation applied to A.
              = 'N':  Solve A * x = s*b  (No transpose)
              = 'T':  Solve A**T* x = s*b  (Transpose)
              = 'C':  Solve A**T* x = s*b  (Conjugate transpose = Transpose)

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    NORMIN

              NORMIN is CHARACTER*1
              Specifies whether CNORM has been set or not.
              = 'Y':  CNORM contains the column norms on entry
              = 'N':  CNORM is not set on entry.  On exit, the norms will
                      be computed and stored in CNORM.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is REAL array, dimension (LDA,N)
              The triangular matrix A.  If UPLO = 'U', the leading n by n
              upper triangular part of the array A contains the upper
              triangular matrix, and the strictly lower triangular part of
              A is not referenced.  If UPLO = 'L', the leading n by n lower
              triangular part of the array A contains the lower triangular
              matrix, and the strictly upper triangular part of A is not
              referenced.  If DIAG = 'U', the diagonal elements of A are
              also not referenced and are assumed to be 1.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max (1,N).

    X

              X is REAL array, dimension (N)
              On entry, the right hand side b of the triangular system.
              On exit, X is overwritten by the solution vector x.

    SCALE

              SCALE is REAL
              The scaling factor s for the triangular system
                 A * x = s*b  or  A**T* x = s*b.
              If SCALE = 0, the matrix A is singular or badly scaled, and
              the vector x is an exact or approximate solution to A*x = 0.

    CNORM

              CNORM is REAL array, dimension (N)
              If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
              contains the norm of the off-diagonal part of the j-th column
              of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
              to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
              must be greater than or equal to the 1-norm.
              If NORMIN = 'N', CNORM is an output argument and CNORM(j)
              returns the 1-norm of the offdiagonal part of the j-th column
              of A.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      A rough bound on x is computed; if that is less than overflow, STRSV
      is called, otherwise, specific code is used which checks for possible
      overflow or divide-by-zero at every operation.
      A columnwise scheme is used for solving A*x = b.  The basic algorithm
      if A is lower triangular is
           x[1:n] := b[1:n]
           for j = 1, ..., n
                x(j) := x(j) / A(j,j)
                x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
           end
      Define bounds on the components of x after j iterations of the loop:
         M(j) = bound on x[1:j]
         G(j) = bound on x[j+1:n]
      Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
      Then for iteration j+1 we have
         M(j+1) <= G(j) / | A(j+1,j+1) |
         G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
      where CNORM(j+1) is greater than or equal to the infinity-norm of
      column j+1 of A, not counting the diagonal.  Hence
         G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                      1<=i<=j
      and
         |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                       1<=i< j
      Since |x(j)| <= M(j), we use the Level 2 BLAS routine STRSV if the
      reciprocal of the largest M(j), j=1,..,n, is larger than
      max(underflow, 1/overflow).
      The bound on x(j) is also used to determine when a step in the
      columnwise method can be performed without fear of overflow.  If
      the computed bound is greater than a large constant, x is scaled to
      prevent overflow, but if the bound overflows, x is set to 0, x(j) to
      1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
      Similarly, a row-wise scheme is used to solve A**T*x = b.  The basic
      algorithm for A upper triangular is
           for j = 1, ..., n
                x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
           end
      We simultaneously compute two bounds
           G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
           M(j) = bound on x(i), 1<=i<=j
      The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
      add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
      Then the bound on x(j) is
           M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
                <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                          1<=i<=j
      and we can safely call STRSV if 1/M(n) and 1/G(n) are both greater
      than max(underflow, 1/overflow).

Definition at line 236 of file slatrs.f.

  subroutine zlatrs (character uplo, character trans, character diag, character normin, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) x, double precision scale, double precision, dimension( * ) cnorm, integer info)

ZLATRS solves a triangular system of equations with the scale factor set to prevent overflow.

Purpose:

     ZLATRS solves one of the triangular systems
        A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,
     with scaling to prevent overflow.  Here A is an upper or lower
     triangular matrix, A**T denotes the transpose of A, A**H denotes the
     conjugate transpose of A, x and b are n-element vectors, and s is a
     scaling factor, usually less than or equal to 1, chosen so that the
     components of x will be less than the overflow threshold.  If the
     unscaled problem will not cause overflow, the Level 2 BLAS routine
     ZTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
     then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the matrix A is upper or lower triangular.
              = 'U':  Upper triangular
              = 'L':  Lower triangular

    TRANS

              TRANS is CHARACTER*1
              Specifies the operation applied to A.
              = 'N':  Solve A * x = s*b     (No transpose)
              = 'T':  Solve A**T * x = s*b  (Transpose)
              = 'C':  Solve A**H * x = s*b  (Conjugate transpose)

    DIAG

              DIAG is CHARACTER*1
              Specifies whether or not the matrix A is unit triangular.
              = 'N':  Non-unit triangular
              = 'U':  Unit triangular

    NORMIN

              NORMIN is CHARACTER*1
              Specifies whether CNORM has been set or not.
              = 'Y':  CNORM contains the column norms on entry
              = 'N':  CNORM is not set on entry.  On exit, the norms will
                      be computed and stored in CNORM.

    N

              N is INTEGER
              The order of the matrix A.  N >= 0.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              The triangular matrix A.  If UPLO = 'U', the leading n by n
              upper triangular part of the array A contains the upper
              triangular matrix, and the strictly lower triangular part of
              A is not referenced.  If UPLO = 'L', the leading n by n lower
              triangular part of the array A contains the lower triangular
              matrix, and the strictly upper triangular part of A is not
              referenced.  If DIAG = 'U', the diagonal elements of A are
              also not referenced and are assumed to be 1.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max (1,N).

    X

              X is COMPLEX*16 array, dimension (N)
              On entry, the right hand side b of the triangular system.
              On exit, X is overwritten by the solution vector x.

    SCALE

              SCALE is DOUBLE PRECISION
              The scaling factor s for the triangular system
                 A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
              If SCALE = 0, the matrix A is singular or badly scaled, and
              the vector x is an exact or approximate solution to A*x = 0.

    CNORM

              CNORM is DOUBLE PRECISION array, dimension (N)
              If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
              contains the norm of the off-diagonal part of the j-th column
              of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
              to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
              must be greater than or equal to the 1-norm.
              If NORMIN = 'N', CNORM is an output argument and CNORM(j)
              returns the 1-norm of the offdiagonal part of the j-th column
              of A.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -k, the k-th argument had an illegal value

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      A rough bound on x is computed; if that is less than overflow, ZTRSV
      is called, otherwise, specific code is used which checks for possible
      overflow or divide-by-zero at every operation.
      A columnwise scheme is used for solving A*x = b.  The basic algorithm
      if A is lower triangular is
           x[1:n] := b[1:n]
           for j = 1, ..., n
                x(j) := x(j) / A(j,j)
                x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
           end
      Define bounds on the components of x after j iterations of the loop:
         M(j) = bound on x[1:j]
         G(j) = bound on x[j+1:n]
      Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
      Then for iteration j+1 we have
         M(j+1) <= G(j) / | A(j+1,j+1) |
         G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
      where CNORM(j+1) is greater than or equal to the infinity-norm of
      column j+1 of A, not counting the diagonal.  Hence
         G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                      1<=i<=j
      and
         |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                       1<=i< j
      Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTRSV if the
      reciprocal of the largest M(j), j=1,..,n, is larger than
      max(underflow, 1/overflow).
      The bound on x(j) is also used to determine when a step in the
      columnwise method can be performed without fear of overflow.  If
      the computed bound is greater than a large constant, x is scaled to
      prevent overflow, but if the bound overflows, x is set to 0, x(j) to
      1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
      Similarly, a row-wise scheme is used to solve A**T *x = b  or
      A**H *x = b.  The basic algorithm for A upper triangular is
           for j = 1, ..., n
                x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
           end
      We simultaneously compute two bounds
           G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
           M(j) = bound on x(i), 1<=i<=j
      The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
      add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
      Then the bound on x(j) is
           M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
                <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                          1<=i<=j
      and we can safely call ZTRSV if 1/M(n) and 1/G(n) are both greater
      than max(underflow, 1/overflow).

Definition at line 237 of file zlatrs.f.

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