latbs - Man Page

latbs: triangular solve with scaling

Synopsis

Functions

subroutine clatbs (uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm, info)
CLATBS solves a triangular banded system of equations.
subroutine dlatbs (uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm, info)
DLATBS solves a triangular banded system of equations.
subroutine slatbs (uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm, info)
SLATBS solves a triangular banded system of equations.
subroutine zlatbs (uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm, info)
ZLATBS solves a triangular banded system of equations.

Detailed Description

Function Documentation

subroutine clatbs (character uplo, character trans, character diag, character normin, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)

CLATBS solves a triangular banded system of equations.  

Purpose:

 CLATBS 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, where A is an upper or lower
 triangular band matrix.  Here 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 CTBSV 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.

KD

          KD is INTEGER
          The number of subdiagonals or superdiagonals in the
          triangular matrix A.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first KD+1 rows of the array. The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

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, CTBSV
  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 CTBSV 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 CTBSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

Definition at line 241 of file clatbs.f.

subroutine dlatbs (character uplo, character trans, character diag, character normin, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) x, double precision scale, double precision, dimension( * ) cnorm, integer info)

DLATBS solves a triangular banded system of equations.  

Purpose:

 DLATBS solves one of the triangular systems

    A *x = s*b  or  A**T*x = s*b

 with scaling to prevent overflow, where A is an upper or lower
 triangular band matrix.  Here 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 DTBSV 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.

KD

          KD is INTEGER
          The number of subdiagonals or superdiagonals in the
          triangular matrix A.  KD >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first KD+1 rows of the array. The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

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, DTBSV
  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 DTBSV 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 DTBSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

Definition at line 240 of file dlatbs.f.

subroutine slatbs (character uplo, character trans, character diag, character normin, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)

SLATBS solves a triangular banded system of equations.  

Purpose:

 SLATBS solves one of the triangular systems

    A *x = s*b  or  A**T*x = s*b

 with scaling to prevent overflow, where A is an upper or lower
 triangular band matrix.  Here 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 STBSV 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.

KD

          KD is INTEGER
          The number of subdiagonals or superdiagonals in the
          triangular matrix A.  KD >= 0.

AB

          AB is REAL array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first KD+1 rows of the array. The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

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, STBSV
  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 STBSV 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 STBSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

Definition at line 240 of file slatbs.f.

subroutine zlatbs (character uplo, character trans, character diag, character normin, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( * ) x, double precision scale, double precision, dimension( * ) cnorm, integer info)

ZLATBS solves a triangular banded system of equations.  

Purpose:

 ZLATBS 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, where A is an upper or lower
 triangular band matrix.  Here 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 ZTBSV 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.

KD

          KD is INTEGER
          The number of subdiagonals or superdiagonals in the
          triangular matrix A.  KD >= 0.

AB

          AB is COMPLEX*16 array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first KD+1 rows of the array. The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

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, ZTBSV
  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 ZTBSV 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 ZTBSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

Definition at line 241 of file zlatbs.f.

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Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK