latbs(3)

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version 3.12.1 · 9 versions
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🎀 latbs(3) — Library Functions Manual 🎀

✨ NAME ✨

latbs — 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 🔎

It's function-doc time, bestie. Let's go. 🫶

💅 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:

Okay so CLATBS solves one of these triangular systems, bestie:

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 — is CHARACTER*1. Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular
  • 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 — is CHARACTER*1. Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular
  • 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 — is INTEGER. The order of the matrix A. N >= 0.
  • KD — is INTEGER. The number of subdiagonals or superdiagonals in the triangular matrix A. KD >= 0.
  • 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 — is INTEGER. The leading dimension of the array AB. LDAB >= KD+1.
  • 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 — 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 — 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 — 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 these triangular systems, bestie:

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 — is CHARACTER*1. Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular
  • 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 — is CHARACTER*1. Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular
  • 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 — is INTEGER. The order of the matrix A. N >= 0.

KD

(the page cuts off right here in part 1 — KD's description continues in part 2 💖)

  • KD — INTEGER. The number of subdiagonals or superdiagonals in the triangular matrix A. KD >= 0. ✨
  • AB — DOUBLE PRECISION array, dimension (LDAB,N). This is where the upper or lower triangular band matrix A lives, packed into 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, like so:
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 — INTEGER. The leading dimension of the array AB. Bestie, this one is non-negotiable: LDAB >= KD+1. 💅
  • X — DOUBLE PRECISION array, dimension (N). On entry, this is the right hand side b of the triangular system. On exit, X gets absolutely steamrolled — it's overwritten by the solution vector x. 🫶
  • SCALE — DOUBLE PRECISION. The scaling factor s for the triangular system
   A * x = s*b  or  A**T* x = s*b.

If SCALE = 0, that means the matrix A is singular or badly scaled, and the vector x is an exact-or-approximate solution to A*x = 0. So like... not the vibe, but it's a valid answer. 😭

  • CNORM — 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. The function does the homework for you, slay.

  • INFO — INTEGER. The receipts:
    • = 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:

okay so, a rough bound on x is computed first. If that bound is less than overflow, DTBSV is called — easy, iconic, done. Otherwise, a whole special code path kicks in that checks for possible overflow or divide-by-zero at every single operation. Fr fr, every one. 🧚

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). That's the vibe check right there. ⭐

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, then 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). Math girlies stay winning. 🌈

Definition at line 240 of file dlatbs.f.

🧮 subroutine slatbs — the single-precision girlie

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. Same energy as DLATBS, just REAL. 💖

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. And 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. No cap. 🔥

Parameters

  • UPLO — CHARACTER*1. Specifies whether the matrix A is upper or lower triangular.
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • TRANS — 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 — CHARACTER*1. Specifies whether or not the matrix A is unit triangular.
    • = 'N': Non-unit triangular
    • = 'U': Unit triangular
  • NORMIN — 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 — INTEGER. The order of the matrix A. N >= 0. 🎀
  • KD — INTEGER. The number of subdiagonals or superdiagonals in the triangular matrix A. KD >= 0.
  • AB — 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 — INTEGER. The leading dimension of the array AB. LDAB >= KD+1. Do not test her on this. 💅
  • X — 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. Gone. Replaced. Main character now. 👑
  • SCALE — 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 — 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 — INTEGER. The verdict:
    • = 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:

🌸 (the details continue in the next part, bestie — we're not done yet)

okay so the rest of the SLATBS bound-talk, bestie: it computes a rough bound on x; if that’s less than overflow, STBSV gets called. If not, there’s special code that checks for possible overflow or divide-by-zero at every operation. 💅

For A*x = b, it uses a columnwise scheme. If A is lower triangular, the basic algorithm 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 checkpoint: line 240 of slatbs.f 🎯

Next up: the zlatbs subroutine, the COMPLEX*16 queen of this whole situation 💅

   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

okay so ZLATBS solves one of these 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 — CHARACTER*1. Specifies whether the matrix A is upper or lower triangular. = 'U': upper triangular; = 'L': lower triangular.
  • TRANS — 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 — CHARACTER*1. Specifies whether or not the matrix A is unit triangular. = 'N': non-unit triangular; = 'U': unit triangular.
  • NORMIN — 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 — INTEGER. The order of the matrix A. N >= 0.
  • KD — INTEGER. The number of subdiagonals or superdiagonals in the triangular matrix A. KD >= 0.
  • AB — 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 — INTEGER. The leading dimension of the array AB. LDAB >= KD+1.
  • X — 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 — 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 — 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; 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 — 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).

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