latps(3)

🌸 NAME 🌸

Okay so this is latps(3) from the Library Functions Manual, and the whole page is about latps — triangular solve with robust scaling 💅. No cap, that's the name and the vibe.

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✨ SYNOPSIS ✨

🎀 Functions 🎀

Bestie, there are four flavors of this routine. Same energy, different precision:

• clatps

subroutine clatps (uplo, trans, diag, normin, n, ap, x, scale, cnorm,
    info)

CLATPS solves a triangular system of equations with the matrix held in packed storage.

• dlatps

subroutine dlatps (uplo, trans, diag, normin, n, ap, x, scale, cnorm,
    info)

DLATPS solves a triangular system of equations with the matrix held in packed storage.

• slatps

subroutine slatps (uplo, trans, diag, normin, n, ap, x, scale, cnorm,
    info)

SLATPS solves a triangular system of equations with the matrix held in packed storage.

• zlatps

subroutine zlatps (uplo, trans, diag, normin, n, ap, x, scale, cnorm,
    info)

ZLATPS solves a triangular system of equations with the matrix held in packed storage.

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🦋 Detailed Description 🦋

This is the overview part, bestie. The actual function tea is right below under Function Documentation. 🫶

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👑 Function Documentation 👑

💖 subroutine clatps 💖

subroutine clatps (character uplo, character trans, character diag,
    character normin, integer n, complex, dimension( * ) ap, complex,
    dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)

CLATPS solves a triangular system of equations with the matrix held in packed storage. Iconic.

🌟 Purpose 🌟

Okay so CLATPS is out here solving 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 matrix stored in packed form. Here 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 CTPSV 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. 😭 That singular-matrix moment is real.

🍓 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.
• AP — COMPLEX array, dimension (N*(N+1)/2). The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
• X — 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 — 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 — 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. = 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, CTPSV 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 CTPSV 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 CTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

Definition at line 229 of file clatps.f.

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💙 subroutine dlatps 💙

subroutine dlatps (character uplo, character trans, character diag,
    character normin, integer n, double precision, dimension( * ) ap, double
    precision, dimension( * ) x, double precision scale, double precision,
    dimension( * ) cnorm, integer info)

DLATPS solves a triangular system of equations with the matrix held in packed storage.

🌟 Purpose 🌟

DLATPS 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 matrix stored in packed form. 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 DTPSV 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. 💀 Singular matrix said "not today."

🍓 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.

💅 AP 💅

AP

AP is a DOUBLE PRECISION array, dimension (N*(N+1)/2). It's the upper or lower triangular matrix A, packed columnwise in a linear array (no cap, that's just how she's stored 💅). The j-th column of A lives in AP like this:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

X

X is a DOUBLE PRECISION array, dimension (N). On entry it's the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x — so yeah, your b is gone, she ate it 💀. Grab a copy first if you're sentimental.

SCALE

SCALE is DOUBLE PRECISION. It's 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 SCALE = 0 is that red flag moment 🚩 — note it and move on.

CNORM

CNORM is a 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. Feed her a bound that's too small and that's on you bestie 🫶.

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. Free real estate, she fills it in herself ✨.

INFO

INFO is INTEGER.

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

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🌈 Author 🌈

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

---

🔥 Further Details: 🔥

okay this next bit is the whole why of it, and it's genuinely iconic — she doesn't just vibe-check the overflow risk, she bounds it 💅

A rough bound on x is computed; if that is less than overflow, DTPSV 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 DTPSV 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. That's the plot twist of the whole routine tbh.

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

Definition at line 227 of file dlatps.f.

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🌸 subroutine slatps 🌸

subroutine slatps (character uplo, character trans, character diag,
    character normin, integer n, real, dimension( * ) ap, real, dimension( *
    ) x, real scale, real, dimension( * ) cnorm, integer info)

SLATPS solves a triangular system of equations with the matrix held in packed storage. Same energy as her double-precision big sister, just single precision 🎀.

✨ Purpose: ✨

SLATPS 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 matrix stored in packed form. 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 STPSV 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. Singular matrix? She still comes through with something. main character energy fr fr.

💖 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. Respect the constraint bestie, she checks.

AP

AP is REAL array, dimension (N*(N+1)/2). The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=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 — again, she takes your b, that's the deal 🙈.

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: ⭐

okay so we're still deep in Further Details, and the first line is literally: A rough bound on x is computed; if that is less than overflow, STPSV 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 STPSV 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 STPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow). 💅

and that's a wrap on this block — Definition at line 227 of file slatps.f. 🎀

---

🧚 subroutine zlatps — the COMPLEX*16 packed-triangular solver 💜

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

ZLATPS solves a triangular system of equations with the matrix held in packed storage. no cap. ✨

✨ Purpose:

ZLATPS 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 matrix stored in packed form. Here 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 ZTPSV 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.
• AP — COMPLEX*16 array, dimension (N*(N+1)/2). The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

• 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, 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.

= 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 the Further Details for ZLATPS are the same energy as above but with ZTPSV and the conjugate-transpose ' notation. here's every line, no skips: 💖

A rough bound on x is computed; if that is less than overflow, ZTPSV
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 ZTPSV 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 ZTPSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).

Definition at line 229 of file zlatps.f. 🦋

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