SRC/dlatbs.f(3)

💅 SRC/dlatbs.f(3) 💅

okay so bestie, this is the SRC/dlatbs.f(3) page from the Library Functions Manual, and we are about to get into DLATBS with all the details intact. It's giving linear algebra with overflow protection, and honestly? Iconic. ✨

---

🌸 NAME 🌸

SRC/dlatbs.f

---

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine dlatbs (uplo, trans, diag, normin, n, kd, ab, ldab, x, scale,
    cnorm, info)

DLATBS solves a triangular banded system of equations.

---

🎀 Function/Subroutine Documentation 🎀

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: 🦋

Okay so DLATBS solves one of the triangular systems:

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

with scaling to prevent overflow. Here's the rundown: A is an upper or lower triangular band 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 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. Periodt. 💅

🍓 Parameters 🍓

💖 UPLO

UPLO is CHARACTER*1. It specifies whether the matrix A is upper or lower triangular.

• 'U': Upper triangular
• 'L': Lower triangular

🦋 TRANS

TRANS is CHARACTER*1. It 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. It specifies whether or not the matrix A is unit triangular.

• 'N': Non-unit triangular
• 'U': Unit triangular

🌈 NORMIN

NORMIN is CHARACTER*1. It 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. It's the order of the matrix A. N >= 0.

👑 KD

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

🦋 AB

AB is DOUBLE PRECISION array, dimension (LDAB,N). It's 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. It's the leading dimension of the array AB. LDAB >= KD+1.

🌸 X

X is 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.

⭐ 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, 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.

💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code.

---

That's the whole page, bestie! 💖 Footer vibes: LAPACK Version 3.12.0, SRC/dlatbs.f(3). Slay. ✨🎀