ZLATBS(3)

🦋 zlatbs.f(3) — like, the Library Functions Manual page, bestie 💅

okay so this is a page straight out of the Library Functions Manual — it's zlatbs.f(3) and it's literally the LAPACK girlie who solves triangular banded systems of equations for you. no cap, she's iconic. let's get into it ✨

---

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlatbs.f

that's the whole name. it's just the file path. she's mysterious like that 💅

---

✨ SYNOPSIS ✨

Functions/Subroutines

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

ZLATBS solves a triangular banded system of equations. periodt. 🔥

---

👑 Function/Subroutine Documentation 👑

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. (yes i said it twice, it's important, let her have this 💖)

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

the whole squad showed up, we love a collab 💖

🔥 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. fr fr, she's thorough.

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). slay. 💅✨

Definition at line 241 of file zlatbs.f.

---

🍓 Author 🍓

Generated automatically by Doxygen for LAPACK from the source code. (the robot did it, not me 🙈)

---

and that's the whole page bestie — no details skipped, iykyk 💖

okay so like, this whole thing is LAPACK, Version 3.12.0, and the footer energy is /home/abui...RC/zlatbs.f(3) — that's the vibe check, we're done here, mwah 🫶🎀