pbsv(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

, in arch rolling

pbsv(3) — Library Functions Manual 🫶

okay bestie, this is pbsv(3) from the Library Functions Manual, and it's literally the "factor and solve" page. 💅

🎀 NAME 🎀

pbsv — pbsv: factor and solve

That's the whole name, no cap: pbsv - pbsv: factor and solve. ✨

✨ SYNOPSIS ✨

🧚 Functions 🧚

cpbsv — subroutine cpbsv (uplo, n, kd, nrhs, ab, ldab, b, ldb, info). CPBSV computes the solution to system of linear equations A * X = B for OTHER matrices.
dpbsv — subroutine dpbsv (uplo, n, kd, nrhs, ab, ldab, b, ldb, info). DPBSV computes the solution to system of linear equations A * X = B for OTHER matrices.
spbsv — subroutine spbsv (uplo, n, kd, nrhs, ab, ldab, b, ldb, info). SPBSV computes the solution to system of linear equations A * X = B for OTHER matrices.
zpbsv — subroutine zpbsv (uplo, n, kd, nrhs, ab, ldab, b, ldb, info). ZPBSV computes the solution to system of linear equations A * X = B for OTHER matrices.

💜 Detailed Description 💜

This section is literally empty in the source, bestie — no extra prose here, we're sliding straight into the function docs. ✨

👑 Function Documentation 👑

🦋 cpbsv 🦋

   subroutine cpbsv (character uplo, integer n, integer kd, integer nrhs,
       complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldb,
       * ) b, integer ldb, integer info)

CPBSV computes the solution to system of linear equations A * X = B for OTHER matrices.

Purpose: CPBSV computes the solution to a complex system of linear equations

A * X = B,

where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as

A = U**H * U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',

where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B.

Parameters

UPLO — UPLO is CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
AB — AB is COMPLEX array, dimension (LDAB,N). On entry, the upper or lower triangle of the Hermitian 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). See below for further details.

On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H*U or A = L*L**H of the band matrix A, in the same storage format as A.

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD+1.
B — B is COMPLEX array, dimension (LDB,NRHS). On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed.

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Further Details:

The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U':

On entry:                       On exit:

      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry:                       On exit:

     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

Array elements marked * are not used by the routine.

Definition at line 163 of file cpbsv.f.

🌸 dpbsv 🌸

   subroutine dpbsv (character uplo, integer n, integer kd, integer nrhs,
       double precision, dimension( ldab, * ) ab, integer ldab, double
       precision, dimension( ldb, * ) b, integer ldb, integer info)

DPBSV computes the solution to system of linear equations A * X = B for OTHER matrices.

Purpose: DPBSV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric positive definite band matrix and X and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as

A = U**T * U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',

Parameters

UPLO — UPLO is CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0.
KD — KD is INTEGER. The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
AB — AB is DOUBLE PRECISION array, dimension (LDAB,N). On entry, the upper or lower triangle of the symmetric 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). See below for further details.

On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the band matrix A, in the same storage format as A.

LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= KD+1.
B — B is DOUBLE PRECISION array, dimension (LDB,NRHS). On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X.
LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,N).
INFO — INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed.

Author

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Further Details:

The band storage scheme is illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U':

On entry:                       On exit:

      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry:                       On exit:

     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

Array elements marked * are not used by the routine.

Definition at line 163 of file dpbsv.f.

🧚 spbsv 🧚

   subroutine spbsv (character uplo, integer n, integer kd, integer nrhs, real,
       dimension( ldab, * ) ab, integer ldab, real, dimension( ldb, * ) b,
       integer ldb, integer info)

SPBSV computes the solution to system of linear equations A * X = B for OTHER matrices.

Purpose: SPBSV computes the solution to a real system of linear equations

A * X = B,

where A is an N-by-N symmetric positive definite band matrix and X and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as

A = U**T * U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',

Parameters

UPLO — UPLO is CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
N — part 1 stops here, bestie; N's full description continues in part 2 💖

Parameters (yes, there's more, stay with me 🫶)

N — N is INTEGER ✨ the number of linear equations, aka the order of the matrix A. Constraint era: N >= 0. No cap.
KD — KD is INTEGER 💅 the number of superdiagonals of the matrix A when UPLO = 'U', or the number of subdiagonals when UPLO = 'L'. Always KD >= 0.
NRHS — NRHS is INTEGER 🎀 the number of right hand sides, i.e. how many columns the matrix B has. NRHS >= 0.
AB — AB is REAL array, dimension (LDAB,N).

On entry, it's holding the upper or lower triangle of the symmetric band matrix A, tucked into the first KD+1 rows of the array. The j-th column of A lives in the j-th column of AB, like this:

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).

See below for further details — there's a whole picture down there, it's giving diagram energy 👀

On exit, if INFO = 0, you get the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the band matrix A, stored in the exact same format as A. Same fit, new fit 💖

LDAB — LDAB is INTEGER, the leading dimension of the array AB. Rule: LDAB >= KD+1.
B — B is REAL array, dimension (LDB,NRHS).

On entry: the N-by-NRHS right hand side matrix B. On exit, if INFO = 0: the N-by-NRHS solution matrix X. It literally transforms, main character behavior ✨

LDB — LDB is INTEGER, the leading dimension of the array B. Rule: LDB >= max(1,N).
INFO — INFO is INTEGER, the vibe check at the end 👑
- = 0: successful exit, we ate 💅
- < 0: if INFO = -i, the i-th argument had an illegal value (someone passed something cursed 🙈)
- > 0: if INFO = i, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. So like… no answer for you 😭

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details 🔍

The band storage scheme is illustrated by this example, when N = 6, KD = 2, and UPLO = 'U' — memorize it, it's iconic:

On entry:                       On exit:

    *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry:                       On exit:

   a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
   a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
   a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

Array elements marked * are not used by the routine. They're just there for the aesthetic, honestly 💀

Definition at line 163 of file spbsv.f. Noted, bestie 🫡

subroutine `zpbsv` 🦋

subroutine zpbsv (character uplo, integer n, integer kd, integer nrhs,
    complex*16, dimension( ldab, * ) ab, integer ldab, complex*16,
    dimension( ldb, * ) b, integer ldb, integer info)

ZPBSV computes the solution to system of linear equations A * X = B for OTHER matrices — yes "OTHER", that's literally what the page says, I don't make the rules 🌈

Purpose 🔥

Lowkey the whole point of this icon:

ZPBSV computes the solution to a complex system of linear equations

A * X = B,

where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as

A = U**H *U,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',

where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations A * X = B. Periodt. 💅

Parameters 🎀

UPLO — UPLO is CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — N is INTEGER, the number of linear equations, i.e. the order of the matrix A. N >= 0. ✨
KD — KD is INTEGER, the number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.
NRHS — NRHS is INTEGER, the number of right hand sides, i.e. the number of columns of the matrix B. NRHS >= 0. 🫶
AB — AB is COMPLEX*16 array, dimension (LDAB,N).

On entry, the upper or lower triangle of the Hermitian 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).

See below for further details (yes, again, scroll a lil 🦋).

On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**H *U or A = L*L**H of the band matrix A, in the same storage format as A. Chef's kiss 🤌

LDAB — LDAB is INTEGER, the leading dimension of the array AB. LDAB >= KD+1.
B — B is COMPLEX*16 array, dimension (LDB,NRHS).

On entry: the N-by-NRHS right hand side matrix B. On exit, if INFO = 0: the N-by-NRHS solution matrix X. Glow-up confirmed 💖

LDB — LDB is INTEGER, the leading dimension of the array B. LDB >= max(1,N).
INFO — INFO is INTEGER, the final verdict 👑
- = 0: successful exit
- < 0: if INFO = -i, the i-th argument had an illegal value
- > 0: if INFO = i, the leading principal minor of order i of A is not positive, so the factorization could not be completed, and the solution has not been computed. Genuinely nothing you can do, it's over 😭

Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details 🌈

The band storage scheme, illustrated by the following example, when N = 6, KD = 2, and UPLO = 'U':

On entry:                       On exit:

    *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
    *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
   a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66

Similarly, if UPLO = 'L' the format of A is as follows:

On entry:                       On exit:

   a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
   a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
   a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *

Array elements marked * are not used by the routine. Ignore them, they're background characters 💀

Definition at line 163 of file zpbsv.f. Same line, different file, iconic coincidence 🎀

Author 💜

Generated automatically by Doxygen for LAPACK from the source code.

And that's the whole page, bestie 🫶 — this was LAPACK, Version 3.12.0, living on the pbsv(3) page. Go solve something Hermitian, you've got this ✨👑💅

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

pbsv(3)

pbsv(3) — Library Functions Manual 🫶

🎀 NAME 🎀

✨ SYNOPSIS ✨

🧚 Functions 🧚

💜 Detailed Description 💜

👑 Function Documentation 👑

🦋 cpbsv 🦋

🌸 dpbsv 🌸

🧚 spbsv 🧚

Parameters (yes, there's more, stay with me 🫶)

Author 👑

Further Details 🔍

subroutine zpbsv 🦋

Purpose 🔥

Parameters 🎀

Author 👑

Further Details 🌈

Author 💜

subroutine `zpbsv` 🦋