pbtrf(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

okay so bestie, welcome to pbtrf(3) from the Library Functions Manual — this is the page for pbtrf, aka triangular factor 🔥. It's giving "Cholesky factorization of a band matrix" and the whole routine squad is lowkey iconic. No cap: every signature, flag, value, formula, example, author credit, and file line is staying exactly where it belongs, just with more sparkle 💅. periodt.

🌸 NAME

pbtrf — pbtrf: triangular factor

✨ SYNOPSIS

🫶 Functions

  • subroutine cpbtrf (uplo, n, kd, ab, ldab, info) — CPBTRF
  • subroutine dpbtrf (uplo, n, kd, ab, ldab, info) — DPBTRF
  • subroutine spbtrf (uplo, n, kd, ab, ldab, info) — SPBTRF
  • subroutine zpbtrf (uplo, n, kd, ab, ldab, info) — ZPBTRF

🦋 Detailed Description

okay so the detailed description is basically the function docs below — that's where cpbtrf, dpbtrf, spbtrf, and zpbtrf each get their main character moment ✨. Scroll for the full purpose, parameters, authors, further details, contributors, and source-file definition. Periodt 💖.

🔥 Function Documentation

💖 subroutine cpbtrf — CPBTRF

subroutine cpbtrf (character uplo, integer n, integer kd, complex,
    dimension( ldab, * ) ab, integer ldab, integer info)

This one is CPBTRF, and she's the complex Hermitian band Cholesky girlie. Obsessed. 💅

✨ Purpose:

CPBTRF computes the Cholesky factorization of a complex Hermitian positive definite band matrix A.

The factorization has the form:

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

where U is an upper triangular matrix and L is lower triangular. Iconic. 💜

💅 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 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.
  • 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).

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.
  • 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 is not positive, and the factorization could not be completed.

🌟 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.

🫶 Contributors:

Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line 141 of file cpbtrf.f.

💖 subroutine dpbtrf — DPBTRF

subroutine dpbtrf (character uplo, integer n, integer kd, double precision,
    dimension( ldab, * ) ab, integer ldab, integer info)

This one is DPBTRF, and she's serving real symmetric band Cholesky realness. Slay. 💙

✨ Purpose:

DPBTRF computes the Cholesky factorization of a real symmetric positive definite band matrix A.

The factorization has the form:

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

where U is an upper triangular matrix and L is lower triangular. 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 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.
  • 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).

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.
  • 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 is not positive, and the factorization could not be completed.

🌟 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.

🫶 Contributors:

Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line 141 of file dpbtrf.f.

💖 subroutine spbtrf — SPBTRF

subroutine spbtrf (character uplo, integer n, integer kd, real, dimension(
    ldab, * ) ab, integer ldab, integer info)

This one is SPBTRF, and she's the real symmetric band Cholesky sister. Gorgeous. ✨

✨ Purpose:

SPBTRF computes the Cholesky factorization of a real symmetric positive definite band matrix A.

The factorization has the form:

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

where U is an upper triangular matrix and L is lower triangular. No cap. 💚

💅 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 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.
  • AB — AB is REAL 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).

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.
  • 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 is not positive, and the factorization could not be completed.

🌟 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.

🫶 Contributors:

Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line 141 of file spbtrf.f.

💖 subroutine zpbtrf — ZPBTRF

subroutine zpbtrf (character uplo, integer n, integer kd, complex*16,
    dimension( ldab, * ) ab, integer ldab, integer info)

This one is ZPBTRF, and she's the complex*16 band Cholesky queen. Main character energy fr fr 👑.

✨ Purpose:

okay so ZPBTRF computes the Cholesky factorization of a complex Hermitian positive definite band matrix A, and the factorization has the form A = U**H * U if UPLO = 'U', or A = L * L**H if UPLO = 'L', where U is upper triangular and L is lower triangular. now the params, bestie 💅

✨ 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 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.

🎀 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).

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.

🔥 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 is not positive, and the factorization could not be completed.

👑 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.

🌸 Contributors: 🌸

Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line 141 of file zpbtrf.f.

🎀 Author 🎀

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