pbstf(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🎀 pbstf(3) 🎀

okay so bestie, welcome to pbstf — that's the pbstf(3) page from the Library Functions Manual, and honestly? she's a whole vibe. this is the split Cholesky factor situation, and the key thing to remember is that you use her with hbgst. iykyk 💅

she's not one function, she's a little family of four: cpbstf, dpbstf, spbstf and zpbstf. same energy, different precisions. we love a consistent queen 👑

✨ SYNOPSIS ✨

🌈 Functions

  • subroutine cpbstf (uplo, n, kd, ab, ldab, info) — CPBSTF
  • subroutine dpbstf (uplo, n, kd, ab, ldab, info) — DPBSTF
  • subroutine spbstf (uplo, n, kd, ab, ldab, info) — SPBSTF
  • subroutine zpbstf (uplo, n, kd, ab, ldab, info) — ZPBSTF

🦋 DESCRIPTION 🦋

Detailed Description

💜 Function Documentation 💜

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

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

CPBSTF

okay so here's the Purpose, and she is important:

CPBSTF computes a split Cholesky factorization of a complex Hermitian positive definite band matrix A. no cap, that's the whole job 💖

this routine is literally designed to be used in conjunction with CHBGST — they're a package deal, besties 🫶

the factorization has the form A = S**H*S where S is a band matrix of the same bandwidth as A and has the following structure:

S = ( U    )
    ( M  L )

where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m. iconic structure tbh ✨

🌟 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 factor S from the split Cholesky factorization A = S**H*S. See Further Details. (yes queen, the array gets overwritten — that's the whole point 💀)

  • 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 factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite.
🚨 real talk: an INFO > 0 means your matrix is not positive definite. that's not a "try again" situation, that's a "your matrix said no" situation 😭

👑 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 = 7, KD = 2:

S = ( s11  s12  s13                     )
    (      s22  s23  s24                )
    (           s33  s34                )
    (                s44                )
    (           s53  s54  s55           )
    (                s64  s65  s66      )
    (                     s75  s76  s77 )

If UPLO = 'U', the array AB holds:

on entry:                          on exit:

 *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53**H s64**H s75**H
 *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54**H s65**H s76**H
a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55    s66    s77

If UPLO = 'L', the array AB holds:

on entry:                          on exit:

a11  a22  a33  a44  a55  a66  a77  s11    s22    s33    s44  s55  s66  s77
a21  a32  a43  a54  a65  a76   *   s12**H s23**H s34**H s54  s65  s76   *
a31  a42  a53  a64  a64   *    *   s13**H s24**H s53    s64  s75   *    *

array elements marked * are not used by the routine; s12**H denotes conjg(s12); the diagonal elements of S are real. periodt 💅

🌟 definition at line 152 of file cpbstf.f.

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

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

DPBSTF

Purpose:

DPBSTF computes a split Cholesky factorization of a real symmetric positive definite band matrix A. same deal, double precision, still slaying ✨

this routine is designed to be used in conjunction with DSBGST 👑

the factorization has the form A = S**T*S where S is a band matrix of the same bandwidth as A and the following structure:

S = ( U    )
    ( M  L )

where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m.

🌟 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 factor S from the split Cholesky factorization A = S**T*S. See Further Details.

  • 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 factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite.
🚨 again for the people in the back: INFO > 0 = A is not positive definite. no fixing that with vibes 🙈

👑 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 = 7, KD = 2:

S = ( s11  s12  s13                     )
    (      s22  s23  s24                )
    (           s33  s34                )
    (                s44                )
    (           s53  s54  s55           )
    (                s64  s65  s66      )
    (                     s75  s76  s77 )

If UPLO = 'U', the array AB holds:

on entry:                          on exit:

 *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53  s64  s75
 *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76
a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77

If UPLO = 'L', the array AB holds:

on entry:                          on exit:

a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
a21  a32  a43  a54  a65  a76   *   s12  s23  s34  s54  s65  s76   *
a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *

array elements marked * are not used by the routine. (no **H here — she's real, not complex, and we respect that 🌸)

🌟 definition at line 151 of file dpbstf.f.

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

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

SPBSTF

Purpose:

SPBSTF computes a split Cholesky factorization of a real symmetric positive definite band matrix A. same routine as DPBSTF but single precision — she's the lowkey version and she still ate 💅

this routine is designed to be used in conjunction with SSBGST 🫶

the factorization has the form A = S**T*S where S is a band matrix of the same bandwidth as A and the following structure:

S = ( U    )
    ( M  L )

where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m.

🌟 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 factor S from the split Cholesky factorization A = S**T*S. See Further Details.

  • 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 factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite.
🚨 yes, same warning, because same rule: INFO > 0 = negative updated diagonal = A is not positive definite. it's giving "the math said no" 😭

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd. — yes bestie, we’re still in the author credits and NAG Ltd. is right there too 💅

✨ Further Details ✨

Okay so the band storage scheme is a whole vibe, and here’s the example from the source when N = 7, KD = 2:

S = ( s11  s12  s13                     )
    (      s22  s23  s24                )
    (           s33  s34                )
    (                s44                )
    (           s53  s54  s55           )
    (                s64  s65  s66      )
    (                     s75  s76  s77 )

If UPLO = 'U', the array AB holds:

on entry:                          on exit:

 *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53  s64  s75
 *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76
a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77

If UPLO = 'L', the array AB holds:

on entry:                          on exit:

a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
a21  a32  a43  a54  a65  a76   *   s12  s23  s34  s54  s65  s76   *
a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *

Array elements marked * are not used by the routine. Not used, not invited, not on the guest list 🚫✨

Definition at line 151 of file spbstf.f.

🦋 subroutine zpbstf — aka ZPBSTF

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

ZPBSTF

🎀 Purpose

Okay so ZPBSTF computes a split Cholesky factorization of a complex Hermitian positive definite band matrix A. It’s giving “band matrix but make it split Cholesky” energy 🔥

This routine is designed to be used in conjunction with ZHBGST. Bestie, they’re a package deal.

The factorization has the form A = S**H*S where S is a band matrix of the same bandwidth as A and the following structure:

S = ( U    )
    ( M  L )

where U is upper triangular of order m = (n+kd)/2, and L is lower triangular of order n-m.

🌸 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 factor S from the split Cholesky factorization A = S**H*S. See Further Details.
  • 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 factorization could not be completed, because the updated element a(i,i) was negative; the matrix A is not positive definite.
The INFO > 0 case is the “nope, not positive definite” alarm, bestie. Don’t ignore it 🚨

👑 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 = 7, KD = 2:

S = ( s11  s12  s13                     )
    (      s22  s23  s24                )
    (           s33  s34                )
    (                s44                )
    (           s53  s54  s55           )
    (                s64  s65  s66      )
    (                     s75  s76  s77 )

If UPLO = 'U', the array AB holds:

on entry:                          on exit:

 *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53**H s64**H s75**H
 *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54**H s65**H s76**H
a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55    s66    s77

If UPLO = 'L', the array AB holds:

on entry:                          on exit:

a11  a22  a33  a44  a55  a66  a77  s11    s22    s33    s44  s55  s66  s77
a21  a32  a43  a54  a65  a76   *   s12**H s23**H s34**H s54  s65  s76   *
a31  a42  a53  a64  a64   *    *   s13**H s24**H s53    s64  s75   *    *

Array elements marked * are not used by the routine; s12**H denotes conjg(s12); the diagonal elements of S are real.

Definition at line 152 of file zpbstf.f.

Author (auto-gen edition)

Generated automatically by Doxygen for LAPACK from the source code.

That footer, bestie: LAPACK, Version 3.12.0, pbstf(3) — no date printed on the line, and that’s the whole page. Mwah 💋💖

📦 where this page comes from