gbtf2(3)

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version 3.12.1 · 9 versions
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🌸 gbtf2 🌸

okay so this is gbtf2(3) from the Library Functions Manual — bestie, it's a whole family of LAPACK routines that do LU factorization on general band matrices using the unblocked version of the algorithm. Four flavors, one page, all the tea below 👑

✨ SYNOPSIS ✨

Functions

  • subroutine cgbtf2 (m, n, kl, ku, ab, ldab, ipiv, info) — CGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm. 💅
  • subroutine dgbtf2 (m, n, kl, ku, ab, ldab, ipiv, info) — DGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm. 💅
  • subroutine sgbtf2 (m, n, kl, ku, ab, ldab, ipiv, info) — SGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm. 💅
  • subroutine zgbtf2 (m, n, kl, ku, ab, ldab, ipiv, info) — ZGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm. 💅

Detailed Description

(that's literally it, she's a heading with no body — the real content lives down in Function Documentation 🫶)

Function Documentation

🦋 subroutine cgbtf2 (integer m, integer n, integer kl, integer ku, complex, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, integer info)

subroutine cgbtf2 (integer m, integer n, integer kl, integer ku, complex,
    dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv,
    integer info)

CGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

Purpose

CGBTF2 computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges.
This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters

  • M — M is INTEGER. The number of rows of the matrix A. M >= 0.
  • N — N is INTEGER. The number of columns of the matrix A. N >= 0.
  • KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
  • KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
  • AB — AB is COMPLEX array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl). On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details 👇
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
  • IPIV — IPIV is INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
  • INFO — INFO is INTEGER
    • = 0 — successful exit ✨
    • < 0 — if INFO = -i, the i-th argument had an illegal value
    • > 0 — if INFO = +i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations. 💀

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 M = N = 6, KL = 2, KU = 1:
On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u14  u25  u36
    *    *    +    +    +    +       *    *   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
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row interchanges.

Definition at line 144 of file cgbtf2.f.

🧚 subroutine dgbtf2 (integer m, integer n, integer kl, integer ku, double precision, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, integer info)

subroutine dgbtf2 (integer m, integer n, integer kl, integer ku, double
    precision, dimension( ldab, * ) ab, integer ldab, integer, dimension( *
    ) ipiv, integer info)

DGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

Purpose

DGBTF2 computes an LU factorization of a real m-by-n band matrix A using partial pivoting with row interchanges.
This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters

  • M — M is INTEGER. The number of rows of the matrix A. M >= 0.
  • N — N is INTEGER. The number of columns of the matrix A. N >= 0.
  • KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
  • KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
  • AB — AB is DOUBLE PRECISION array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl). On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details 👇
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
  • IPIV — IPIV is INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
  • INFO — INFO is INTEGER
    • = 0 — successful exit ✨
    • < 0 — if INFO = -i, the i-th argument had an illegal value
    • > 0 — if INFO = +i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations. 💀

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 M = N = 6, KL = 2, KU = 1:
On entry:                       On exit:

    *    *    *    +    +    +       *    *    *   u14  u25  u36
    *    *    +    +    +    +       *    *   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
   a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
   a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row interchanges.

Definition at line 144 of file dgbtf2.f.

🎀 subroutine sgbtf2 (integer m, integer n, integer kl, integer ku, real, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, integer info)

subroutine sgbtf2 (integer m, integer n, integer kl, integer ku, real,
    dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv,
    integer info)

SGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

Purpose

SGBTF2 computes an LU factorization of a real m-by-n band matrix A using partial pivoting with row interchanges.
This is the unblocked version of the algorithm, calling Level 2 BLAS.

Parameters

  • M — M is INTEGER. The number of rows of the matrix A. M >= 0.
  • N — N is INTEGER. The number of columns of the matrix A. N >= 0.
  • KL — KL is INTEGER. The number of subdiagonals within the band of A. KL >= 0.
  • KU — KU is INTEGER. The number of superdiagonals within the band of A. KU >= 0.
  • AB — AB is REAL array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows: AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl). On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details 👇
  • LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
  • IPIV — IPIV is INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i).
  • INFO — INFO is INTEGER … and that's exactly where part 1 runs out of page, bestie 😭 the rest of INFO's glow-up rolls over into part 2 🌀
  • INFO — INTEGER, the exit-status bestie, and she is very important:
    • = 0: successful exit, we ate and left no crumbs 💖
    • < 0: if INFO = -i, the i-th argument had an illegal value — rude of it honestly 🙈
    • > 0: if INFO = +i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations 💀 like it will literally break, no cap

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 Further Details:

ok so this is the part where the band storage scheme gets illustrated, and honestly it's giving diagram-of-the-year. Watch this, bestie — this is the case when M = N = 6, KL = 2, KU = 1:

           !>
           !>  The band storage scheme is illustrated by the following example, when
           !>  M = N = 6, KL = 2, KU = 1:
           !>
           !>  On entry:                       On exit:
           !>
           !>      *    *    *    +    +    +       *    *    *   u14  u25  u36
           !>      *    *    +    +    +    +       *    *   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
           !>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
           !>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
           !>
           !>  Array elements marked * are not used by the routine; elements marked
           !>  + need not be set on entry, but are required by the routine to store
           !>  elements of U, because of fill-in resulting from the row
           !>  interchanges.
           !>

tldr, so you don't have to squint: array elements marked * are not used by the routine, and elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row interchanges. That's the whole tea ☕

Definition at line 144 of file sgbtf2.f.

🌈 subroutine zgbtf2 (integer m, integer n, integer kl, integer ku, complex*16, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, integer info)

ZGBTF2 computes the LU factorization of a general band matrix using the unblocked version of the algorithm.

💜 Purpose:

ZGBTF2 computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges.

This is the unblocked version of the algorithm, calling Level 2 BLAS. (yes, the plainer sibling of the blocked one — she still slays, just one column-ish at a time 💅)

🌸 Parameters

  • M — INTEGER. The number of rows of the matrix A. M >= 0.
  • N — INTEGER. The number of columns of the matrix A. N >= 0.
  • KL — INTEGER. The number of subdiagonals within the band of A. KL >= 0.
  • KU — INTEGER. The number of superdiagonals within the band of A. KU >= 0.
  • AB — COMPLEX*16 array, dimension (LDAB,N). On entry, the matrix A in band storage, in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set. The j-th column of A is stored in the j-th column of the array AB as follows:
AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)

On exit, details of the factorization: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. See below for further details. (the "see below" is real, the diagram is literally right there, keep scrolling 🫶)

  • LDAB — INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1. No exceptions, bestie, she's strict about her leading dimension.
  • IPIV — INTEGER array, dimension (min(M,N)). The pivot indices; for 1 <= i <= min(M,N), row i of the matrix was interchanged with row IPIV(i). That's the receipts of every row swap 📋
  • INFO — INTEGER, the vibe check at the end:
    • = 0: successful exit ✨
    • < 0: if INFO = -i, the i-th argument had an illegal value 😭
    • > 0: if INFO = +i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations 💀

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 Further Details:

once more for the people in the back — the band storage scheme, illustrated for the case when M = N = 6, KL = 2, KU = 1:

           !>
           !>  The band storage scheme is illustrated by the following example, when
           !>  M = N = 6, KL = 2, KU = 1:
           !>
           !>  On entry:                       On exit:
           !>
           !>      *    *    *    +    +    +       *    *    *   u14  u25  u36
           !>      *    *    +    +    +    +       *    *   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
           !>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
           !>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
           !>
           !>  Array elements marked * are not used by the routine; elements marked
           !>  + need not be set on entry, but are required by the routine to store
           !>  elements of U, because of fill-in resulting from the row
           !>  interchanges.
           !>

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U, because of fill-in resulting from the row interchanges. Same deal as before, just the complex edition 🧚

Definition at line 144 of file zgbtf2.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole page, bestie — you made it to the end 🎀 this iconic little artifact is from LAPACK, Version 3.12.0, dated gbtf2(3) — go forth and factorize responsibly, ily, bye 💖🦋✨

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