/home/abui...RC/cgbsv.f(3)

🌸 CGBSV — cgbsv.f ✨

ok so bestie, this page lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/cgbsv.f and it's from the LAPACK Library Functions Manual, section 3 💅 like it's literally a Fortran source doc, cgbsv.f(3), and we're about to get into it.

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✨ SYNOPSIS ✨

🎀 Functions/Subroutines

subroutine CGBSV (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)

CGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver) — yes she's a simple driver, she's lowkey the main character of band-matrix solving 🫶

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👑 Function/Subroutine Documentation 👑

subroutine CGBSV (integer n, integer kl, integer ku, integer nrhs, complex,
    dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv,
    complex, dimension( ldb, * ) b, integer ldb, integer info)

CGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver) — okay so same energy as the synopsis, it's giving consistency, we love that for her ✨

💖 Purpose:

!>
!> CGBSV computes the solution to a complex system of linear equations
!> A * X = B, where A is a band matrix of order N with KL subdiagonals
!> and KU superdiagonals, and X and B are N-by-NRHS matrices.
!>
!> The LU decomposition with partial pivoting and row interchanges is
!> used to factor A as A = L * U, where L is a product of permutation
!> and unit lower triangular matrices with KL subdiagonals, and U is
!> upper triangular with KL+KU superdiagonals.  The factored form of A
!> is then used to solve the system of equations A * X = B.
!>

so basically: band matrix A, order N, with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices, periodt 💅 She factors A = L * U with LU decomposition + partial pivoting + row interchanges, where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals. Then she uses the factored form to solve A * X = B. iconic behavior, no cap 🔥

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🦋 Parameters 🦋

• N

!>          N is INTEGER
!>          The number of linear equations, i.e., the order of the
!>          matrix A.  N >= 0.
!>

the number of linear equations aka the order of matrix A, and N >= 0 fr fr 🙈

• KL

!>          KL is INTEGER
!>          The number of subdiagonals within the band of A.  KL >= 0.
!>

how many subdiagonals live inside the band of A, KL >= 0 🎀

• KU

!>          KU is INTEGER
!>          The number of superdiagonals within the band of A.  KU >= 0.
!>

how many superdiagonals live inside the band of A, KU >= 0 ✨

• NRHS

!>          NRHS is INTEGER
!>          The number of right hand sides, i.e., the number of columns
!>          of the matrix B.  NRHS >= 0.
!>

the number of right hand sides aka the number of columns of matrix B, NRHS >= 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(N,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.
!>

AB is a COMPLEX array, dimension (LDAB,N). On entry it's matrix A in band storage, living in rows KL+1 to 2*KL+KU+1; rows 1 to KL of the array need not be set (so don't stress about those 💀). The j-th column of A is stored in the j-th column of AB like this: AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL). On exit you get the factorization deets: U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers from the factorization chill 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.
!>

the leading dimension of the array AB, and LDAB >= 2*KL+KU+1 — do not skip that one bestie 😭

• IPIV

!>          IPIV is INTEGER array, dimension (N)
!>          The pivot indices that define the permutation matrix P;
!>          row i of the matrix was interchanged with row IPIV(i).
!>

INTEGER array, dimension (N) — the pivot indices defining permutation matrix P; row i of the matrix was interchanged with row IPIV(i) 👑

• 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.
!>

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. glow-up moment 💖

• LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B.  LDB >= max(1,N).
!>

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, U(i,i) is exactly zero.  The factorization
!>                has been completed, but the factor U is exactly
!>                singular, and the solution has not been computed.
!>

INTEGER, and here's the vibe check:

• = 0: successful exit, we slayed ✨
• < 0: if INFO = -i, the i-th argument had an illegal value, rude 😤
• > 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and the solution has not been computed 💀

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🌟 Author 🌟

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

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🧚 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.
!>

so the band storage scheme is illustrated above when M = N = 6, KL = 2, KU = 1. Array elements marked * are not used by the routine (ignore them, they're not the vibe 🙈); 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 🔥

bestie please clock the difference between entry and exit — that fill-in from row interchanges is literally the whole point of the + slots, don't come for me 😭

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Definition at line 161 of file cgbsv.f. 📍

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💫 Author 💫

Generated automatically by Doxygen for LAPACK from the source code 🫶

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ok that's the whole page, mwah 💋 — LAPACK, Version 3.12.0, and this little beauty is /home/abui...RC/cgbsv.f(3), see you in the next manpage bestie 🎀✨🌈