SRC/zgbsv.f(3)

💖 NAME 💖

okay bestie, you're on the SRC/zgbsv.f(3) page from the Library Functions Manual, and the main character is literally SRC/zgbsv.f — aka the home of the zgbsv subroutine. slay. 💅

✨ SYNOPSIS ✨

Functions/Subroutines
       subroutine zgbsv (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
            ZGBSV computes the solution to system of linear equations A * X = B
           for GB matrices (simple driver)

🦋 Function/Subroutine Documentation 🦋

   subroutine zgbsv (integer n, integer kl, integer ku, integer nrhs,
       complex*16, dimension( ldab, * ) ab, integer ldab, integer, dimension( *
       ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver) — and yes, that's the whole vibe of this little icon. 🫶

🌸 Purpose 🌸

Okay so ZGBSV 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. no cap.

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. Then the factored form of A is used to solve the system of equations A * X = B. iconic behavior, honestly. ✨

🎀 Parameters 🎀

• N — N is INTEGER. The number of linear equations, i.e., the order 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.
• 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 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 like this: 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.
• LDAB — LDAB is INTEGER. The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
• 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).
• 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.
• 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, 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.

bestie, that INFO > 0 case is the dramatic one: U(i,i) is exactly zero, factorization finished, but U is exactly singular and the solution has not been computed. not me getting chills. 😭

👑 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 161 of file zgbsv.f.

💜 Author 💜

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💖 and that's the whole page, bestie! Footer source/version vibes: LAPACK Version 3.12.0, SRC/zgbsv.f(3). No separate date is listed in this footer, so I'm not making one up — slay. 🎀