SRC/zsysv_rk.f(3)

Hey bestie 💖 welcome to the SRC/zsysv_rk.f(3) page — it's part of the Library Functions Manual, and it's all about the zsysv_rk subroutine. This is giving linear-algebra main character energy, and we're gonna get through every single detail together. 🦋✨

🌸 NAME 🌸

SRC/zsysv_rk.f — that's the page, bestie. The icon of the hour is the zsysv_rk routine. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine zsysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work,
    lwork, info)

ZSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices — periodt. 🎀

🦋 Function/Subroutine Documentation 🦋

subroutine zsysv_rk

subroutine zsysv_rk (character uplo, integer n, integer nrhs, complex*16,
    dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e,
    integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer
    ldb, complex*16, dimension( * ) work, integer lwork, integer info)

ZSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices. No cap. ✨

Purpose:

Okay so ZSYSV_RK computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. 💖

The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as:

A = P*U*D*(U**T)*(P**T),  if UPLO = 'U', or
A = P*L*D*(L**T)*(P**T),  if UPLO = 'L',

where U (or L) is unit upper (or lower) triangular matrix, U**T (or L**T) is the transpose of U (or L), P is a permutation matrix, P**T is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. 🧚

ZSYTRF_RK is called to compute the factorization of a complex symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine ZSYTRS_3. Slay. 🔥

Parameters

• UPLO — UPLO is CHARACTER*1. It specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — N is INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. 💜
• NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 🍓
• A — A is COMPLEX*16 array, dimension (LDA,N). On entry, the symmetric matrix A.
  • If UPLO = 'U': the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced.
  • If UPLO = 'L': the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
  • On exit, if INFO = 0, diagonal of the block diagonal matrix D and factors U or L as computed by ZSYTRF_RK:
    • a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); (superdiagonal (or subdiagonal) elements of D are stored on exit in array E), and
    • b) If UPLO = 'U': factor U in the superdiagonal part of A. If UPLO = 'L': factor L in the subdiagonal part of A.
  • For more info see the description of ZSYTRF_RK routine.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🎀
• E — E is COMPLEX*16 array, dimension (N). On exit, contains the output computed by the factorization routine ZSYTRF_RK, i.e. the superdiagonal (or subdiagonal) elements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, where:
  • If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
  • If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
  • NOTE: For 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both UPLO = 'U' or UPLO = 'L' cases.
  • For more info see the description of ZSYTRF_RK routine.
• IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D, as determined by ZSYTRF_RK.
  • For more info see the description of ZSYTRF_RK routine. 👑
• 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). 🌈
• WORK — WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ). Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ⭐
• LWORK — LWORK is INTEGER. The length of WORK. LWORK >= 1. For best performance of factorization stage LWORK >= max(1,N*NB), where NB is the optimal blocksize for ZSYTRF_RK.
  • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array for factorization stage, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. 🫶
• INFO — INFO is INTEGER.
  • = 0: successful exit
  • < 0: If INFO = -k, the k-th argument had an illegal value
  • > 0: If INFO = k, the matrix A is singular, because:
    • If UPLO = 'U': column k in the upper triangular part of A contains all zeros.
    • If UPLO = 'L': column k in the lower triangular part of A contains all zeros.
    • Therefore D(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.
    • NOTE: INFO only stores the first occurrence of a singularity, any subsequent occurrence of singularity is not stored in INFO even though the factorization always completes. 😭

Author

The legends behind this routine are:

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

Contributors:

• December 2016, Igor Kozachenko, Computer Science Division, University of California, Berkeley
• September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, School of Mathematics, University of Manchester

Definition at line 226 of file zsysv_rk.f.

👑 Author 👑

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