SRC/csysv_rk.f(3)

🌸 NAME 🌸

SRC/csysv_rk.f — okay bestie, this is the SRC/csysv_rk.f page from the Library Functions Manual. 💖 It is literally the manual-page moment for csysv_rk, and yes, we are going to get through every parameter together.

---

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

subroutine csysv_rk (uplo, n, nrhs, a, lda, e, ipiv, b, ldb, work,
    lwork, info)
     CSYSV_RK computes the solution to system of linear equations A * X
    = B for SY matrices

---

💫 Function/Subroutine Documentation 💫

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

CSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices. 🦋

🌟 Purpose 🌟

CSYSV_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. ✨

CSYTRF_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 CSYTRS_3. 🫶

🧚 Parameters 🧚

• UPLO — CHARACTER*1. 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 — INTEGER. The number of linear equations, i.e., the order of the matrix A. N >= 0. 🍓
• NRHS — INTEGER. The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. 🎀
• A — COMPLEX 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 CSYTRF_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 CSYTRF_RK routine. 💜
• LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N). ⭐
• E — COMPLEX array, dimension (N). On exit, contains the output computed by the factorization routine CSYTRF_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 CSYTRF_RK routine. 🦋
• IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D, as determined by CSYTRF_RK. For more info see the description of CSYTRF_RK routine. 🔥
• B — 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. 🌸
• LDB — INTEGER. The leading dimension of the array B. LDB >= max(1,N). 💖
• WORK — COMPLEX array, dimension ( MAX(1,LWORK) ). Work array used in the factorization stage. On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ✨
• LWORK — 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 CSYTRF_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 — 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. 😭

⚠️ Bestie, if INFO > 0, the matrix is singular and division by zero will occur if you use it to solve a system of equations. This is not a drill. 💀

👑 Author 👑

• 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 csysv_rk.f. 🫶

---

💅 Author 💅

Generated automatically by Doxygen for LAPACK from the source code. ✨

---

mwah 💋 — the footer energy is LAPACK Version 3.12.0, source SRC/csysv_rk.f(3). Periodt. 👑