SRC/ssytf2_rk.f(3)

💖 SRC/ssytf2_rk.f(3) — Library Functions Manual 💖

🌸 NAME 🌸

SRC/ssytf2_rk.f — the page for the ssytf2_rk subroutine, bestie. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine ssytf2_rk (uplo, n, a, lda, e, ipiv, info)
    SSYTF2_RK computes the factorization of a real symmetric indefinite
    matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting
    method (BLAS2 unblocked algorithm).

💅 Function/Subroutine Documentation 💅

subroutine ssytf2_rk (character uplo, integer n, real, dimension( lda, * )
     a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv,
     integer info)

SSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). No cap.

🎯 Purpose 🎯

Okay so bestie, SSYTF2_RK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is a 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.

This is the unblocked version of the algorithm, calling Level 2 BLAS. For more information see the Further Details section. 🫶

🎀 Parameters 🎀

UPLO

UPLO is CHARACTER*1. It picks whether the upper or lower triangular part of the symmetric matrix A is stored:

• = 'U': Upper triangular
• = 'L': Lower triangular

N

N is INTEGER. It’s the order of the matrix A, and N >= 0.

A

A is a REAL array, dimension (LDA,N).

On entry, it’s the symmetric matrix A, and which triangle gets read depends on UPLO:

• 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, it contains:

• a) ONLY diagonal elements of the symmetric block diagonal matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); the 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.

LDA

LDA is INTEGER. It’s the leading dimension of the array A, and LDA >= max(1,N).

E

E is a REAL array, dimension (N).

On exit, it contains 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, and E(1) is set to 0;
• If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, and E(N) is set to 0.

NOTE: For a 1-by-1 diagonal block D(k), where 1 <= k <= N, the element E(k) is set to 0 in both the UPLO = 'U' and UPLO = 'L' cases. 💖

IPIV

IPIV is an INTEGER array, dimension (N). It describes the permutation matrix P in the factorization of matrix A like this: the absolute value of IPIV(k) represents the index of the row and column that were interchanged with the k-th row and column. The value of UPLO describes the order in which the interchanges were applied. Also, the sign of IPIV represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, which correspond to 1 or 2 interchanges at each factorization step. For more info, see the Further Details section. 🦋

If UPLO = 'U' (in factorization order, k decreases from N to 1):

• a) A single positive entry IPIV(k) > 0 means D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N); if IPIV(k) = k, no interchange occurred.
• b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k-1) < 0 means D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs.)
  • 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
  • 2) If -IPIV(k-1) != k-1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k-1) = k-1, no interchange occurred.
• c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
• d) NOTE: Any entry IPIV(k) is always NONZERO on output.

If UPLO = 'L' (in factorization order, k increases from 1 to N):

• a) A single positive entry IPIV(k) > 0 means D(k,k) is a 1-by-1 diagonal block. If IPIV(k) != k, rows and columns k and IPIV(k) were interchanged in the matrix A(1:N,1:N). If IPIV(k) = k, no interchange occurred.
• b) A pair of consecutive negative entries IPIV(k) < 0 and IPIV(k+1) < 0 means D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (NOTE: negative entries in IPIV appear ONLY in pairs.)
  • 1) If -IPIV(k) != k, rows and columns k and -IPIV(k) were interchanged in the matrix A(1:N,1:N). If -IPIV(k) = k, no interchange occurred.
  • 2) If -IPIV(k+1) != k+1, rows and columns k-1 and -IPIV(k-1) were interchanged in the matrix A(1:N,1:N). If -IPIV(k+1) = k+1, no interchange occurred.
• c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
• d) NOTE: Any entry IPIV(k) is always NONZERO on output.

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 🫶

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

🚧 Further Details 🚧

TODO: put further details

Yes bestie, that’s literally what it says. 🙈

🌈 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
• 01-01-96 - Based on modifications by J. Lewis, Boeing Computer Services Company A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA

Definition at line 240 of file ssytf2_rk.f.

🫶 Author (Doxygen-generated) 🫶

Generated automatically by Doxygen for LAPACK from the source code.

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