SRC/ssytrf.f(3)

🌸 NAME

SRC/ssytrf.f — okay bestie, this is the page for the ssytrf subroutine from the Library Functions Manual. It’s giving “symmetric matrix factorization main character” energy, and honestly? Iconic. 💅

✨ SYNOPSIS

📚 Functions/Subroutines

• subroutine ssytrf (uplo, n, a, lda, ipiv, work, lwork, info) — the one and only SSYTRF, here to factor a real symmetric matrix.

💖 Function/Subroutine Documentation

🌈 subroutine ssytrf

subroutine ssytrf (character uplo, integer n, real, dimension( lda, * ) a,
    integer lda, integer, dimension( * ) ipiv, real, dimension( * ) work,
    integer lwork, integer info)

SSYTRF — yes, that’s the name, no cap.

🎀 Purpose

SSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is

A = U**T*D*U  or  A = L*D*L**T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.

🌟 Parameters

• UPLO — CHARACTER*1
  • = 'U': Upper triangle of A is stored;
  • = 'L': Lower triangle of A is stored.
• N — INTEGER. The order of the matrix A. N >= 0.
• A — REAL 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details). 💖
• LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• IPIV — INTEGER array, dimension (N). Details of the interchanges and the block structure of D.
  • If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block.
  • If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
  • If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
• WORK — REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — INTEGER. The length of WORK. LWORK >=1. For best performance LWORK >= N*NB, where NB is the block size returned by ILAENV.
  • If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, 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 = -i, the i-th argument had an illegal value
  • > 0: if INFO = i, D(i,i) is exactly zero. 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.

👑 Author

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

🔍 Further Details

If UPLO = 'U', then A = U**T*D*U, where

   U = P(n)*U(n)* ... *P(k)U(k)* ...,

i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    v    0   )   k-s
   U(k) =  (   0    I    0   )   s
           (   0    0    I   )   n-k
              k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = 'L', then A = L*D*L**T, where

   L = P(1)*L(1)* ... *P(k)*L(k)* ...,

i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    0     0   )  k-1
   L(k) =  (   0    I     0   )  s
           (   0    v     I   )  n-k-s+1
              k-1   s  n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

📌 Definition

Definition at line 181 of file ssytrf.f.

👑 Author

Generated automatically by Doxygen for LAPACK from the source code.

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💖 And that’s the whole vibe, bestie — this page is from LAPACK Version 3.12.0, source file SRC/ssytrf.f(3). Slay. 🫶