SRC/dsytf2.f(3)

💖 SRC/dsytf2.f 💖

Okay so this is the Library Functions Manual page for SRC/dsytf2.f(3) — and she is serving real symmetric matrix factorization energy. ✨

🎀 NAME 🎀

SRC/dsytf2.f

✨ SYNOPSIS ✨

🐚 Functions/Subroutines

• subroutine dsytf2 (uplo, n, a, lda, ipiv, info) — DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). No cap, that's the whole vibe check. 💅

💅 Function/Subroutine Documentation 💅

Here's the full signature, bestie — copy it exactly or the compiler will side-eye you:

subroutine dsytf2 (character uplo, integer n, double precision, dimension(
    lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

And the one-liner again, because she's iconic: DSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

🎯 Purpose

Okay so DSYTF2 takes a real symmetric matrix A and factorizes it with the Bunch-Kaufman diagonal pivoting method. The tea is:

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

Where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**T is the transpose of U, 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. It's giving "small but mighty". 🔥

🧾 Parameters

• UPLO — UPLO is CHARACTER*1. It says which triangle of the symmetric matrix A is stored:
  • = 'U': Upper triangular
  • = 'L': Lower triangular
• N — N is INTEGER. The order of the matrix A. N >= 0. Periodt.
• A — A is DOUBLE PRECISION array, dimension (LDA,N). On entry, it's 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 are chilling in there (see below for further details). 💖
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). Don't skip this one, bestie.
• IPIV — IPIV is INTEGER array, dimension (N). It holds the details of the interchanges and the block structure of D. Here's the breakdown:
  • If UPLO = 'U':
    • 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 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':
    • 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 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.
• INFO — INFO is INTEGER. The exit-status bestie:
  • = 0: successful exit
  • < 0: if INFO = -k, the k-th argument had an illegal value
  • > 0: if INFO = k, D(k,k) 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

Okay, deep lore time. If UPLO = 'U', then:

A = U*D*U**T

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). Slay.

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). Iconic behavior. 💅

🫶 Contributors

• 09-29-06 - patch from
  • Bobby Cheng, MathWorks
  • Replace l.204 and l.372
  • IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
  • by
  • IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
• 01-01-96 - Based on modifications by
  • J. Lewis, Boeing Computer Services Company
  • A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
• 1-96 - Based on modifications by J. Lewis, Boeing Computer Services Company

Definition at line 193 of file dsytf2.f.

👑 Author 👑

And here's the auto-generated bit:

Generated automatically by Doxygen for LAPACK from the source code.

Bye bestie! 💖 That's the whole SRC/dsytf2.f(3) page from LAPACK Version 3.12.0. The footer didn't hand me a separate date, so I'm not inventing one — just source and version, periodt. 🫶