ZSYTF2(3)

🌸 zsytf2.f(3) — LAPACK Library Functions Manual 🌸

okay bestie, this page is zsytf2.f(3) from the LAPACK Library Functions Manual, and it's giving serious "I do linear algebra and I know it" energy. Let's get into it 💅

---

✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zsytf2.f

---

🎀 SYNOPSIS 🎀

🦋 Functions/Subroutines 🦋

• subroutine ZSYTF2 (uplo, n, a, lda, ipiv, info) — ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

---

💅 Function/Subroutine Documentation 💅

✨ subroutine ZSYTF2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info) ✨

ZSYTF2 computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).

yes, the synopsis says real symmetric indefinite and the Purpose says complex symmetric — both are literally what the page says, no cap 🫶

🌟 Purpose 🌟

okay so here's the whole vibe: ZSYTF2 computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method:

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 🔥

---

💖 Parameters 💖

• UPLO — UPLO is CHARACTER*1. Specifies whether the upper or lower triangular part 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.
• 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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• IPIV — IPIV is INTEGER array, dimension (N). Details of the interchanges and the block structure of D.
  • 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.
  • = 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 🔥

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).

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).

---

🧚 Contributors 🧚

09-29-06 - patch from
  Bobby Cheng, MathWorks

  Replace l.209 and l.377
       IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
  by
       IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN

1-96 - Based on modifications by J. Lewis, Boeing Computer Services
      Company

Definition at line 190 of file zsytf2.f.

---

💀 Author 💀

Generated automatically by Doxygen for LAPACK from the source code.

---

bye bestie 💖 — signing off from LAPACK Version 3.12.0, page path /home/abui...RC/zsytf2.f(3). That's the footer energy, periodt 🎀✨