hetf2_rook(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💖 NAME 💖

Manual name: hetf2_rook(3) — Library Functions Manual. The page is hetf2_rook and the family it’s documenting is {he,sy}tf2_rook: triangular factor, level 2. Bestie, this is the LAPACK page for the rook-pivoting triangular factorization routines — no cap, it’s a whole little squad of them. 💅

✨ SYNOPSIS ✨

Here’s the function lineup, with every signature exactly as it appears. 🫶

  • subroutine chetf2_rook (uplo, n, a, lda, ipiv, info) — computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).
  • subroutine csytf2_rook (uplo, n, a, lda, ipiv, info) — computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).
  • subroutine dsytf2_rook (uplo, n, a, lda, ipiv, info) — computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).
  • subroutine ssytf2_rook (uplo, n, a, lda, ipiv, info) — computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).
  • subroutine zhetf2_rook (uplo, n, a, lda, ipiv, info) — computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).
  • subroutine zsytf2_rook (uplo, n, a, lda, ipiv, info) — computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).

🌸 DETAILED DESCRIPTION 🌸

No extra preamble here, bestie — it goes straight into the function docs. ✨

🎀 FUNCTION DOCUMENTATION 🎀

🦋 subroutine chetf2_rook

subroutine chetf2_rook (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

CHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).

💖 Purpose:

Okay so this icon computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:

A = U*D*U**H  or  A = L*D*L**H

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**H is the conjugate transpose of U, and D is Hermitian 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. Literally the unblocked one, periodt. 💅

🌸 Parameters

  • UPLO — UPLO is CHARACTER*1. It specifies whether the upper or lower triangular part of the Hermitian 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 array, dimension (LDA,N).
    • On entry, the Hermitian 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) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, 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) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, 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.
Bestie, that > 0 case is the “it finished, but the math is about to get real messy” warning. Division by zero is giving chaos. 💀

🌟 Author

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

🔥 Further Details:

If UPLO = 'U', then A = U*D*U**H, 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**H, 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:

  • November 2013, 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, USA

Definition at line 193 of file chetf2_rook.f.

🦋 subroutine csytf2_rook

subroutine csytf2_rook (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

CSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm).

💖 Purpose:

This one computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman ("rook") 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. Same energy as before, but for the complex symmetric case. ✨

🌸 Parameters

  • UPLO — UPLO is CHARACTER*1. It 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 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

so IPIV? yeah it's an INTEGER array, dimension (N) and it's literally the receipts ✨ — it holds all the deets about the interchanges and the block structure of D. bestie, read it carefully because it's giving lore:

  • 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) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. two-for-one, iconic 🎀
  • 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. same energy, different triangle 🦋
    • if IPIV(k) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. going down this time 🫶
  • INFO — INFO is INTEGER, and it's the vibe check at the end 👑
    • = 0: successful exit, we ate, no crumbs 🍓
    • < 0: if INFO = -k, the k-th argument had an illegal value — so somebody passed something unserious 😭
    • > 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. bestie that's a whole crash, do NOT ignore it 💀
so yeah, INFO > 0 isn't an error in running the routine — the factorization finished — but the matrix D is straight-up singular. solving with it = division by zero. no cap. 🙈

👑 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). one little cell, slay 💅
  • 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). two cells plus their little corner, still 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) 🌈

🫶 Contributors:

  • November 2013, 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 193 of file csytf2_rook.f.

💜 subroutine dsytf2_rook ( character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info ) 💜

subroutine dsytf2_rook (character uplo, integer n, double precision,
    dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv,
    integer info)

DSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm). okay double-precision bestie, this is your moment 🧚

⭐ Purpose:

  • DSYTF2_ROOK computes the factorization of a real symmetric matrix A using the bounded 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. no blocking, just vibes 🔥

🌸 Parameters

  • UPLO — UPLO is CHARACTER*1, and it 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. (yes, zero is allowed, we don't judge 🫶)
  • A — A is DOUBLE PRECISION 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. so only half the matrix is even invited to the party 💅
    • on exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details). it comes back changed, main character energy ✨
  • LDA — LDA is INTEGER, the leading dimension of the array A. LDA >= max(1,N). don't skimp on it, fr fr
  • 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) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, 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) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, 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. again: singular D, zero division, chaos 💀

👑 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:

  • November 2013, 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 193 of file dsytf2_rook.f.

🦋 subroutine ssytf2_rook ( character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info ) 🦋

subroutine ssytf2_rook (character uplo, integer n, real, dimension( lda, * )
    a, integer lda, integer, dimension( * ) ipiv, integer info)

SSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm). single precision edition, same drama, half the bytes 💖

⭐ Purpose:

okay so SSYTF2_ROOK is the real-symmetric bestie and it computes the factorization of a real symmetric matrix A using the bounded 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. no cap, it’s serving unblocked realness 💅

🌸 Parameters

  • UPLO — CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • 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 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) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, 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) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • INFO — 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:

  • November 2013, 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 193 of file ssytf2_rook.f.

🦋 zhetf2_rook — complex Hermitian edition

subroutine zhetf2_rook (character uplo, integer n, complex*16, dimension(
    lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZHETF2_ROOK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm). she’s giving complex, she’s giving Hermitian, she’s giving unblocked 💜

✨ Purpose:

ZHETF2_ROOK computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman () diagonal pivoting method:

A = U*D*U**H  or  A = L*D*L**H

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, U**H is the conjugate transpose of U, and D is Hermitian 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 — CHARACTER*1. Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
    • = 'U': Upper triangular
    • = 'L': Lower triangular
  • N — INTEGER. The order of the matrix A. N >= 0.
  • A — COMPLEX*16 array, dimension (LDA,N).
    • On entry, the Hermitian 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 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) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged, 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) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • INFO — 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:

okay bestie, back to zhetf2_rook for the final lore drop 💅 the factorization tea continues, and yes every overwrite rule is coming with us.

✨ Further Details ✨

If UPLO = 'U', then A = U*D*U**H, 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**H, 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 🫶

  • November 2013 — 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, USA

Definition at line 193 of file zhetf2_rook.f.

zsytf2_rook 🦋

   subroutine zsytf2_rook (character uplo, integer n, complex*16, dimension(
       lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)

ZSYTF2_ROOK computes the factorization of a complex symmetric indefinite matrix using the bounded Bunch-Kaufman "rook" diagonal pivoting method (unblocked algorithm).

Purpose 🎯

ZSYTF2_ROOK computes the factorization of a complex symmetric matrix A using the bounded 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 — CHARACTER*1. Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
    • 'U': Upper triangular
    • 'L': Lower triangular
  • N — INTEGER. The order of the matrix A. N >= 0.
  • A — 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
bestie, do not play with LDA — it has to be at least max(1,N) 💀
  • IPIV — 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) < 0 and IPIV(k-1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k-1 and -IPIV(k-1) were inerchaged — yes, that is the exact source spelling 😭 — 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) < 0 and IPIV(k+1) < 0, then rows and columns k and -IPIV(k) were interchanged and rows and columns k+1 and -IPIV(k+1) were inerchaged — again, exact source spelling 😭 — and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
  • INFO — 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.
INFO > 0 is the “we finished, but D is singular” warning — solving with it is gonna divide by zero, no cap 💀

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 🫶

  • November 2013 — 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 — yep, abd is literally in the source 😭

Definition at line 193 of file zsytf2_rook.f.

Author ✨

Generated automatically by Doxygen for LAPACK from the source code.

okay that is the whole hetf2_rook(3) saga, bestie 💖 the footer says source LAPACK, version 3.12.0, and the date/tag line is hetf2_rook(3) — mwah, go factor something safely 🫶✨

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