SRC/zhptrf.f(3)

✨ SRC/zhptrf.f ✨

okay so bestie, this is the manual page for SRC/zhptrf.f and it's literally a LAPACK legend. buckle up 💖

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🎀 SYNOPSIS 🎀

Functions/Subroutines

• subroutine zhptrf (uplo, n, ap, ipiv, info) — ZHPTRF

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🦋 Function/Subroutine Documentation 🦋

subroutine zhptrf (character uplo, integer n, complex*16, dimension( * ) ap, integer, dimension( * ) ipiv, integer info)

ZHPTRF — the main character, no cap 👑

🌸 Purpose:

!>
!> ZHPTRF computes the factorization of a complex Hermitian packed
!> matrix A using the 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, and D is Hermitian and block diagonal with
!> 1-by-1 and 2-by-2 diagonal blocks.
!>

like, it takes your Hermitian packed matrix and factorizes it with the Bunch-Kaufman diagonal pivoting method. it's giving math girlboss 💅

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💜 Parameters

UPLO

!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored.
!>

which triangle are we storing, bestie? 'U' for the upper one, 'L' for the lower one. that's the whole vibe check 🌈

N

!>          N is INTEGER
!>          The order of the matrix A.  N >= 0.
!>

the order of matrix A. and yes, N >= 0 — don't come at it with negative numbers 🙈

AP

!>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
!>          On entry, the upper or lower triangle of the Hermitian matrix
!>          A, packed columnwise in a linear array.  The j-th column of A
!>          is stored in the array AP as follows:
!>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
!>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
!>
!>          On exit, the block diagonal matrix D and the multipliers used
!>          to obtain the factor U or L, stored as a packed triangular
!>          matrix overwriting A (see below for further details).
!>

COMPLEX*16 array of dimension (N*(N+1)/2) — packed columnwise, iconic. on entry it's your triangle of the Hermitian matrix A; on exit the block diagonal matrix D and the multipliers just casually overwrite A. she ate and left no crumbs 🔥

IPIV

!>          IPIV is 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.
!>

INTEGER array of dimension (N) — all the tea about the interchanges and the block structure of D ☕ read it carefully, iykyk.

INFO

!>          INFO is 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.
!>

the receipts 💀 0 means slay, < 0 means argument -i was illegal, and > 0 means D(i,i) is exactly zero — the factorization finished but D is exactly singular, so division by zero happens if you use it to solve a system. be so careful, fr fr.

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👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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⭐ 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).
!>

this is the deep lore, bestie — the whole decomposition laid out with the permutation matrices, the unit triangular matrices, and exactly which entries of A get overwritten. read it twice, it's a lot but it's worth it 🧚

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🌟 Contributors:

J. Lewis, Boeing Computer Services Company

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Definition at line 158 of file zhptrf.f.

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💖 Author

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