SRC/chptrf.f(3)

💖 SRC/chptrf.f 💖

okay bestie, welcome to the Library Functions Manual page for SRC/chptrf.f. this is the CHPTRF era, and we're doing complex Hermitian packed matrix factorization with full main-character energy. no cap. 💅

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✨ NAME ✨

SRC/chptrf.f

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🌸 SYNOPSIS 🌸

🌼 Functions/Subroutines 🌼

• subroutine chptrf (uplo, n, ap, ipiv, info) — CHPTRF

subroutine chptrf (uplo, n, ap, ipiv, info)
       CHPTRF

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💅 Function/Subroutine Documentation 💅

subroutine chptrf (character uplo, integer n, complex, dimension( * ) ap,
       integer, dimension( * ) ipiv, integer info)
       CHPTRF

🎀 Purpose 🎀

okay so CHPTRF 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. periodt. 💖

💜 Parameters 💜

UPLO — bestie, UPLO is CHARACTER*1, and it tells you which triangle is living its truth:

• = 'U': Upper triangle of A is stored;
• = 'L': Lower triangle of A is stored.

N — N is INTEGER. it's the order of the matrix A, and N >= 0. that's the whole vibe check. ✨

AP — AP is COMPLEX array, dimension (N*(N+1)/2).

on entry, it's 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 are stored as a packed triangular matrix overwriting A — see below for further details, bestie. it's giving packed triangular realness. 🦋

IPIV — IPIV is INTEGER array, dimension (N). it has the 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.

INFO — INFO is INTEGER, and this is the tea ☕:

• = 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.

⚠️ bestie, that > 0 case is not a cute little glitch. D(i,i) being exactly zero means D is exactly singular, and division by zero will occur if it's used to solve a system of equations. literally do not ignore it. 😭

👑 Author 👑

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

🔥 Further Details 🔥

okay so 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).

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and 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 🌟

J. Lewis, Boeing Computer Services Company

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Definition at line 158 of file chptrf.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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and that's the whole page, bestie! the footer says LAPACK Version 3.12.0 and SRC/chptrf.f(3) — so that's the source/version sign-off, and yes, the Doxygen-generated author credit is right there too. slay. 💖✨🎀