SRC/csptrf.f(3)

💅 SRC/csptrf.f 💅

okay bestie, welcome to the Library Functions Manual page for SRC/csptrf.f — header folded in, vibes immaculate. ✨

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🌸 NAME 🌸

SRC/csptrf.f — the file, the function source, the main character. 💖

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

• subroutine csptrf (uplo, n, ap, ipiv, info) — CSPTRF. That's the whole call, no cap.

🦋 Function/Subroutine Documentation 🦋

💅 subroutine csptrf 💅

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

CSPTRF. periodt.

🌟 Purpose 🌟

okay so CSPTRF computes the factorization of a complex symmetric matrix A stored in packed format 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, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. literally the whole glow-up. 🔥

🌈 Parameters 🌈

🎀 UPLO 🎀

UPLO is CHARACTER*1.

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

🎀 N 🎀

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

🎀 AP 🎀

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

On entry, the upper or lower triangle of the symmetric 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). 💖

🎀 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.

🎀 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.

⚠️ bestie, that last one is not a drill: D(i,i) being exactly zero means 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 🧚

5-96 - Based on modifications by J. Lewis, Boeing Computer Services Company

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). 🫶

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Definition at line 157 of file csptrf.f.

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code.

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okay that's the page, bestie. final sign-off from LAPACK Version 3.12.0 and SRC/csptrf.f(3) — mwah 💋✨