SRC/dsptrf.f(3)

SRC/dsptrf.f(3) 💖 — from the Library Functions Manual, bestie

Okay so this is the DSPTRF page, and it’s giving LAPACK main character energy. We’re keeping every single detail, no cap. 💅

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NAME ✨

SRC/dsptrf.f — literally the name, bestie. This is the page for the DSPTRF routine, and we’re about to go through it like the responsible Unix bestie I am. 🫶

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SYNOPSIS 🦋

Functions/Subroutines 🧩

subroutine dsptrf (uplo, n, ap, ipiv, info)
    DSPTRF

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Function/Subroutine Documentation 🌸

Here’s the full signature, copied exactly because we do not mess with code, periodt:

subroutine dsptrf (character uplo, integer n, double precision, dimension( *
       ) ap, integer, dimension( * ) ipiv, integer info)
       DSPTRF

Purpose 🎯

DSPTRF computes the factorization of a real 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.

Parameters 💅

Okay bestie, parameters time — every one of them, in order, no skipping:

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 DOUBLE PRECISION 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.

Author 🫶

Shoutout to the authors, bestie:

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

Further Details 🔍

Okay this part is the deep lore, and it’s lowkey iconic. 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 🧚

J. Lewis, Boeing Computer Services Company — thank you for your service, icon. 💖

📍 Definition at line 158 of file dsptrf.f.

Author (the generator one) 💻

Generated automatically by Doxygen for LAPACK from the source code.

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💖 Cute sign-off time: this page is from LAPACK Version 3.12.0, source SRC/dsptrf.f(3). The footer didn’t give me a date, so I’m not inventing one — byeee, bestie! 💅✨