CSPTRF(3)

💅 CSPTRF(3) — the one where we factor a complex symmetric matrix, bestie

okay so this is a page from the Library Functions Manual, and it's all about CSPTRF, a LAPACK subroutine. it lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/csptrf.f and honestly? she's iconic. let's get into it 🫶

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✨ SYNOPSIS ✨

Functions/Subroutines 🌸

subroutine CSPTRF (uplo, n, ap, ipiv, info)
    CSPTRF

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🦋 Function/Subroutine Documentation 🦋

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

🎀 Purpose 🎀

!> !> CSPTRF computes the factorization of a complex symmetric matrix A !> stored in packed format using the Bunch-Kaufman diagonal pivoting !> method: ! !> A = U*D*UT or A = L*D*LT ! !> 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. !

like, literally the whole vibe: Bunch-Kaufman diagonal pivoting, packed symmetric complex matrix, and you get A = U*D*U**T or A = L*D*L**T out of it. U (or L) is a product of permutation and unit upper (or lower) triangular matrices, and D is symmetric + block diagonal with 1-by-1 and 2-by-2 diagonal blocks. slay 🔥

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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 of A we're keeping, bestie. 'U' = upper triangle stored, 'L' = lower triangle stored. pick one, no cap 💅

N

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

the order of the matrix A. must be N >= 0 — zero is allowed, negative is not, don't try it 🙈

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). !

COMPLEX array, dimension (N*(N+1)/2). on entry it's the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array — the j-th column of A goes in like this:

• 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

and on exit? it gets overwritten 😭 the block diagonal matrix D plus the multipliers used to get U or L live there now, stored as a packed triangular matrix overwriting A (more deets below, keep reading).

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, dimension (N). this is the receipts on 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. !

INTEGER, the vibe check at the end:

• = 0: successful exit, we ate 💖
• < 0: if INFO = -i, the i-th argument had an illegal value — someone passed a bad arg, tragic
• > 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's used to solve a system of equations 💀 that's a whole situation, be careful

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👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

(the whole academic Avengers lineup, we stan 🫶)

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🌈 Further Details: 🌈

!> !> 5-96 - Based on modifications by J. Lewis, Boeing Computer Services !> Company ! !> If UPLO = 'U', then A = U*D*UT, 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 = LD*LT, 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). !

first, the lore drop: 5-96 - Based on modifications by J. Lewis, Boeing Computer Services Company 🛩️

then, the actual math, which is honestly so much:

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). ate and left no crumbs 🔥

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🧚 Definition 🧚

Definition at line 157 of file csptrf.f.

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ngl this whole page was generated automatically by Doxygen for LAPACK from the source code — so if something looks weird, blame the robot, not me 🙈

okay that's the whole thing, mwah 💜 — LAPACK, Version 3.12.0 🦋 and the file is /home/abui...RC/csptrf.f(3), see u next time bestie ✨🎀