DSPTRD(3)

💅 DSPTRD(3) — Library Functions Manual 💅

okay bestie, this is the DSPTRD(3) page from the Library Functions Manual, and the source path is /home/abui...RC/dsptrd.f(3). Strap in, we're doing packed symmetric matrix lore. 💖

💖 NAME 💖

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dsptrd.f

🌸 SYNOPSIS 🌸

🧸 Functions/Subroutines 🧸

subroutine DSPTRD (uplo, n, ap, d, e, tau, info)
           DSPTRD

📚 Function/Subroutine Documentation 📚

subroutine DSPTRD (character uplo, integer n, double precision, dimension( *
       ) ap, double precision, dimension( * ) d, double precision, dimension( *
       ) e, double precision, dimension( * ) tau, integer info)
       DSPTRD

🎯 Purpose 🎯

DSPTRD reduces a real symmetric matrix A stored in packed form to symmetric tridiagonal form T by an orthogonal similarity transformation:

Q**T * A * Q = T

That's the whole main-character arc, no cap. ✨

✨ 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 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)*(2*n-j)/2) = A(i,j) for j<=i<=n.

On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.

packed storage is lowkey doing the most, but it's iconic. 🫶

🦋 D

D is DOUBLE PRECISION array, dimension (N).

The diagonal elements of the tridiagonal matrix T:

D(i) = A(i,i).

🦋 E

E is DOUBLE PRECISION array, dimension (N-1).

The off-diagonal elements of the tridiagonal matrix T:

E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

🦋 TAU

TAU is DOUBLE PRECISION array, dimension (N-1).

The scalar factors of the elementary reflectors (see Further Details).

🦋 INFO

INFO is INTEGER.

• = 0: successful exit
• < 0: if INFO = -i, the i-th argument had an illegal value

👑 AUTHOR 👑

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

🌈 FURTHER DETAILS 🌈

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors:

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form:

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP, overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form:

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP, overwriting A(i+2:n,i), and tau is stored in TAU(i).

📍 DEFINITION 📍

Definition at line 149 of file dsptrd.f.

🧚 AUTHOR (GENERATED-BY VIBES) 🧚

Generated automatically by Doxygen for LAPACK from the source code.

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okay bestie, final footer tea: this is LAPACK Version 3.12.0, page path /home/abui...RC/dsptrd.f(3) — and yeah, the footer doesn't give a date, just the version and the path, periodt. 💅✨