SRC/ssptrd.f(3)

💅 SRC/ssptrd.f 💅

okay so bestie, this page is literally SRC/ssptrd.f, straight out of the Library Functions Manual, and it's here to explain one very specific LAPACK subroutine. Buckle up 🎀

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🌸 NAME 🌸

SRC/ssptrd.f — that's the file, that's the name, periodt 💅

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✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine ssptrd (uplo, n, ap, d, e, tau, info)
    SSPTRD

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🎀 Function/Subroutine Documentation 🎀

okay so we're talking about this subroutine, and it's giving main-character Fortran energy:

subroutine ssptrd (character uplo, integer n, real, dimension( * ) ap, real,
     dimension( * ) d, real, dimension( * ) e, real, dimension( * ) tau,
     integer info)

SSPTRD

💖 Purpose 💖

SSPTRD 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 vibe: packed symmetric matrix goes in, tridiagonal form comes out, orthogonal similarity transformation does the slaying ✨

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🧸 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 REAL array, dimension (N*(N+1)/2).

On entry, it holds 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 overwritten 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. 👀

D

D is REAL array, dimension (N).

• The diagonal elements of the tridiagonal matrix T:
• D(i) = A(i,i).

E

E is REAL 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 REAL 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.

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👑 Author 👑

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

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🔍 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 at line 149 of file ssptrd.f.

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🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code.

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That's the whole page, bestie — LAPACK Version 3.12.0, SRC/ssptrd.f(3), and we are so done 💅✨🦋