SSPTRD(3)

💅 SSPTRD(3) — like, the whole glow-up where a symmetric matrix becomes tridiagonal

okay so first things first, bestie: this page lives in the Library Functions Manual under the name ssptrd.f(3) (the full path situation is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/ssptrd.f, and yes, they truncated it in the header to /home/abui...RC/ssptrd.f(3) — it's giving "we ran out of room at the function name, periodt"). Anyway. Let's get into it 🎀

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✨ NAME ✨

SSPTRD — the routine that 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.* no cap, that's literally the whole serve 🔥

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🦋 SYNOPSIS 🦋

Functions/Subroutines

• subroutine SSPTRD (uplo, n, ap, d, e, tau, info) — the one and only, the main character herself 💅

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👑 Function/Subroutine Documentation 👑

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

Just SSPTRD. That's it. That's the name. Iconic behaviour, honestly ✨

🌸 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.

Yeah so she takes your packed-away symmetric matrix A, does a whole orthogonal similarity transformation moment, and hands you back a tridiagonal T. It's giving makeover montage 🧚

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⭐ Parameters ⭐

💜 UPLO

!> UPLO is CHARACTER*1

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

Translation: which half of the matrix are you handing her? Upper or lower. Pick one, she'll slay either way 🫶

🍓 N

!> N is INTEGER

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

Zero is allowed, bestie, don't stress. Negative is not the vibe.

🌈 AP

!> AP is REAL 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.

Lowkey the most important thing here: AP gets overwritten. Like, she moves in and redecorates. You don't get your original A back. You get T's diagonal + first off-diagonal, plus the reflector data (with TAU) that encodes Q. The exact index maths is above — copy it, don't vibe-guess it 😭

🦋 D

!> D is REAL array, dimension (N)

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

The main-character diagonal. Fresh out of the transformation 🎀

💖 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'.

The little side pieces of the tridiagonal. Note the dimension is (N-1), because obviously a tridiagonal has one fewer off-diagonal than diagonal. Math said so 💅

🧚 TAU

!> TAU is REAL array, dimension (N-1)

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

These are the tau scalars, and they're how Q gets reconstructed later. Keep them safe, they're the receipts 🧾

🔥 INFO

!> INFO is INTEGER

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

So 0 = we ate and left no crumbs. Negative = somebody passed something cursed, and she's telling you which argument by index. Very considerate of her, honestly 🫶

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🌟 Author 🌟

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

Four whole institutions. Group project but make it actually work ✨

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🌸 Further Details: 🌸

okay this is the part where she explains exactly how Q is built, and you're gonna want to actually read it because it's the whole recipe:

!> 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).

Notice the order is reversed between the two cases — H(n-1) . . . H(2) H(1) for 'U' vs H(1) H(2) . . . H(n-1) for 'L'. Not me almost missing that. Don't be me. Read it twice 🙈

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⭐ Definition at line 149 of file ssptrd.f. ⭐

That's literally where she lives. Line one-four-nine. 🏠

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💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code.

Not a human. A robot did this. And honestly? She ate 🤖✨

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ok bestie that's the whole page, every flag, every index formula, every tau, preserved with love 💖 go forth and reduce your symmetric packed matrices

LAPACK · Version 3.12.0 · signed off with a kiss from /home/abui...RC/ssptrd.f(3) 🫶🎀