DDTTRF(l)

💖 DDTTRF — bestie, we're doing an LU factorization 💖

okay so this is DDTTRF from the LAPACK crew (a modified LAPACK routine, if we're being precise, version 2.0, dated 12 May 1997 — vintage, iconic, no notes). She lives in section l of the manual, and she computes an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting. Yes, you read that right. No pivoting. We're raw-dogging this factorization. Buckle up 🎀

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✨ SYNOPSIS ✨

SUBROUTINE DDTTRF( N, DL, D, DU, INFO )

    INTEGER        INFO, N

    DOUBLE         PRECISION D( * ), DL( * ), DU( * )

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🌸 PURPOSE 🌸

DDTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination without partial pivoting. (Say it with me: no. partial. pivoting. 💅)

The factorization has the form

A = L * U

where L is a product of unit lower bidiagonal matrices, and U is upper triangular with nonzeros in only the main diagonal and first superdiagonal. So like… L is doing the bare minimum structurally, and U is keeping it tight. main character energy from both of them honestly 👑

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🦋 ARGUMENTS 🦋

• N (input) INTEGER — the order of the matrix A. Constraint check bestie: N >= 0. That's it. That's the rule.
• DL (input/output) COMPLEX array, dimension (N-1) — on entry, DL must contain the (n-1) subdiagonal elements of A. On exit, DL gets overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A. So she comes in as your data and leaves as somebody else's data. glow up 💫
• D (input/output) COMPLEX array, dimension (N) — on entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A. Same deal, same glow up.
• DU (input/output) COMPLEX array, dimension (N-1) — on entry, DU must contain the (n-1) superdiagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first superdiagonal of U. bye original values, hope you had a nice time ✌️
• INFO (output) INTEGER — the vibe check on how it all went down:
  • = 0: successful exit. slay. everything ate 🎉
  • < 0: if INFO = -i, the i-th argument had an illegal value. not me getting called out on argument number i 😭
  • > 0: if INFO = i, U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations. 🚨

⚠️ bestie, listen to me for one second: that > 0 case is not a cute little warning, it's a real one. U is exactly singular. If you take this factorization and try to solve a system with it, you are going to divide by zero. That is a 💀 moment, not a 🌸 moment. Check INFO before you use anything, fr fr.

🙈 also lowkey worth saying out loud: there's no partial pivoting here. That's the whole design of this routine, it's not a bug, but it does mean the factorization is what it is — and if U(i,i) hits exactly zero, you get told about it via INFO, and that's your lot. No rescue pivot incoming.

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💜 and that's the whole page, bestie! this was DDTTRF(l), a modified LAPACK routine, LAPACK routine version 2.0, dated 12 May 1997. go forth and factorize responsibly 🫶✨