SLAHILB(3)

✨ SLAHILB(3) ✨ — the scaled Hilbert matrix generator, bestie edition 💅

okay so you found SLAHILB(3) in the LAPACK manual — it lives at /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/MATGEN/slahilb.f and it's literally the main character of matrix generators. no cap. 🎀

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🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine SLAHILB (n, nrhs, a, lda, x, ldx, b, ldb, work, info)

SLAHILB — she's the whole function, periodt.

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💖 Function/Subroutine Documentation 💖

subroutine SLAHILB (integer n, integer nrhs, real, dimension(lda, n) a,
    integer lda, real, dimension(ldx, nrhs) x, integer ldx, real,
    dimension(ldb, nrhs) b, integer ldb, real, dimension(n) work, integer
    info)

SLAHILB 🫶

Purpose:

!>
!> SLAHILB generates an N by N scaled Hilbert matrix in A along with
!> NRHS right-hand sides in B and solutions in X such that A*X=B.
!>
!> The Hilbert matrix is scaled by M = LCM(1, 2, ..., 2*N-1) so that all
!> entries are integers.  The right-hand sides are the first NRHS
!> columns of M * the identity matrix, and the solutions are the
!> first NRHS columns of the inverse Hilbert matrix.
!>
!> The condition number of the Hilbert matrix grows exponentially with
!> its size, roughly as O(e ** (3.5*N)).  Additionally, the inverse
!> Hilbert matrices beyond a relatively small dimension cannot be
!> generated exactly without extra precision.  Precision is exhausted
!> when the largest entry in the inverse Hilbert matrix is greater than
!> 2 to the power of the number of bits in the fraction of the data type
!> used plus one, which is 24 for single precision.
!>
!> In single, the generated solution is exact for N <= 6 and has
!> small componentwise error for 7 <= N <= 11.
!>

so basically: she builds you an N × N scaled Hilbert matrix in A, plus NRHS right-hand sides in B and the matching solutions in X, all satisfying A*X=B. iconique. 🔥

the scaling factor is M = LCM(1, 2, ..., 2*N-1) so every entry stays an integer — no messy fractions, we love a clean queen. 💅 the right-hand sides are the first NRHS columns of M * the identity matrix, and the solutions are the first NRHS columns of the inverse Hilbert matrix.

⚠️ real talk: the condition number of the Hilbert matrix grows exponentially with N, roughly O(e ** (3.5*N)). and inverse Hilbert matrices past a relatively small dimension cannot be generated exactly without extra precision. precision is exhausted when the largest entry in the inverse Hilbert matrix is greater than 2 to the power of the number of bits in the fraction of the data type used plus one — which is 24 for single precision. 😭

in single precision, the generated solution is exact for N <= 6 and has small componentwise error for 7 <= N <= 11. so cute, so fragile. 🦋

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🌸 Parameters 🌸

• N — N is INTEGER. the dimension of the matrix A. slay. 👑
• NRHS — NRHS is INTEGER. the requested number of right-hand sides. the more the merrier, bestie. 💖
• A — A is REAL array, dimension (LDA, N). the generated scaled Hilbert matrix. she's the moment. ✨
• LDA — LDA is INTEGER. the leading dimension of the array A. LDA >= N. keep it valid or it's giving illegal. 💀
• X — X is REAL array, dimension (LDX, NRHS). the generated exact solutions. currently, the first NRHS columns of the inverse Hilbert matrix. lowkey iconic. 🎀
• LDX — LDX is INTEGER. the leading dimension of the array X. LDX >= N. periodt. 🫶
• B — B is REAL array, dimension (LDB, NRHS). the generated right-hand sides. currently, the first NRHS columns of LCM(1, 2, ..., 2*N-1) times the identity matrix. no cap. 🌈
• LDB — LDB is INTEGER. the leading dimension of the array B. LDB >= N. don't play with it. 💅
• WORK — WORK is REAL array, dimension (N). just vibes and workspace. 🧚
• INFO — INFO is INTEGER.
  • = 0: successful exit. we ate. 🍓
  • = 1: N is too large; the data is still generated but may not be not exact. (yes, the original says "may not be not exact" — we're just relaying the message, bestie. 🙈)
  • < 0: if INFO = -i, the i-th argument had an illegal value. red flag fr fr. 💀

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👑 Author 👑

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

📌 (definition at line 123 of file slahilb.f.)

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💜 Author (of this doc) 💜

Generated automatically by Doxygen for LAPACK from the source code. our little documentation fairy did that. 🧚✨

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LAPACK — Version 3.12.0 — and the source file path to remember: /home/abui...N/slahilb.f(3) 🌸

okay bye bestie, go generate some scaled Hilbert matrices responsibly. 🫶💖🎀