SLAHILB(3)

💖 NAME

okay so this is SLAHILB(3), and the file/name situation is literally TESTING/MATGEN/slahilb.f — a Library Functions Manual page, no cap. ✨

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✨ SYNOPSIS

Functions/Subroutines:

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

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🦋 Function/Subroutine Documentation

Here’s the full typed-out signature, bestie:

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

okay so 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. 🫶

🌸 Parameters

• N — N is INTEGER; the dimension of the matrix A.
• NRHS — NRHS is INTEGER; the requested number of right-hand sides.
• A — A is REAL array, dimension (LDA, N); the generated scaled Hilbert matrix.
• LDA — LDA is INTEGER; the leading dimension of the array A. LDA >= N.
• X — X is REAL array, dimension (LDX, NRHS); the generated exact solutions. Currently, the first NRHS columns of the inverse Hilbert matrix.
• LDX — LDX is INTEGER; the leading dimension of the array X. LDX >= N.
• 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) * the identity matrix.
• LDB — LDB is INTEGER; the leading dimension of the array B. LDB >= N.
• WORK — WORK is REAL array, dimension (N).
• INFO — INFO is INTEGER; = 0: successful exit; = 1: N is too large; the data is still generated but may not be not exact (the source really says it like that 😭); < 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.

Definition at line 123 of file slahilb.f.

🫶 Author

Generated automatically by Doxygen for LAPACK from the source code.

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mwah 💋 — this page is stamped LAPACK Version 3.12.0, and the footer vibe is TESTING/MATGEN/slahilb.f(3).