CLAHILB(3)

📦 version 3.12.0, in opensuse-leap 16.0

CLAHILB(3) is here, bestie, and it’s serving from the Library Functions Manual 💖 — the page path up top is /home/abui...N/clahilb.f(3).

🎀 NAME 🎀

The NAME line is literally a whole path, no cute one-word nickname: /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/MATGEN/clahilb.f.

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

  • subroutine CLAHILB (n, nrhs, a, lda, x, ldx, b, ldb, work, info, path) — CLAHILB.
subroutine CLAHILB (n, nrhs, a, lda, x, ldx, b, ldb, work, info, path)
    CLAHILB

🌸 Function/Subroutine Documentation 🌸

subroutine CLAHILB (integer n, integer nrhs, complex, dimension(lda,n) a,
    integer lda, complex, dimension(ldx, nrhs) x, integer ldx, complex,
    dimension(ldb, nrhs) b, integer ldb, real, dimension(n) work, integer
    info, character*3 path)

CLAHILB

💜 Purpose 💜

Okay so CLAHILB 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. Iconic linear algebra behavior, honestly.

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 COMPLEX 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 COMPLEX 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.
  • < 0: if INFO = -i, the i-th argument had an illegal value.

PATH

PATH is CHARACTER*3. The LAPACK path name.

🫶 Author 🫶

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

Definition at line 132 of file clahilb.f.

👑 Author 👑

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