# zlahilb.f - Man Page

TESTING/MATGEN/zlahilb.f

## Synopsis

### Functions/Subroutines

subroutine zlahilb (n, nrhs, a, lda, x, ldx, b, ldb, work, info, path)
ZLAHILB

## Function/Subroutine Documentation

### subroutine zlahilb (integer n, integer nrhs, complex*16, dimension(lda,n) a, integer lda, complex*16, dimension(ldx, nrhs) x, integer ldx, complex*16, dimension(ldb, nrhs) b, integer ldb, double precision, dimension(n) work, integer info, character*3 path)

ZLAHILB

Purpose:

ZLAHILB 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 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

NAG Ltd.

Definition at line 132 of file zlahilb.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zlahilb(3) is an alias of zlahilb.f(3).

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK