# 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

Univ. of Colorado Denver

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