# ?hesvx

Uses the diagonal pivoting factorization to compute the solution to the complex system of linear equations with a Hermitian matrix A, and provides error bounds on the solution.

## Syntax

FORTRAN 77:

call chesvx( fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info )

call zhesvx( fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, lwork, rwork, info )

Fortran 95:

call hesvx( a, b, x [,uplo] [,af] [,ipiv] [,fact] [,ferr] [,berr] [,rcond] [,info] )

## Description

This routine uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations `A*X = B`, where A is an n-by-n Hermitian matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

Error bounds on the solution and a condition estimate are also provided.

The routine ?hesvx performs the following steps:

1. If `fact = 'N'`, the diagonal pivoting method is used to factor the matrix A. The form of the factorization is `A = U*D*UH` or `A = L*D*LH`, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

2. If some `di,i` = 0, so that D is exactly singular, then the routine returns with `info = i`. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, `info = n+1` is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

3. The system of equations is solved for X using the factored form of A.

4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

## Input Parameters

fact

  CHARACTER*1. Must be 'F' or 'N'.

  Specifies whether or not the factored form of the matrix A has been supplied on entry.

  If `fact = 'F'`: on entry, af and ipiv contain the factored form of A. Arrays a, af, and ipiv are not modified.

  If `fact = 'N'`, the matrix A is copied to af and factored.

uplo

  CHARACTER*1. Must be 'U' or 'L'.

  Indicates whether the upper or lower triangular part of A is stored and how A is factored:

  If `uplo = 'U'`, the array a stores the upper triangular part of the Hermitian matrix A, and A is factored as `U*D*UH`.

  If `uplo = 'L'`, the array a stores the lower triangular part of the Hermitian matrix A; A is factored as `L*D*LH`.

n

  INTEGER. The order of matrix A; n ≥ 0.

nrhs

  INTEGER. The number of right-hand sides, the number of columns in `B; nrhs ≥ 0`.

a,af,b,work

  COMPLEX for chesvx

  DOUBLE COMPLEX for zhesvx.

  Arrays: `a(lda,*)`, `af(ldaf,*)`, `b(ldb,*)`, `work(*)`.

  The array a contains the upper or the lower triangular part of the Hermitian matrix A (see uplo). The second dimension of a must be at least `max(1,n)`.

  The array af is an input argument if `fact = 'F'`. It contains he block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization `A = U*D*UH` or `A = L*D*LH` as computed by ?hetrf. The second dimension of af must be at least `max(1,n)`.

  The array b contains the matrix B whose columns are the right-hand sides for the systems of equations. The second dimension of b must be at least `max(1, nrhs)`.

  `work(*)` is a workspace array of dimension at least `max(1, lwork)`.

lda

  INTEGER. The first dimension of a; `lda ≥ max(1, n)`.

ldaf

  INTEGER. The first dimension of af; `ldaf ≥ max(1, n)`.

ldb

  INTEGER. The first dimension of b; `ldb ≥ max(1, n)`.

ipiv

  INTEGER.

  Array, DIMENSION at least `max(1, n)`. The array ipiv is an input argument if `fact = 'F'`. It contains details of the interchanges and the block structure of D, as determined by ?hetrf.

  If `ipiv(i) = k > 0`, then `dii` is a 1-by-1 diagonal block, and the i-th row and column of A was interchanged with the k-th row and column.

  If `uplo = 'U'` and ipiv(i) =ipiv(i-1) = -m < 0, then D has a 2-by-2 block in rows/columns i and `i-1`, and `(i-1)`-th row and column of A was interchanged with the m-th row and column.

  If `uplo = 'L'` and ipiv(i) =ipiv(i+1) = -m < 0, then D has a 2-by-2 block in rows/columns i and `i+1`, and `(i+1)`-th row and column of A was interchanged with the m-th row and column.

ldx

  INTEGER. The leading dimension of the output array x; `ldx ≥ max(1, n)`.

lwork

  INTEGER. The size of the work array.

  If `lwork = -1`, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla. See Application Notes below for details and for the suggested value of lwork.

rwork

  REAL for chesvx

  DOUBLE PRECISION for zhesvx.

  Workspace array, DIMENSION at least `max(1, n)`.

## Output Parameters

x

  COMPLEX for chesvx

  DOUBLE COMPLEX for zhesvx.

  Array, DIMENSION `(ldx,*)`.

  If `info = 0` or `info = n+1`, the array x contains the solution matrix X to the system of equations. The second dimension of x must be at least `max(1,nrhs)`.

af, ipiv

  These arrays are output arguments if `fact = 'N'`. See the description of af, ipiv in Input Arguments section.

rcond

  REAL for chesvx

  DOUBLE PRECISION for zhesvx.

  An estimate of the reciprocal condition number of the matrix A. If rcond is less than the machine precision (in particular, if `rcond = 0)`, the matrix is singular to working precision. This condition is indicated by a return code of info > 0.

ferr

  REAL for chesvx

  DOUBLE PRECISION for zhesvx.

  Array, DIMENSION at least `max(1, nrhs)`. Contains the estimated forward error bound for each solution vector `x(j)` (the j-th column of the solution matrix X). If xtrue is the true solution corresponding to `x(j)`, `ferr(j)` is an estimated upper bound for the magnitude of the largest element in `(x(j) - xtrue)` divided by the magnitude of the largest element in `x(j)`. The estimate is as reliable as the estimate for rcon, and is almost always a slight overestimate of the true error.

berr

  REAL for chesvx

  DOUBLE PRECISION for zhesvx.

  Array, DIMENSION at least `max(1, nrhs)`. Contains the component-wise relative backward error for each solution vector `x(j)`, that is, the smallest relative change in any element of A or B that makes `x(j)` an exact solution.

`work(1)`

  If info = 0, on exit `work(1)` contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

  INTEGER. If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

  If `info = i`, and `i ≤ n`, then `dii` is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.

  If `info = i`, and i = n + 1, then D is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their Fortran 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see Fortran 95 Interface Conventions.

Specific details for the routine hesvx interface are as follows:

a

  Holds the matrix A of size (`n,n`).

b

  Holds the matrix B of size (`n,nrhs`).

x

  Holds the matrix X of size (`n,nrhs`).

af

  Holds the matrix AF of size (`n,n`).

ipiv

  Holds the vector of length (n).

ferr

  Holds the vector of length (nrhs).

berr

  Holds the vector of length (nrhs).

uplo

  Must be 'U' or 'L'. The default value is 'U'.

fact

  Must be 'N' or 'F'. The default value is 'N'. If `fact = 'F'`, then both arguments af and ipiv must be present; otherwise, an error is returned.

## Application Notes

The value of lwork must be at least `2*n`. For better performance, try using `lwork = n*blocksize`, where blocksize is the optimal block size for ?hetrf.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set `lwork = -1`.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If you set `lwork = -1`, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: Driver Routines