# ?gesvx

Computes the solution to the system of linear equations with a square matrix A and multiple right-hand sides, and provides error bounds on the solution.

## Syntax

FORTRAN 77:

call sgesvx( fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info )

call dgesvx( fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info )

call cgesvx( fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info )

call zgesvx( fact, trans, n, nrhs, a, lda, af, ldaf, ipiv, equed, r, c, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info )

Fortran 95:

call gesvx( a, b, x [,af] [,ipiv] [,fact] [,trans] [,equed] [,r] [,c] [,ferr] [,berr] [,rcond] [,rpvgrw] [,info] )

## Description

This routine uses the LU factorization to compute the solution to a real or complex system of linear equations `A*X = B`, where A is an n-by-n 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 ?gesvx performs the following steps:

1. If `fact = 'E'`, real scaling factors r and c are computed to equilibrate the system:

   `trans = 'N'`: `diag(r)*A*diag(c)*inv(diag(c))*X = diag(r)*B`

   `trans = 'T'`: `(diag(r)*A*diag(c))T*inv(diag(r))*X = diag(c)*B`

   `trans = 'C'`: `(diag(r)*A*diag(c))H*inv(diag(r))*X = diag(c)*B`

   Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by `diag(r)*A*diag(c)` and B by `diag(r)*B` (if `trans='N')` or `diag(c)*B` (if `trans = 'T'` or 'C').

2. If `fact = 'N'` or 'E', the LU decomposition is used to factor the matrix A (after equilibration if `fact = 'E'`) as `A = P*L*U`, where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular.

3. If some `Ui,i`= 0, so that U 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.

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

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

6. If equilibration was used, the matrix X is premultiplied by `diag(c)` (if `trans = 'N'`) or `diag(r)` (if `trans = 'T'` or 'C') so that it solves the original system before equilibration.

## Input Parameters

fact

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

  Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored.

  If `fact = 'F'`: on entry, af and ipiv contain the factored form of A. If equed is not 'N', the matrix A has been equilibrated with scaling factors given by r and c.
  a, af, and ipiv are not modified.

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

  If `fact = 'E'`, the matrix A will be equilibrated if necessary, then copied to af and factored.

trans

  CHARACTER*1. Must be 'N', 'T', or 'C'.

  Specifies the form of the system of equations:

  If `trans = 'N'`, the system has the form `A*X = B` (No transpose).

  If `trans = 'T'`, the system has the form AT*X = B (Transpose).

  If `trans = 'C'`, the system has the form AH*X = B (Transpose for real flavors, conjugate transpose for complex flavors).

n

  INTEGER. The number of linear equations; the order of the matrix A; n ≥ 0.

nrhs

  INTEGER. The number of right hand sides; the number of columns of the matrices B and X; nrhs ≥ 0.

a,af,b,work

  REAL for sgesvx

  DOUBLE PRECISION for dgesvx

  COMPLEX for cgesvx

  DOUBLE COMPLEX for zgesvx.

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

  The array a contains the matrix A. If `fact = 'F'` and equed is not 'N', then A must have been equilibrated by the scaling factors in r and/or c. The second dimension of a must be at least `max(1,n)`.

  The array af is an input argument if `fact = 'F'`. It contains the factored form of the matrix A, that is, the factors L and U from the factorization `A = P*L*U` as computed by ?getrf. If equed is not 'N', then af is the factored form of the equilibrated matrix A. 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. The dimension of work must be at least `max(1,4*n)` for real flavors, and at least `max(1,2*n)` for complex flavors.

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 the pivot indices from the factorization `A = P*L*U` as computed by ?getrf; row i of the matrix was interchanged with row `ipiv(i)`.

equed

  CHARACTER*1. Must be 'N', 'R', 'C', or 'B'.

  equed is an input argument if `fact = 'F'`. It specifies the form of equilibration that was done:

  If `equed = 'N'`, no equilibration was done (always true if `fact = 'N'`).

  If `equed = 'R'`, row equilibration was done, that is, A has been premultiplied by diag(r).

  If `equed = 'C'`, column equilibration was done, that is, A has been postmultiplied by diag(c).

  If `equed = 'B'`, both row and column equilibration was done, that is, A has been replaced by `diag(r)*A*diag(c)`.

r, c

  REAL for single precision flavors

  DOUBLE PRECISION for double precision flavors.

  Arrays: `r(n)`, `c(n)`. The array r contains the row scale factors for A, and the array c contains the column scale factors for A. These arrays are input arguments if `fact = 'F'` only; otherwise they are output arguments.

  If `equed = 'R'` or 'B', A is multiplied on the left by diag(r); if `equed = 'N'` or 'C', r is not accessed.

  If `fact = 'F'` and `equed = 'R'` or 'B', each element of r must be positive.

  If `equed = 'C'` or 'B', A is multiplied on the right by diag(c); if `equed = 'N'` or 'R', c is not accessed.

  If `fact = 'F'` and `equed = 'C'` or 'B', each element of c must be positive.

ldx

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

iwork

  INTEGER. Workspace array, DIMENSION at least `max(1, n)`; used in real flavors only.

rwork

  REAL for single precision flavors

  DOUBLE PRECISION for double precision flavors.

  Workspace array, DIMENSION at least `max(1, 2*n)`; used in complex flavors only.

## Output Parameters

x

  REAL for sgesvx

  DOUBLE PRECISION for dgesvx

  COMPLEX for cgesvx

  DOUBLE COMPLEX for zgesvx.

  Array, DIMENSION `(ldx,*)`.

  If `info = 0` or `info = n+1`, the array x contains the solution matrix X to the original system of equations. Note that A and B are modified on exit if `equed ≠ 'N'`, and the solution to the equilibrated system is:

  `diag(C)-1*X`, if `trans = 'N'` and `equed = 'C'` or 'B'; `diag(R)-1*X`, if `trans = 'T'` or 'C' and `equed = 'R'` or 'B'. The second dimension of x must be at least `max(1,nrhs)`.

a

  Array a is not modified on exit if `fact = 'F'` or 'N', or if `fact = 'E'` and equed = 'N'. If `equed ≠ 'N'`, A is scaled on exit as follows:

  equed = 'R': `A = diag(R)*A`

  equed = 'C': `A = A*diag(c)`

  equed = 'B': `A = diag(R)*A*diag(c)`.

af

  If `fact = 'N'` or 'E', then af is an output argument and on exit returns the factors L and U from the factorization `A = PLU` of the original matrix A (if `fact = 'N'`) or of the equilibrated matrix A (if `fact = 'E'`). See the description of a for the form of the equilibrated matrix.

b

  Overwritten by `diag(r)*B` if `trans = 'N'` and `equed = 'R'`or `'B'`;

  overwritten by `diag(c)*B` if `trans = 'T'` or `'C'` and `equed = 'C'` or `'B'`;

  not changed if equed = 'N'.

r, c

  These arrays are output arguments if `fact ≠ 'F'`. See the description of r, c in Input Arguments section.

rcond

  REAL for single precision flavors

  DOUBLE PRECISION for double precision flavors.

  An estimate of the reciprocal condition number of the matrix A after equilibration (if done). 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 single precision flavors

  DOUBLE PRECISION for double precision flavors.

  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 rcond, and is almost always a slight overestimate of the true error.

berr

  REAL for single precision flavors

  DOUBLE PRECISION for double precision flavors.

  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.

ipiv

  If `fact = 'N'` or 'E', then ipiv is an output argument and on exit contains the pivot indices from the factorization `A = P*L*U` of the original matrix A (if `fact = 'N'`) or of the equilibrated matrix A (if `fact = 'E'`).

equed

  If `fact ≠ 'F'` , then equed is an output argument. It specifies the form of equilibration that was done (see the description of equed in Input Arguments section).

work, rwork

  On exit, `work(1)` for real flavors, or `rwork(1)` for complex flavors, contains the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If `work(1)` for real flavors, or `rwork(1)` for complex flavors is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution x, condition estimator rcond, and forward error bound ferr could be unreliable. If factorization fails with `0 < info ≤ n`, then `work(1)` for real flavors, or `rwork(1)` for complex flavors contains the reciprocal pivot growth factor for the leading info columns of A.

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 U(i, i) is exactly zero. The factorization has been completed, but the factor U 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 U 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 gesvx 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).

r

  Holds the vector of length (n). Default value for each element is `r(i) = 1.0_WP`.

c

  Holds the vector of length (n). Default value for each element is `c(i) = 1.0_WP`.

ferr

  Holds the vector of length (nrhs).

berr

  Holds the vector of length (nrhs).

fact

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

trans

  Must be 'N', 'C', or 'T'. The default value is 'N'.

equed

  Must be 'N', 'B', 'C', or 'R'. The default value is 'N'.

rpvgrw

  Real value that contains the reciprocal pivot growth factor norm(A)/norm(U).

Parent topic: Driver Routines