CGESVX uses the LU factorization to compute the solution to a complex


 system of linear equations


    A * X = B,


 where A is an N-by-N matrix and X and B are N-by-NRHS matrices.


 Error bounds on the solution and a condition estimate are also


 provided.

 The following steps are performed:


 1. If FACT = 'E', real scaling factors 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 U(i,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.

FACT

          FACT is CHARACTER*1


          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.


          = '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.


          = 'N':  The matrix A will be copied to AF and factored.


          = 'E':  The matrix A will be equilibrated if necessary, then


                  copied to AF and factored.

TRANS

          TRANS is CHARACTER*1


          Specifies the form of the system of equations:


          = 'N':  A * X = B     (No transpose)


          = 'T':  A**T * X = B  (Transpose)


          = 'C':  A**H * X = B  (Conjugate transpose)

N

          N is INTEGER


          The number of linear equations, i.e., the order of the


          matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrices B and X.  NRHS >= 0.

A

          A is COMPLEX array, dimension (LDA,N)


          On entry, the N-by-N 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.  A is not modified if FACT = 'F' or


          'N', or if FACT = 'E' and EQUED = 'N' on exit.


          On exit, if EQUED .ne. 'N', A is scaled as follows:


          EQUED = 'R':  A := diag(R) * A


          EQUED = 'C':  A := A * diag(C)


          EQUED = 'B':  A := diag(R) * A * diag(C).

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AF

          AF is COMPLEX array, dimension (LDAF,N)


          If FACT = 'F', then AF is an input argument and on entry


          contains the factors L and U from the factorization


          A = P*L*U as computed by CGETRF.  If EQUED .ne. 'N', then


          AF is the factored form of the equilibrated matrix A.


          If FACT = 'N', then AF is an output argument and on exit


          returns the factors L and U from the factorization A = P*L*U


          of the original matrix A.


          If FACT = 'E', then AF is an output argument and on exit


          returns the factors L and U from the factorization A = P*L*U


          of the equilibrated matrix A (see the description of A for


          the form of the equilibrated matrix).

LDAF

          LDAF is INTEGER


          The leading dimension of the array AF.  LDAF >= max(1,N).

IPIV

          IPIV is INTEGER array, dimension (N)


          If FACT = 'F', then IPIV is an input argument and on entry


          contains the pivot indices from the factorization A = P*L*U


          as computed by CGETRF; row i of the matrix was interchanged


          with row IPIV(i).


          If FACT = 'N', 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 = 'E', then IPIV is an output argument and on exit


          contains the pivot indices from the factorization A = P*L*U


          of the equilibrated matrix A.

EQUED

          EQUED is CHARACTER*1


          Specifies the form of equilibration that was done.


          = 'N':  No equilibration (always true if FACT = 'N').


          = 'R':  Row equilibration, i.e., A has been premultiplied by


                  diag(R).


          = 'C':  Column equilibration, i.e., A has been postmultiplied


                  by diag(C).


          = 'B':  Both row and column equilibration, i.e., A has been


                  replaced by diag(R) * A * diag(C).


          EQUED is an input argument if FACT = 'F'; otherwise, it is an


          output argument.

R

          R is REAL array, dimension (N)


          The row scale factors for A.  If EQUED = 'R' or 'B', A is


          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R


          is not accessed.  R is an input argument if FACT = 'F';


          otherwise, R is an output argument.  If FACT = 'F' and


          EQUED = 'R' or 'B', each element of R must be positive.

C

          C is REAL array, dimension (N)


          The column scale factors for A.  If EQUED = 'C' or 'B', A is


          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C


          is not accessed.  C is an input argument if FACT = 'F';


          otherwise, C is an output argument.  If FACT = 'F' and


          EQUED = 'C' or 'B', each element of C must be positive.

B

          B is COMPLEX array, dimension (LDB,NRHS)


          On entry, the N-by-NRHS right hand side matrix B.


          On exit,


          if EQUED = 'N', B is not modified;


          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by


          diag(R)*B;


          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is


          overwritten by diag(C)*B.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX array, dimension (LDX,NRHS)


          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X


          to the original system of equations.  Note that A and B are


          modified on exit if EQUED .ne. 'N', and the solution to the


          equilibrated system is inv(diag(C))*X if TRANS = 'N' and


          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'


          and EQUED = 'R' or 'B'.

LDX

          LDX is INTEGER


          The leading dimension of the array X.  LDX >= max(1,N).

RCOND

          RCOND is REAL


          The 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

          FERR is REAL array, dimension (NRHS)


          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

          BERR is REAL array, dimension (NRHS)


          The componentwise relative backward error of each solution


          vector X(j) (i.e., the smallest relative change in


          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is COMPLEX array, dimension (2*N)

RWORK

          RWORK is REAL array, dimension (2*N)


          On exit, RWORK(1) contains the reciprocal pivot growth


          factor norm(A)/norm(U). The "max absolute element" norm is


          used. If RWORK(1) 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


          RWORK(1) contains the reciprocal pivot growth factor for the


          leading INFO columns of A.

INFO

          INFO is INTEGER


          = 0:  successful exit


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


          > 0:  if INFO = i, and i is


                <= N:  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.


                = N+1: 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.

Univ. of Tennessee

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NAG Ltd.

April 2012