/*! @file sgsrfs.c

 * \brief Improves computed solution to a system of inear equations

 * 

 * 

 * -- SuperLU routine (version 3.0) --

 * Univ. of California Berkeley, Xerox Palo Alto Research Center,

 * and Lawrence Berkeley National Lab.

 * October 15, 2003

 *

 * Modified from lapack routine SGERFS

 * 

 */

/*

 * File name:	sgsrfs.c

 * History:     Modified from lapack routine SGERFS

 */

#include <math.h></math.h>

#include "slu_sdefs.h"

/*! \brief

 *

 * 

 *   Purpose   

 *   =======   

 *

 *   SGSRFS improves the computed solution to a system of linear   

 *   equations and provides error bounds and backward error estimates for 

 *   the solution.   

 *

 *   If equilibration was performed, the system becomes:

 *           (diag(R)*A_original*diag(C)) * X = diag(R)*B_original.

 *

 *   See supermatrix.h for the definition of 'SuperMatrix' structure.

 *

 *   Arguments   

 *   =========   

 *

 * trans   (input) trans_t

 *          Specifies the form of the system of equations:

 *          = NOTRANS: A * X = B  (No transpose)

 *          = TRANS:   A'* X = B  (Transpose)

 *          = CONJ:    A**H * X = B  (Conjugate transpose)

 *   

 *   A       (input) SuperMatrix*

 *           The original matrix A in the system, or the scaled A if

 *           equilibration was done. The type of A can be:

 *           Stype = SLU_NC, Dtype = SLU_S, Mtype = SLU_GE.

 *    

 *   L       (input) SuperMatrix*

 *	     The factor L from the factorization Pr*A*Pc=L*U. Use

 *           compressed row subscripts storage for supernodes, 

 *           i.e., L has types: Stype = SLU_SC, Dtype = SLU_S, Mtype = SLU_TRLU.

 * 

 *   U       (input) SuperMatrix*

 *           The factor U from the factorization Pr*A*Pc=L*U as computed by

 *           sgstrf(). Use column-wise storage scheme, 

 *           i.e., U has types: Stype = SLU_NC, Dtype = SLU_S, Mtype = SLU_TRU.

 *

 *   perm_c  (input) int*, dimension (A->ncol)

 *	     Column permutation vector, which defines the 

 *           permutation matrix Pc; perm_c[i] = j means column i of A is 

 *           in position j in A*Pc.

 *

 *   perm_r  (input) int*, dimension (A->nrow)

 *           Row permutation vector, which defines the permutation matrix Pr;

 *           perm_r[i] = j means row i of A is in position j in Pr*A.

 *

 *   equed   (input) Specifies the form of equilibration that was done.

 *           = 'N': No equilibration.

 *           = 'R': Row equilibration, i.e., A was premultiplied by diag(R).

 *           = 'C': Column equilibration, i.e., A was postmultiplied by

 *                  diag(C).

 *           = 'B': Both row and column equilibration, i.e., A was replaced 

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

 *

 *   R       (input) float*, dimension (A->nrow)

 *           The row scale factors for A.

 *           If equed = 'R' or 'B', A is premultiplied by diag(R).

 *           If equed = 'N' or 'C', R is not accessed.

 * 

 *   C       (input) float*, dimension (A->ncol)

 *           The column scale factors for A.

 *           If equed = 'C' or 'B', A is postmultiplied by diag(C).

 *           If equed = 'N' or 'R', C is not accessed.

 *

 *   B       (input) SuperMatrix*

 *           B has types: Stype = SLU_DN, Dtype = SLU_S, Mtype = SLU_GE.

 *           The right hand side matrix B.

 *           if equed = 'R' or 'B', B is premultiplied by diag(R).

 *

 *   X       (input/output) SuperMatrix*

 *           X has types: Stype = SLU_DN, Dtype = SLU_S, Mtype = SLU_GE.

 *           On entry, the solution matrix X, as computed by sgstrs().

 *           On exit, the improved solution matrix X.

 *           if *equed = 'C' or 'B', X should be premultiplied by diag(C)

 *               in order to obtain the solution to the original system.

 *

 *   FERR    (output) float*, dimension (B->ncol)   

 *           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    (output) float*, dimension (B->ncol)   

 *           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).

 *

 *   stat     (output) SuperLUStat_t*

 *            Record the statistics on runtime and floating-point operation count.

 *            See util.h for the definition of 'SuperLUStat_t'.

 *

 *   info    (output) int*   

 *           = 0:  successful exit   

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

 *

 *    Internal Parameters   

 *    ===================   

 *

 *    ITMAX is the maximum number of steps of iterative refinement.   

 *

 * 

 */

void

sgsrfs(trans_t trans, SuperMatrix *A, SuperMatrix *L, SuperMatrix *U,

       int *perm_c, int *perm_r, char *equed, float *R, float *C,

       SuperMatrix *B, SuperMatrix *X, float *ferr, float *berr,

       SuperLUStat_t *stat, int *info)

{

#define ITMAX 5

    /* Table of constant values */

    int    ione = 1;

    float ndone = -1.;

    float done = 1.;

    /* Local variables */

    NCformat *Astore;

    float   *Aval;

    SuperMatrix Bjcol;

    DNformat *Bstore, *Xstore, *Bjcol_store;

    float   *Bmat, *Xmat, *Bptr, *Xptr;

    int      kase;

    float   safe1, safe2;

    int      i, j, k, irow, nz, count, notran, rowequ, colequ;

    int      ldb, ldx, nrhs;

    float   s, xk, lstres, eps, safmin;

    char     transc[1];

    trans_t  transt;

    float   *work;

    float   *rwork;

    int      *iwork;

    extern int slacon_(int *, float *, float *, int *, float *, int *);

#ifdef _CRAY

    extern int SCOPY(int *, float *, int *, float *, int *);

    extern int SSAXPY(int *, float *, float *, int *, float *, int *);

#else

    extern int scopy_(int *, float *, int *, float *, int *);

    extern int saxpy_(int *, float *, float *, int *, float *, int *);

#endif

    Astore = A->Store;

    Aval   = Astore->nzval;

    Bstore = B->Store;

    Xstore = X->Store;

    Bmat   = Bstore->nzval;

    Xmat   = Xstore->nzval;

    ldb    = Bstore->lda;

    ldx    = Xstore->lda;

    nrhs   = B->ncol;

    /* Test the input parameters */

    *info = 0;

    notran = (trans == NOTRANS);

    if ( !notran && trans != TRANS && trans != CONJ ) *info = -1;

    else if ( A->nrow != A->ncol || A->nrow < 0 ||

	      A->Stype != SLU_NC || A->Dtype != SLU_S || A->Mtype != SLU_GE )

	*info = -2;

    else if ( L->nrow != L->ncol || L->nrow < 0 ||

 	      L->Stype != SLU_SC || L->Dtype != SLU_S || L->Mtype != SLU_TRLU )

	*info = -3;

    else if ( U->nrow != U->ncol || U->nrow < 0 ||

 	      U->Stype != SLU_NC || U->Dtype != SLU_S || U->Mtype != SLU_TRU )

	*info = -4;

    else if ( ldb < SUPERLU_MAX(0, A->nrow) ||

 	      B->Stype != SLU_DN || B->Dtype != SLU_S || B->Mtype != SLU_GE )

        *info = -10;

    else if ( ldx < SUPERLU_MAX(0, A->nrow) ||

 	      X->Stype != SLU_DN || X->Dtype != SLU_S || X->Mtype != SLU_GE )

	*info = -11;

    if (*info != 0) {

	i = -(*info);

	xerbla_("sgsrfs", &i);

	return;

    }

    /* Quick return if possible */

    if ( A->nrow == 0 || nrhs == 0) {

	for (j = 0; j < nrhs; ++j) {

	    ferr[j] = 0.;

	    berr[j] = 0.;

	}

	return;

    }

    rowequ = lsame_(equed, "R") || lsame_(equed, "B");

    colequ = lsame_(equed, "C") || lsame_(equed, "B");

    /* Allocate working space */

    work = floatMalloc(2*A->nrow);

    rwork = (float *) SUPERLU_MALLOC( A->nrow * sizeof(float) );

    iwork = intMalloc(2*A->nrow);

    if ( !work || !rwork || !iwork ) 

        ABORT("Malloc fails for work/rwork/iwork.");

    if ( notran ) {

	*(unsigned char *)transc = 'N';

        transt = TRANS;

    } else {

	*(unsigned char *)transc = 'T';

	transt = NOTRANS;

    }

    /* NZ = maximum number of nonzero elements in each row of A, plus 1 */

    nz     = A->ncol + 1;

    eps    = slamch_("Epsilon");

    safmin = slamch_("Safe minimum");

    /* Set SAFE1 essentially to be the underflow threshold times the

       number of additions in each row. */

    safe1  = nz * safmin;

    safe2  = safe1 / eps;

    /* Compute the number of nonzeros in each row (or column) of A */

    for (i = 0; i < A->nrow; ++i) iwork[i] = 0;

    if ( notran ) {

	for (k = 0; k < A->ncol; ++k)

	    for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i) 

		++iwork[Astore->rowind[i]];

    } else {

	for (k = 0; k < A->ncol; ++k)

	    iwork[k] = Astore->colptr[k+1] - Astore->colptr[k];

    }	

    /* Copy one column of RHS B into Bjcol. */

    Bjcol.Stype = B->Stype;

    Bjcol.Dtype = B->Dtype;

    Bjcol.Mtype = B->Mtype;

    Bjcol.nrow  = B->nrow;

    Bjcol.ncol  = 1;

    Bjcol.Store = (void *) SUPERLU_MALLOC( sizeof(DNformat) );

    if ( !Bjcol.Store ) ABORT("SUPERLU_MALLOC fails for Bjcol.Store");

    Bjcol_store = Bjcol.Store;

    Bjcol_store->lda = ldb;

    Bjcol_store->nzval = work; /* address aliasing */

    /* Do for each right hand side ... */

    for (j = 0; j < nrhs; ++j) {

	count = 0;

	lstres = 3.;

	Bptr = &Bmat[j*ldb];

	Xptr = &Xmat[j*ldx];

	while (1) { /* Loop until stopping criterion is satisfied. */

	    /* Compute residual R = B - op(A) * X,   

	       where op(A) = A, A**T, or A**H, depending on TRANS. */

#ifdef _CRAY

	    SCOPY(&A->nrow, Bptr, &ione, work, &ione);

#else

	    scopy_(&A->nrow, Bptr, &ione, work, &ione);

#endif

	    sp_sgemv(transc, ndone, A, Xptr, ione, done, work, ione);

	    /* Compute componentwise relative backward error from formula 

	       max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )   

	       where abs(Z) is the componentwise absolute value of the matrix

	       or vector Z.  If the i-th component of the denominator is less

	       than SAFE2, then SAFE1 is added to the i-th component of the   

	       numerator before dividing. */

	    for (i = 0; i < A->nrow; ++i) rwork[i] = fabs( Bptr[i] );

	    /* Compute abs(op(A))*abs(X) + abs(B). */

	    if (notran) {

		for (k = 0; k < A->ncol; ++k) {

		    xk = fabs( Xptr[k] );

		    for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i)

			rwork[Astore->rowind[i]] += fabs(Aval[i]) * xk;

		}

	    } else {

		for (k = 0; k < A->ncol; ++k) {

		    s = 0.;

		    for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i) {

			irow = Astore->rowind[i];

			s += fabs(Aval[i]) * fabs(Xptr[irow]);

		    }

		    rwork[k] += s;

		}

	    }

	    s = 0.;

	    for (i = 0; i < A->nrow; ++i) {

		if (rwork[i] > safe2) {

		    s = SUPERLU_MAX( s, fabs(work[i]) / rwork[i] );

		} else if ( rwork[i] != 0.0 ) {

                    /* Adding SAFE1 to the numerator guards against

                       spuriously zero residuals (underflow). */

		    s = SUPERLU_MAX( s, (safe1 + fabs(work[i])) / rwork[i] );

                }

                /* If rwork[i] is exactly 0.0, then we know the true 

                   residual also must be exactly 0.0. */

	    }

	    berr[j] = s;

	    /* Test stopping criterion. Continue iterating if   

	       1) The residual BERR(J) is larger than machine epsilon, and   

	       2) BERR(J) decreased by at least a factor of 2 during the   

	          last iteration, and   

	       3) At most ITMAX iterations tried. */

	    if (berr[j] > eps && berr[j] * 2. <= lstres && count < ITMAX) {

		/* Update solution and try again. */

		sgstrs (trans, L, U, perm_c, perm_r, &Bjcol, stat, info);

#ifdef _CRAY

		SAXPY(&A->nrow, &done, work, &ione,

		       &Xmat[j*ldx], &ione);

#else

		saxpy_(&A->nrow, &done, work, &ione,

		       &Xmat[j*ldx], &ione);

#endif

		lstres = berr[j];

		++count;

	    } else {

		break;

	    }

	} /* end while */

	stat->RefineSteps = count;

	/* Bound error from formula:

	   norm(X - XTRUE) / norm(X) .le. FERR = norm( abs(inv(op(A)))*   

	   ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)   

          where   

            norm(Z) is the magnitude of the largest component of Z   

            inv(op(A)) is the inverse of op(A)   

            abs(Z) is the componentwise absolute value of the matrix or

	       vector Z   

            NZ is the maximum number of nonzeros in any row of A, plus 1   

            EPS is machine epsilon   

          The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))   

          is incremented by SAFE1 if the i-th component of   

          abs(op(A))*abs(X) + abs(B) is less than SAFE2.   

          Use SLACON to estimate the infinity-norm of the matrix   

             inv(op(A)) * diag(W),   

          where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */

	for (i = 0; i < A->nrow; ++i) rwork[i] = fabs( Bptr[i] );

	/* Compute abs(op(A))*abs(X) + abs(B). */

	if ( notran ) {

	    for (k = 0; k < A->ncol; ++k) {

		xk = fabs( Xptr[k] );

		for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i)

		    rwork[Astore->rowind[i]] += fabs(Aval[i]) * xk;

	    }

	} else {

	    for (k = 0; k < A->ncol; ++k) {

		s = 0.;

		for (i = Astore->colptr[k]; i < Astore->colptr[k+1]; ++i) {

		    irow = Astore->rowind[i];

		    xk = fabs( Xptr[irow] );

		    s += fabs(Aval[i]) * xk;

		}

		rwork[k] += s;

	    }

	}

	for (i = 0; i < A->nrow; ++i)

	    if (rwork[i] > safe2)

		rwork[i] = fabs(work[i]) + (iwork[i]+1)*eps*rwork[i];

	    else

		rwork[i] = fabs(work[i])+(iwork[i]+1)*eps*rwork[i]+safe1;

	kase = 0;

	do {

	    slacon_(&A->nrow, &work[A->nrow], work,

		    &iwork[A->nrow], &ferr[j], &kase);

	    if (kase == 0) break;

	    if (kase == 1) {

		/* Multiply by diag(W)*inv(op(A)**T)*(diag(C) or diag(R)). */

		if ( notran && colequ )

		    for (i = 0; i < A->ncol; ++i) work[i] *= C[i];

		else if ( !notran && rowequ )

		    for (i = 0; i < A->nrow; ++i) work[i] *= R[i];

		sgstrs (transt, L, U, perm_c, perm_r, &Bjcol, stat, info);

		for (i = 0; i < A->nrow; ++i) work[i] *= rwork[i];

	    } else {

		/* Multiply by (diag(C) or diag(R))*inv(op(A))*diag(W). */

		for (i = 0; i < A->nrow; ++i) work[i] *= rwork[i];

		sgstrs (trans, L, U, perm_c, perm_r, &Bjcol, stat, info);

		if ( notran && colequ )

		    for (i = 0; i < A->ncol; ++i) work[i] *= C[i];

		else if ( !notran && rowequ )

		    for (i = 0; i < A->ncol; ++i) work[i] *= R[i];  

	    }

	} while ( kase != 0 );

	/* Normalize error. */

	lstres = 0.;

 	if ( notran && colequ ) {

	    for (i = 0; i < A->nrow; ++i)

	    	lstres = SUPERLU_MAX( lstres, C[i] * fabs( Xptr[i]) );

  	} else if ( !notran && rowequ ) {

	    for (i = 0; i < A->nrow; ++i)

	    	lstres = SUPERLU_MAX( lstres, R[i] * fabs( Xptr[i]) );

	} else {

	    for (i = 0; i < A->nrow; ++i)

	    	lstres = SUPERLU_MAX( lstres, fabs( Xptr[i]) );

	}

	if ( lstres != 0. )

	    ferr[j] /= lstres;

    } /* for each RHS j ... */

    SUPERLU_FREE(work);

    SUPERLU_FREE(rwork);

    SUPERLU_FREE(iwork);

    SUPERLU_FREE(Bjcol.Store);

    return;

} /* sgsrfs */