# SRC/zgsisx.c File Reference

Computes an approximate solutions of linear equations A*X=B or A'*X=B. More...

`#include "slu_zdefs.h"`

## Functions
void | zgsisx (superlu_options_t *options, SuperMatrix *A, int *perm_c, int *perm_r, int *etree, char *equed, double *R, double *C, SuperMatrix *L, SuperMatrix *U, void *work, int lwork, SuperMatrix *B, SuperMatrix *X, double *recip_pivot_growth, double *rcond, mem_usage_t *mem_usage, SuperLUStat_t *stat, int *info)

## Detailed Description

     -- SuperLU routine (version 4.1) --
     Lawrence Berkeley National Laboratory.
     November, 2010

## Function Documentation

void zgsisx | ( | superlu_options_t * | options,
|  | SuperMatrix * | A,
|  | int * | perm_c,
|  | int * | perm_r,
|  | int * | etree,
|  | char * | equed,
|  | double * | R,
|  | double * | C,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | void * | work,
|  | int | lwork,
|  | SuperMatrix * | B,
|  | SuperMatrix * | X,
|  | double * | recip_pivot_growth,
|  | double * | rcond,
|  | mem_usage_t * | mem_usage,
|  | SuperLUStat_t * | stat,
|  | int * | info |
| ) |  |  |

     Purpose
     =======

     ZGSISX computes an approximate solutions of linear equations
     A*X=B or A'*X=B, using the ILU factorization from zgsitrf().
     An estimation of the condition number is provided.
     The routine performs the following steps:

       1. If A is stored column-wise (A->Stype = SLU_NC):

    	1.1. If options->Equil = YES or options->RowPerm = LargeDiag, scaling
    	     factors are computed to equilibrate the system:
    	     options->Trans = NOTRANS:
    		 diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
    	     options->Trans = TRANS:
    		 (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
    	     options->Trans = CONJ:
    		 (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 options->Trans=NOTRANS) or diag(C)*B (if options->Trans
    	     = TRANS or CONJ).

    	1.2. Permute columns of A, forming A*Pc, where Pc is a permutation
    	     matrix that usually preserves sparsity.
    	     For more details of this step, see sp_preorder.c.

    	1.3. If options->Fact != FACTORED, the LU decomposition is used to
    	     factor the matrix A (after equilibration if options->Equil = YES)
    	     as Pr*A*Pc = L*U, with Pr determined by partial pivoting.

    	1.4. Compute the reciprocal pivot growth factor.

    	1.5. If some U(i,i) = 0, so that U is exactly singular, then the
    	     routine fills a small number on the diagonal entry, that is
    		U(i,i) = ||A(:,i)||_oo * options->ILU_FillTol ** (1 - i / n),
    	     and info will be increased by 1. The factored form of A is used
    	     to estimate the condition number of the preconditioner. If the
    	     reciprocal of the condition number is less than machine precision,
    	     info = A->ncol+1 is returned as a warning, but the routine still
    	     goes on to solve for X.

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

    	1.7. options->IterRefine is not used

    	1.8. If equilibration was used, the matrix X is premultiplied by
    	     diag(C) (if options->Trans = NOTRANS) or diag(R)
    	     (if options->Trans = TRANS or CONJ) so that it solves the
    	     original system before equilibration.

    	1.9. options for ILU only
    	     1) If options->RowPerm = LargeDiag, MC64 is used to scale and
    		permute the matrix to an I-matrix, that is Pr*Dr*A*Dc has
    		entries of modulus 1 on the diagonal and off-diagonal entries
    		of modulus at most 1. If MC64 fails, dgsequ() is used to
    		equilibrate the system.
                  ( Default: LargeDiag )
    	     2) options->ILU_DropTol = tau is the threshold for dropping.
    		For L, it is used directly (for the whole row in a supernode);
    		For U, ||A(:,i)||_oo * tau is used as the threshold
    	        for the	i-th column.
    		If a secondary dropping rule is required, tau will
    	        also be used to compute the second threshold.
                  ( Default: 1e-4 )
    	     3) options->ILU_FillFactor = gamma, used as the initial guess
    		of memory growth.
    		If a secondary dropping rule is required, it will also
                  be used as an upper bound of the memory.
                  ( Default: 10 )
    	     4) options->ILU_DropRule specifies the dropping rule.
    		Option	      Meaning
    		======	      ===========
    		DROP_BASIC:   Basic dropping rule, supernodal based ILUTP(tau).
    		DROP_PROWS:   Supernodal based ILUTP(p,tau), p = gamma*nnz(A)/n.
    		DROP_COLUMN:  Variant of ILUTP(p,tau), for j-th column,
    			      p = gamma * nnz(A(:,j)).
    		DROP_AREA:    Variation of ILUTP, for j-th column, use
    			      nnz(F(:,1:j)) / nnz(A(:,1:j)) to control memory.
    		DROP_DYNAMIC: Modify the threshold tau during factorizaion:
    			      If nnz(L(:,1:j)) / nnz(A(:,1:j)) > gamma
    				  tau_L(j) := MIN(tau_0, tau_L(j-1) * 2);
    			      Otherwise
    				  tau_L(j) := MAX(tau_0, tau_L(j-1) / 2);
    			      tau_U(j) uses the similar rule.
    			      NOTE: the thresholds used by L and U are separate.
    		DROP_INTERP:  Compute the second dropping threshold by
    			      interpolation instead of sorting (default).
    			      In this case, the actual fill ratio is not
    			      guaranteed smaller than gamma.
    		DROP_PROWS, DROP_COLUMN and DROP_AREA are mutually exclusive.
    		( Default: DROP_BASIC | DROP_AREA )
    	     5) options->ILU_Norm is the criterion of measuring the magnitude
    		of a row in a supernode of L. ( Default is INF_NORM )
    		options->ILU_Norm	RowSize(x[1:n])
    		=================	===============
    		ONE_NORM		||x||_1 / n
    		TWO_NORM		||x||_2 / sqrt(n)
    		INF_NORM		max{|x[i]|}
    	     6) options->ILU_MILU specifies the type of MILU's variation.
    		= SILU: do not perform Modified ILU;
    		= SMILU_1 (not recommended):
    		    U(i,i) := U(i,i) + sum(dropped entries);
    		= SMILU_2:
    		    U(i,i) := U(i,i) + SGN(U(i,i)) * sum(dropped entries);
    		= SMILU_3:
    		    U(i,i) := U(i,i) + SGN(U(i,i)) * sum(|dropped entries|);
    		NOTE: Even SMILU_1 does not preserve the column sum because of
    		late dropping.
                  ( Default: SILU )
    	     7) options->ILU_FillTol is used as the perturbation when
    		encountering zero pivots. If some U(i,i) = 0, so that U is
    		exactly singular, then
    		   U(i,i) := ||A(:,i)|| * options->ILU_FillTol ** (1 - i / n).
                  ( Default: 1e-2 )

       2. If A is stored row-wise (A->Stype = SLU_NR), apply the above algorithm
    	to the transpose of A:

    	2.1. If options->Equil = YES or options->RowPerm = LargeDiag, scaling
    	     factors are computed to equilibrate the system:
    	     options->Trans = NOTRANS:
    		 diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
    	     options->Trans = TRANS:
    		 (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
    	     options->Trans = CONJ:
    		 (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').

    	2.2. Permute columns of transpose(A) (rows of A),
    	     forming transpose(A)*Pc, where Pc is a permutation matrix that
    	     usually preserves sparsity.
    	     For more details of this step, see sp_preorder.c.

    	2.3. If options->Fact != FACTORED, the LU decomposition is used to
    	     factor the transpose(A) (after equilibration if
    	     options->Fact = YES) as Pr*transpose(A)*Pc = L*U with the
    	     permutation Pr determined by partial pivoting.

    	2.4. Compute the reciprocal pivot growth factor.

    	2.5. If some U(i,i) = 0, so that U is exactly singular, then the
    	     routine fills a small number on the diagonal entry, that is
    		 U(i,i) = ||A(:,i)||_oo * options->ILU_FillTol ** (1 - i / n).
    	     And info will be increased by 1. The factored form of A is used
    	     to estimate the condition number of the preconditioner. If the
    	     reciprocal of the condition number is less than machine precision,
    	     info = A->ncol+1 is returned as a warning, but the routine still
    	     goes on to solve for X.

    	2.6. The system of equations is solved for X using the factored form
    	     of transpose(A).

    	2.7. If options->IterRefine is not used.

    	2.8. If equilibration was used, the matrix X is premultiplied by
    	     diag(C) (if options->Trans = NOTRANS) or diag(R)
    	     (if options->Trans = TRANS or CONJ) so that it solves the
    	     original system before equilibration.

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

     Arguments
     =========

     options (input) superlu_options_t*
    	   The structure defines the input parameters to control
    	   how the LU decomposition will be performed and how the
    	   system will be solved.

     A	   (input/output) SuperMatrix*
    	   Matrix A in A*X=B, of dimension (A->nrow, A->ncol). The number
    	   of the linear equations is A->nrow. Currently, the type of A can be:
    	   Stype = SLU_NC or SLU_NR, Dtype = SLU_Z, Mtype = SLU_GE.
    	   In the future, more general A may be handled.

    	   On entry, If options->Fact = FACTORED and equed is not 'N',
    	   then A must have been equilibrated by the scaling factors in
    	   R and/or C.
    	   On exit, A is not modified
             if options->Equil = NO, or
             if options->Equil = YES but equed = 'N' on exit, or
             if options->RowPerm = NO.

    	   Otherwise, if options->Equil = YES and equed is not 'N',
    	   A is scaled as follows:
    	   If A->Stype = SLU_NC:
    	     equed = 'R':  A := diag(R) * A
    	     equed = 'C':  A := A * diag(C)
    	     equed = 'B':  A := diag(R) * A * diag(C).
    	   If A->Stype = SLU_NR:
    	     equed = 'R':  transpose(A) := diag(R) * transpose(A)
    	     equed = 'C':  transpose(A) := transpose(A) * diag(C)
    	     equed = 'B':  transpose(A) := diag(R) * transpose(A) * diag(C).

             If options->RowPerm = LargeDiag, MC64 is used to scale and permute
                the matrix to an I-matrix, that is A is modified as follows:
                P*Dr*A*Dc has entries of modulus 1 on the diagonal and
                off-diagonal entries of modulus at most 1. P is a permutation
                obtained from MC64.
                If MC64 fails, zgsequ() is used to equilibrate the system,
                and A is scaled as above, there is no permutation involved.

     perm_c  (input/output) int*
    	   If A->Stype = SLU_NC, Column permutation vector of size A->ncol,
    	   which defines the permutation matrix Pc; perm_c[i] = j means
    	   column i of A is in position j in A*Pc.
    	   On exit, perm_c may be overwritten by the product of the input
    	   perm_c and a permutation that postorders the elimination tree
    	   of Pc'*A'*A*Pc; perm_c is not changed if the elimination tree
    	   is already in postorder.

    	   If A->Stype = SLU_NR, column permutation vector of size A->nrow,
    	   which describes permutation of columns of transpose(A)
    	   (rows of A) as described above.

     perm_r  (input/output) int*
    	   If A->Stype = SLU_NC, row permutation vector of size A->nrow,
    	   which defines the permutation matrix Pr, and is determined
    	   by partial pivoting.  perm_r[i] = j means row i of A is in
    	   position j in Pr*A.

    	   If A->Stype = SLU_NR, permutation vector of size A->ncol, which
    	   determines permutation of rows of transpose(A)
    	   (columns of A) as described above.

    	   If options->Fact = SamePattern_SameRowPerm, the pivoting routine
    	   will try to use the input perm_r, unless a certain threshold
    	   criterion is violated. In that case, perm_r is overwritten by a
    	   new permutation determined by partial pivoting or diagonal
    	   threshold pivoting.
    	   Otherwise, perm_r is output argument.

     etree   (input/output) int*,  dimension (A->ncol)
    	   Elimination tree of Pc'*A'*A*Pc.
    	   If options->Fact != FACTORED and options->Fact != DOFACT,
    	   etree is an input argument, otherwise it is an output argument.
    	   Note: etree is a vector of parent pointers for a forest whose
    	   vertices are the integers 0 to A->ncol-1; etree[root]==A->ncol.

     equed   (input/output) char*
    	   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).
    	   If options->Fact = FACTORED, equed is an input argument,
    	   otherwise it is an output argument.

     R	   (input/output) double*, dimension (A->nrow)
    	   The row scale factors for A or transpose(A).
    	   If equed = 'R' or 'B', A (if A->Stype = SLU_NC) or transpose(A)
    	       (if A->Stype = SLU_NR) is multiplied on the left by diag(R).
    	   If equed = 'N' or 'C', R is not accessed.
    	   If options->Fact = FACTORED, R is an input argument,
    	       otherwise, R is output.
    	   If options->zFact = FACTORED and equed = 'R' or 'B', each element
    	       of R must be positive.

     C	   (input/output) double*, dimension (A->ncol)
    	   The column scale factors for A or transpose(A).
    	   If equed = 'C' or 'B', A (if A->Stype = SLU_NC) or transpose(A)
    	       (if A->Stype = SLU_NR) is multiplied on the right by diag(C).
    	   If equed = 'N' or 'R', C is not accessed.
    	   If options->Fact = FACTORED, C is an input argument,
    	       otherwise, C is output.
    	   If options->Fact = FACTORED and equed = 'C' or 'B', each element
    	       of C must be positive.

     L	   (output) SuperMatrix*
    	   The factor L from the factorization
    	       Pr*A*Pc=L*U		(if A->Stype SLU_= NC) or
    	       Pr*transpose(A)*Pc=L*U	(if A->Stype = SLU_NR).
    	   Uses compressed row subscripts storage for supernodes, i.e.,
    	   L has types: Stype = SLU_SC, Dtype = SLU_Z, Mtype = SLU_TRLU.

     U	   (output) SuperMatrix*
    	   The factor U from the factorization
    	       Pr*A*Pc=L*U		(if A->Stype = SLU_NC) or
    	       Pr*transpose(A)*Pc=L*U	(if A->Stype = SLU_NR).
    	   Uses column-wise storage scheme, i.e., U has types:
    	   Stype = SLU_NC, Dtype = SLU_Z, Mtype = SLU_TRU.

     work    (workspace/output) void*, size (lwork) (in bytes)
    	   User supplied workspace, should be large enough
    	   to hold data structures for factors L and U.
    	   On exit, if fact is not 'F', L and U point to this array.

     lwork   (input) int
    	   Specifies the size of work array in bytes.
    	   = 0:  allocate space internally by system malloc;
    	   > 0:  use user-supplied work array of length lwork in bytes,
    		 returns error if space runs out.
    	   = -1: the routine guesses the amount of space needed without
    		 performing the factorization, and returns it in
    		 mem_usage->total_needed; no other side effects.

    	   See argument 'mem_usage' for memory usage statistics.

     B	   (input/output) SuperMatrix*
    	   B has types: Stype = SLU_DN, Dtype = SLU_Z, Mtype = SLU_GE.
    	   On entry, the right hand side matrix.
    	   If B->ncol = 0, only LU decomposition is performed, the triangular
    			   solve is skipped.
    	   On exit,
    	      if equed = 'N', B is not modified; otherwise
    	      if A->Stype = SLU_NC:
    		 if options->Trans = NOTRANS and equed = 'R' or 'B',
    		    B is overwritten by diag(R)*B;
    		 if options->Trans = TRANS or CONJ and equed = 'C' of 'B',
    		    B is overwritten by diag(C)*B;
    	      if A->Stype = SLU_NR:
    		 if options->Trans = NOTRANS and equed = 'C' or 'B',
    		    B is overwritten by diag(C)*B;
    		 if options->Trans = TRANS or CONJ and equed = 'R' of 'B',
    		    B is overwritten by diag(R)*B.

             If options->RowPerm = LargeDiag, MC64 is used to scale and permute
                the matrix A to an I-matrix. Then, in addition to the scaling
                above, B is further permuted by P*B if options->Trans = NOTRANS,
                where P is obtained from MC64.

     X	   (output) SuperMatrix*
    	   X has types: Stype = SLU_DN, Dtype = SLU_Z, Mtype = SLU_GE.
    	   If info = 0 or info = A->ncol+1, X contains the solution matrix
    	   to the original system of equations. Note that A and B are modified
    	   on exit if equed is not 'N', and the solution to the equilibrated
    	   system is inv(diag(C))*X if options->Trans = NOTRANS and
    	   equed = 'C' or 'B', or inv(diag(R))*X if options->Trans = 'T' or 'C'
    	   and equed = 'R' or 'B'.

     recip_pivot_growth (output) double*
    	   The reciprocal pivot growth factor max_j( norm(A_j)/norm(U_j) ).
    	   The infinity norm is used. If recip_pivot_growth is much less
    	   than 1, the stability of the LU factorization could be poor.

     rcond   (output) double*
    	   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.

     mem_usage (output) mem_usage_t*
    	   Record the memory usage statistics, consisting of following fields:
    for_lu (float)
    	     The amount of space used in bytes for L data structures.total_needed (float)
    	     The amount of space needed in bytes to perform factorization.expansions (int)
    	     The number of memory expansions during the LU factorization.

     stat   (output) SuperLUStat_t*
    	  Record the statistics on runtime and floating-point operation count.
    	  See slu_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
    	   > 0: if info = i, and i is
    		<= A->ncol: number of zero pivots. They are replaced by small
    		      entries due to options->ILU_FillTol.
    		= A->ncol+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.
    		> A->ncol+1: number of bytes allocated when memory allocation
    		      failure occurred, plus A->ncol.