slu_ddefs.h File Reference

Header file for real operations. More...

`#include <math.h>`
`#include <limits.h>`
`#include <stdio.h>`
`#include <stdlib.h>`
`#include <stdint.h>`
`#include <string.h>`
`#include "slu_Cnames.h"`
`#include "supermatrix.h"`
`#include "slu_util.h"`
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## Typedefs

typedef int | int_t

## Functions

void | dgssv (superlu_options_t *, SuperMatrix *, int *, int *, SuperMatrix *, SuperMatrix *, SuperMatrix *, SuperLUStat_t *, int *)
| Driver routines. More...
void | dgssvx (superlu_options_t *, SuperMatrix *, int *, int *, int *, char *, double *, double *, SuperMatrix *, SuperMatrix *, void *, int, SuperMatrix *, SuperMatrix *, double *, double *, double *, double *, GlobalLU_t *, mem_usage_t *, SuperLUStat_t *, int *)
void | dgsisv (superlu_options_t *, SuperMatrix *, int *, int *, SuperMatrix *, SuperMatrix *, SuperMatrix *, SuperLUStat_t *, int *)
void | dgsisx (superlu_options_t *, SuperMatrix *, int *, int *, int *, char *, double *, double *, SuperMatrix *, SuperMatrix *, void *, int, SuperMatrix *, SuperMatrix *, double *, double *, GlobalLU_t *, mem_usage_t *, SuperLUStat_t *, int *)
void | dCreate_CompCol_Matrix (SuperMatrix *, int, int, int, double *, int *, int *, Stype_t, Dtype_t, Mtype_t)
| Supernodal LU factor related. More...
void | dCreate_CompRow_Matrix (SuperMatrix *, int, int, int, double *, int *, int *, Stype_t, Dtype_t, Mtype_t)
void | dCopy_CompCol_Matrix (SuperMatrix *, SuperMatrix *)
| Copy matrix A into matrix B. More...
void | dCreate_Dense_Matrix (SuperMatrix *, int, int, double *, int, Stype_t, Dtype_t, Mtype_t)
void | dCreate_SuperNode_Matrix (SuperMatrix *, int, int, int, double *, int *, int *, int *, int *, int *, Stype_t, Dtype_t, Mtype_t)
void | dCopy_Dense_Matrix (int, int, double *, int, double *, int)
void | countnz (const int, int *, int *, int *, GlobalLU_t *)
| Count the total number of nonzeros in factors L and U, and in the symmetrically reduced L. More...
void | ilu_countnz (const int, int *, int *, GlobalLU_t *)
| Count the total number of nonzeros in factors L and U. More...
void | fixupL (const int, const int *, GlobalLU_t *)
| Fix up the data storage lsub for L-subscripts. It removes the subscript sets for structural pruning, and applies permuation to the remaining subscripts. More...
void | dallocateA (int, int, double **, int **, int **)
| Allocate storage for original matrix A. More...
void | dgstrf (superlu_options_t *, SuperMatrix *, int, int, int *, void *, int, int *, int *, SuperMatrix *, SuperMatrix *, GlobalLU_t *, SuperLUStat_t *, int *)
int | dsnode_dfs (const int, const int, const int *, const int *, const int *, int *, int *, GlobalLU_t *)
int | dsnode_bmod (const int, const int, const int, double *, double *, GlobalLU_t *, SuperLUStat_t *)
| Performs numeric block updates within the relaxed snode. More...
void | dpanel_dfs (const int, const int, const int, SuperMatrix *, int *, int *, double *, int *, int *, int *, int *, int *, int *, int *, GlobalLU_t *)
void | dpanel_bmod (const int, const int, const int, const int, double *, double *, int *, int *, GlobalLU_t *, SuperLUStat_t *)
int | dcolumn_dfs (const int, const int, int *, int *, int *, int *, int *, int *, int *, int *, int *, GlobalLU_t *)
int | dcolumn_bmod (const int, const int, double *, double *, int *, int *, int, GlobalLU_t *, SuperLUStat_t *)
int | dcopy_to_ucol (int, int, int *, int *, int *, double *, GlobalLU_t *)
int | dpivotL (const int, const double, int *, int *, int *, int *, int *, GlobalLU_t *, SuperLUStat_t *)
void | dpruneL (const int, const int *, const int, const int, const int *, const int *, int *, GlobalLU_t *)
void | dreadmt (int *, int *, int *, double **, int **, int **)
void | dGenXtrue (int, int, double *, int)
void | dFillRHS (trans_t, int, double *, int, SuperMatrix *, SuperMatrix *)
| Let rhs[i] = sum of i-th row of A, so the solution vector is all 1's. More...
void | dgstrs (trans_t, SuperMatrix *, SuperMatrix *, int *, int *, SuperMatrix *, SuperLUStat_t *, int *)
void | dgsitrf (superlu_options_t *, SuperMatrix *, int, int, int *, void *, int, int *, int *, SuperMatrix *, SuperMatrix *, GlobalLU_t *, SuperLUStat_t *, int *)
int | dldperm (int, int, int, int[], int[], double[], int[], double[], double[])
int | ilu_dsnode_dfs (const int, const int, const int *, const int *, const int *, int *, GlobalLU_t *)
void | ilu_dpanel_dfs (const int, const int, const int, SuperMatrix *, int *, int *, double *, double *, int *, int *, int *, int *, int *, int *, GlobalLU_t *)
int | ilu_dcolumn_dfs (const int, const int, int *, int *, int *, int *, int *, int *, int *, int *, GlobalLU_t *)
int | ilu_dcopy_to_ucol (int, int, int *, int *, int *, double *, int, milu_t, double, int, double *, int *, GlobalLU_t *, double *)
int | ilu_dpivotL (const int, const double, int *, int *, int, int *, int *, int *, int *, double, milu_t, double, GlobalLU_t *, SuperLUStat_t *)
int | ilu_ddrop_row (superlu_options_t *, int, int, double, int, int *, double *, GlobalLU_t *, double *, double *, int)
void | dgsequ (SuperMatrix *, double *, double *, double *, double *, double *, int *)
| Driver related. More...
void | dlaqgs (SuperMatrix *, double *, double *, double, double, double, char *)
void | dgscon (char *, SuperMatrix *, SuperMatrix *, double, double *, SuperLUStat_t *, int *)
double | dPivotGrowth (int, SuperMatrix *, int *, SuperMatrix *, SuperMatrix *)
void | dgsrfs (trans_t, SuperMatrix *, SuperMatrix *, SuperMatrix *, int *, int *, char *, double *, double *, SuperMatrix *, SuperMatrix *, double *, double *, SuperLUStat_t *, int *)
int | sp_dtrsv (char *, char *, char *, SuperMatrix *, SuperMatrix *, double *, SuperLUStat_t *, int *)
| Solves one of the systems of equations A*x = b, or A'*x = b. More...
int | sp_dgemv (char *, double, SuperMatrix *, double *, int, double, double *, int)
| Performs one of the matrix-vector operations y := alpha*A*x + beta*y, or y := alpha*A'*x + beta*y,. More...
int | sp_dgemm (char *, char *, int, int, int, double, SuperMatrix *, double *, int, double, double *, int)
double | dmach (char *)
int | dLUMemInit (fact_t, void *, int, int, int, int, int, double, SuperMatrix *, SuperMatrix *, GlobalLU_t *, int **, double **)
| Memory-related. More...
void | dSetRWork (int, int, double *, double **, double **)
| Set up pointers for real working arrays. More...
void | dLUWorkFree (int *, double *, GlobalLU_t *)
| Free the working storage used by factor routines. More...
int | dLUMemXpand (int, int, MemType, int *, GlobalLU_t *)
| Expand the data structures for L and U during the factorization. More...
double * | doubleMalloc (int)
double * | doubleCalloc (int)
int | dmemory_usage (const int, const int, const int, const int)
int | dQuerySpace (SuperMatrix *, SuperMatrix *, mem_usage_t *)
int | ilu_dQuerySpace (SuperMatrix *, SuperMatrix *, mem_usage_t *)
void | dreadhb (FILE *, int *, int *, int *, double **, int **, int **)
| Auxiliary routines. More...
void | dreadrb (int *, int *, int *, double **, int **, int **)
void | dreadtriple (int *, int *, int *, double **, int **, int **)
void | dreadMM (FILE *, int *, int *, int *, double **, int **, int **)
void | dCompRow_to_CompCol (int, int, int, double *, int *, int *, double **, int **, int **)
| Convert a row compressed storage into a column compressed storage. More...
void | dfill (double *, int, double)
| Fills a double precision array with a given value. More...
void | dinf_norm_error (int, SuperMatrix *, double *)
| Check the inf-norm of the error vector. More...
double | dqselect (int, double *, int)
void | dPrint_CompCol_Matrix (char *, SuperMatrix *)
| Routines for debugging. More...
void | dPrint_SuperNode_Matrix (char *, SuperMatrix *)
void | dPrint_Dense_Matrix (char *, SuperMatrix *)
void | dprint_lu_col (char *, int, int, int *, GlobalLU_t *)
| Diagnostic print of column "jcol" in the U/L factor. More...
int | print_double_vec (char *, int, double *)
void | dcheck_tempv (int, double *)
| Check whether tempv[] == 0. This should be true before and after calling any numeric routines, i.e., "panel_bmod" and "column_bmod". More...
int | dgemm_ (const char *, const char *, const int *, const int *, const int *, const double *, const double *, const int *, const double *, const int *, const double *, double *, const int *)
| BLAS. More...
int | dtrsv_ (char *, char *, char *, int *, double *, int *, double *, int *)
int | dtrsm_ (char *, char *, char *, char *, int *, int *, double *, double *, int *, double *, int *)
int | dgemv_ (char *, int *, int *, double *, double *a, int *, double *, int *, double *, double *, int *)

## Detailed Description

Copyright (c) 2003, The Regents of the University of California, through Lawrence Berkeley National Laboratory (subject to receipt of any required approvals from U.S. Dept. of Energy)

All rights reserved.

The source code is distributed under BSD license, see the file License.txt at the top-level directory.

    – SuperLU routine (version 4.1) –
    Univ. of California Berkeley, Xerox Palo Alto Research Center,
    and Lawrence Berkeley National Lab.
    November, 2010

    Global data structures used in LU factorization -

      nsuper: #supernodes = nsuper + 1, numbered [0, nsuper].
      (xsup,supno): supno[i] is the supernode no to which i belongs;
         xsup(s) points to the beginning of the s-th supernode.
         e.g.   supno 0 1 2 2 3 3 3 4 4 4 4 4   (n=12)
                 xsup 0 1 2 4 7 12
         Note: dfs will be performed on supernode rep. relative to the new
               row pivoting ordering

      (xlsub,lsub): lsub[*] contains the compressed subscript of
         rectangular supernodes; xlsub[j] points to the starting
         location of the j-th column in lsub[*]. Note that xlsub
         is indexed by column.
         Storage: original row subscripts

         During the course of sparse LU factorization, we also use
         (xlsub,lsub) for the purpose of symmetric pruning. For each
         supernode {s,s+1,...,t=s+r} with first column s and last
         column t, the subscript set
            lsub[j], j=xlsub[s], .., xlsub[s+1]-1
         is the structure of column s (i.e. structure of this supernode).
         It is used for the storage of numerical values.
         Furthermore,
            lsub[j], j=xlsub[t], .., xlsub[t+1]-1
         is the structure of the last column t of this supernode.
         It is for the purpose of symmetric pruning. Therefore, the
         structural subscripts can be rearranged without making physical
         interchanges among the numerical values.

         However, if the supernode has only one column, then we
         only keep one set of subscripts. For any subscript interchange
         performed, similar interchange must be done on the numerical
         values.

         The last column structures (for pruning) will be removed
         after the numercial LU factorization phase.

      (xlusup,lusup): lusup[*] contains the numerical values of the
         rectangular supernodes; xlusup[j] points to the starting
         location of the j-th column in storage vector lusup[*]
         Note: xlusup is indexed by column.
         Each rectangular supernode is stored by column-major
         scheme, consistent with Fortran 2-dim array storage.

      (xusub,ucol,usub): ucol[*] stores the numerical values of
         U-columns outside the rectangular supernodes. The row
         subscript of nonzero ucol[k] is stored in usub[k].
         xusub[i] points to the starting location of column i in ucol.
         Storage: new row subscripts; that is subscripts of PA.

## Typedef Documentation

typedef int int_t

## Function Documentation

void countnz | ( | const int | ,
|  | int * | ,
|  | int * | ,
|  | int * | ,
|  | GlobalLU_t * |
| ) |  |

void dallocateA | ( | int | ,
|  | int | ,
|  | double ** | ,
|  | int ** | ,
|  | int ** |
| ) |  |

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void dcheck_tempv | ( | int | ,
|  | double * |
| ) |  |

int dcolumn_bmod | ( | const int | jcol,
|  | const int | nseg,
|  | double * | dense,
|  | double * | tempv,
|  | int * | segrep,
|  | int * | repfnz,
|  | int | fpanelc,
|  | GlobalLU_t * | Glu,
|  | SuperLUStat_t * | stat
| ) |  |

#

    Purpose:

    Performs numeric block updates (sup-col) in topological order.
    It features: col-col, 2cols-col, 3cols-col, and sup-col updates.
    Special processing on the supernodal portion of L[*,j]
    Return value:   0 - successful return
                  > 0 - number of bytes allocated when run out of space

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int dcolumn_dfs | ( | const int | m,
|  | const int | jcol,
|  | int * | perm_r,
|  | int * | nseg,
|  | int * | lsub_col,
|  | int * | segrep,
|  | int * | repfnz,
|  | int * | xprune,
|  | int * | marker,
|  | int * | parent,
|  | int * | xplore,
|  | GlobalLU_t * | Glu
| ) |  |

#

    Purpose

      DCOLUMN_DFS performs a symbolic factorization on column jcol, and
      decide the supernode boundary.

      This routine does not use numeric values, but only use the RHS
      row indices to start the dfs.

      A supernode representative is the last column of a supernode.
      The nonzeros in U[*,j] are segments that end at supernodal
      representatives. The routine returns a list of such supernodal
      representatives in topological order of the dfs that generates them.
      The location of the first nonzero in each such supernodal segment
      (supernodal entry location) is also returned.

#

    Local parameters

      nseg: no of segments in current U[*,j]
      jsuper: jsuper=EMPTY if column j does not belong to the same
         supernode as j-1. Otherwise, jsuper=nsuper.

      marker2: A-row –> A-row/col (0/1)
      repfnz: SuperA-col –> PA-row
      parent: SuperA-col –> SuperA-col
      xplore: SuperA-col –> index to L-structure

#

    Return value

        0  success;

    0  number of bytes allocated when run out of space.

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void dCompRow_to_CompCol | ( | int | ,
|  | int | ,
|  | int | ,
|  | double * | ,
|  | int * | ,
|  | int * | ,
|  | double ** | ,
|  | int ** | ,
|  | int ** |
| ) |  |

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void dCopy_CompCol_Matrix | ( | SuperMatrix * | ,
|  | SuperMatrix * |
| ) |  |

void dCopy_Dense_Matrix | ( | int | ,
|  | int | ,
|  | double * | ,
|  | int | ,
|  | double * | ,
|  | int |
| ) |  |

Copies a two-dimensional matrix X to another matrix Y.

int dcopy_to_ucol | ( | int | ,
|  | int | ,
|  | int * | ,
|  | int * | ,
|  | int * | ,
|  | double * | ,
|  | GlobalLU_t * |
| ) |  |

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void dCreate_CompCol_Matrix | ( | SuperMatrix * | ,
|  | int | ,
|  | int | ,
|  | int | ,
|  | double * | ,
|  | int * | ,
|  | int * | ,
|  | Stype_t | ,
|  | Dtype_t | ,
|  | Mtype_t |
| ) |  |

void dCreate_CompRow_Matrix | ( | SuperMatrix * | ,
|  | int | ,
|  | int | ,
|  | int | ,
|  | double * | ,
|  | int * | ,
|  | int * | ,
|  | Stype_t | ,
|  | Dtype_t | ,
|  | Mtype_t |
| ) |  |

void dCreate_Dense_Matrix | ( | SuperMatrix * | ,
|  | int | ,
|  | int | ,
|  | double * | ,
|  | int | ,
|  | Stype_t | ,
|  | Dtype_t | ,
|  | Mtype_t |
| ) |  |

void dCreate_SuperNode_Matrix | ( | SuperMatrix * | ,
|  | int | ,
|  | int | ,
|  | int | ,
|  | double * | ,
|  | int * | ,
|  | int * | ,
|  | int * | ,
|  | int * | ,
|  | int * | ,
|  | Stype_t | ,
|  | Dtype_t | ,
|  | Mtype_t |
| ) |  |

void dfill | ( | double * | ,
|  | int | ,
|  | double |
| ) |  |

void dFillRHS | ( | trans_t | ,
|  | int | ,
|  | double * | ,
|  | int | ,
|  | SuperMatrix * | ,
|  | SuperMatrix * |
| ) |  |

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int dgemm_ | ( | const char * | ,
|  | const char * | ,
|  | const int * | ,
|  | const int * | ,
|  | const int * | ,
|  | const double * | ,
|  | const double * | ,
|  | const int * | ,
|  | const double * | ,
|  | const int * | ,
|  | const double * | ,
|  | double * | ,
|  | const int * |
| ) |  |

int dgemv_ | ( | char * | ,
|  | int * | ,
|  | int * | ,
|  | double * | ,
|  | double * | a,
|  | int * | ,
|  | double * | ,
|  | int * | ,
|  | double * | ,
|  | double * | ,
|  | int * |
| ) |  |

void dGenXtrue | ( | int | ,
|  | int | ,
|  | double * | ,
|  | int |
| ) |  |

void dgscon | ( | char * | norm,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | double | anorm,
|  | double * | rcond,
|  | SuperLUStat_t * | stat,
|  | int * | info
| ) |  |

#

    Purpose

      DGSCON estimates the reciprocal of the condition number of a general
      real matrix A, in either the 1-norm or the infinity-norm, using
      the LU factorization computed by DGETRF.   *

      An estimate is obtained for norm(inv(A)), and the reciprocal of the
      condition number is computed as
         RCOND = 1 / ( norm(A) * norm(inv(A)) ).

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

#

    Arguments

       NORM    (input) char*
               Specifies whether the 1-norm condition number or the
               infinity-norm condition number is required:
               = '1' or 'O':  1-norm;
               = 'I':         Infinity-norm.

       L       (input) SuperMatrix*
               The factor L from the factorization Pr*A*Pc=L*U as computed by
               dgstrf(). Use compressed row subscripts storage for supernodes,
               i.e., L has types: Stype = SLU_SC, Dtype = SLU_D, Mtype = SLU_TRLU.

       U       (input) SuperMatrix*
               The factor U from the factorization Pr*A*Pc=L*U as computed by
               dgstrf(). Use column-wise storage scheme, i.e., U has types:
               Stype = SLU_NC, Dtype = SLU_D, Mtype = SLU_TRU.

       ANORM   (input) double
               If NORM = '1' or 'O', the 1-norm of the original matrix A.
               If NORM = 'I', the infinity-norm of the original matrix A.

       RCOND   (output) double*
              The reciprocal of the condition number of the matrix A,
              computed as RCOND = 1/(norm(A) * norm(inv(A))).

       INFO    (output) int*
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value

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void dgsequ | ( | SuperMatrix * | A,
|  | double * | r,
|  | double * | c,
|  | double * | rowcnd,
|  | double * | colcnd,
|  | double * | amax,
|  | int * | info
| ) |  |

#

    Purpose

      DGSEQU computes row and column scalings intended to equilibrate an
      M-by-N sparse matrix A and reduce its condition number. R returns the row
      scale factors and C the column scale factors, chosen to try to make
      the largest element in each row and column of the matrix B with
      elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.

      R(i) and C(j) are restricted to be between SMLNUM = smallest safe
      number and BIGNUM = largest safe number.  Use of these scaling
      factors is not guaranteed to reduce the condition number of A but
      works well in practice.

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

#

    Arguments

      A       (input) SuperMatrix*
              The matrix of dimension (A->nrow, A->ncol) whose equilibration
              factors are to be computed. The type of A can be:
              Stype = SLU_NC; Dtype = SLU_D; Mtype = SLU_GE.

      R       (output) double*, size A->nrow
              If INFO = 0 or INFO > M, R contains the row scale factors
              for A.

      C       (output) double*, size A->ncol
              If INFO = 0,  C contains the column scale factors for A.

      ROWCND  (output) double*
              If INFO = 0 or INFO > M, ROWCND contains the ratio of the
              smallest R(i) to the largest R(i).  If ROWCND >= 0.1 and
              AMAX is neither too large nor too small, it is not worth
              scaling by R.

      COLCND  (output) double*
              If INFO = 0, COLCND contains the ratio of the smallest
              C(i) to the largest C(i).  If COLCND >= 0.1, it is not
              worth scaling by C.

      AMAX    (output) double*
              Absolute value of largest matrix element.  If AMAX is very
              close to overflow or very close to underflow, the matrix
              should be scaled.

      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->nrow:  the i-th row of A is exactly zero
                    >  A->ncol:  the (i-M)-th column of A is exactly zero

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void dgsisv | ( | superlu_options_t * | ,
|  | SuperMatrix * | ,
|  | int * | ,
|  | int * | ,
|  | SuperMatrix * | ,
|  | SuperMatrix * | ,
|  | SuperMatrix * | ,
|  | SuperLUStat_t * | ,
|  | int * |
| ) |  |

void dgsisx | ( | 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,
|  | GlobalLU_t * | Glu,
|  | mem_usage_t * | mem_usage,
|  | SuperLUStat_t * | stat,
|  | int * | info
| ) |  |

#

    Purpose

    DGSISX computes an approximate solutions of linear equations
    A*X=B or A'*X=B, using the ILU factorization from dgsitrf().
    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_MC64, 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, 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_MC64 )
              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_MC64, 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_D, 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, 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, dgsequ() is used to equilibrate the system,
               and A is scaled as above, but no permutation is involved.
               On exit, A is restored to the orginal row numbering, so
               Dr*A*Dc is returned.

    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 MC64 first then followed 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->Fact = 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_D, 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_D, 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_D, 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.

    X          (output) SuperMatrix*
            X has types: Stype = SLU_DN, Dtype = SLU_D, 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.

Here is the call graph for this function:

void dgsitrf | ( | superlu_options_t * | options,
|  | SuperMatrix * | A,
|  | int | relax,
|  | int | panel_size,
|  | int * | etree,
|  | void * | work,
|  | int | lwork,
|  | int * | perm_c,
|  | int * | perm_r,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | GlobalLU_t * | Glu,
|  | SuperLUStat_t * | stat,
|  | int * | info
| ) |  |

#

    Purpose

    DGSITRF computes an ILU factorization of a general sparse m-by-n
    matrix A using partial pivoting with row interchanges.
    The factorization has the form
        Pr * A = L * U
    where Pr is a row permutation matrix, L is lower triangular with unit
    diagonal elements (lower trapezoidal if A->nrow > A->ncol), and U is upper
    triangular (upper trapezoidal if A->nrow < A->ncol).

    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 ILU decomposition will be performed.

    A           (input) SuperMatrix*
             Original matrix A, permuted by columns, of dimension
             (A->nrow, A->ncol). The type of A can be:
             Stype = SLU_NCP; Dtype = SLU_D; Mtype = SLU_GE.

    relax    (input) int
             To control degree of relaxing supernodes. If the number
             of nodes (columns) in a subtree of the elimination tree is less
             than relax, this subtree is considered as one supernode,
             regardless of the row structures of those columns.

    panel_size (input) int
             A panel consists of at most panel_size consecutive columns.

    etree    (input) int*, dimension (A->ncol)
             Elimination tree of A'*A.
             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.
             On input, the columns of A should be permuted so that the
             etree is in a certain postorder.

    work     (input/output) void*, size (lwork) (in bytes)
             User-supplied work space and space for the output data structures.
             Not referenced if lwork = 0;

    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
             *info; no other side effects.

    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.
             When searching for diagonal, perm_c[*] is applied to the
             row subscripts of A, so that diagonal threshold pivoting
             can find the diagonal of A, rather than that of A*Pc.

    perm_r   (input/output) 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.
             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;

    L           (output) SuperMatrix*
             The factor L from the factorization Pr*A=L*U; use compressed row
             subscripts storage for supernodes, i.e., L has type:
             Stype = SLU_SC, Dtype = SLU_D, Mtype = SLU_TRLU.

    U           (output) SuperMatrix*
             The factor U from the factorization Pr*A*Pc=L*U. Use column-wise
             storage scheme, i.e., U has types: Stype = SLU_NC,
             Dtype = SLU_D, Mtype = SLU_TRU.

    Glu      (input/output) GlobalLU_t *
             If options->Fact == SamePattern_SameRowPerm, it is an input;
                 The matrix A will be factorized assuming that a
                 factorization of a matrix with the same sparsity pattern
                 and similar numerical values was performed prior to this one.
                 Therefore, this factorization will reuse both row and column
            scaling factors R and C, both row and column permutation
            vectors perm_r and perm_c, and the L & U data structures
            set up from the previous factorization.
             Otherwise, it is an output.

    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 according to options->ILU_FillTol.
                > A->ncol: number of bytes allocated when memory allocation
              failure occurred, plus A->ncol. If lwork = -1, it is
              the estimated amount of space needed, plus A->ncol.

#

    Local Working Arrays:

      m = number of rows in the matrix
      n = number of columns in the matrix

      marker[0:3*m-1]: marker[i] = j means that node i has been
         reached when working on column j.
         Storage: relative to original row subscripts
         NOTE: There are 4 of them:
               marker/marker1 are used for panel dfs, see (ilu_)dpanel_dfs.c;
               marker2 is used for inner-factorization, see (ilu)_dcolumn_dfs.c;
               marker_relax(has its own space) is used for relaxed supernodes.

      parent[0:m-1]: parent vector used during dfs
         Storage: relative to new row subscripts

      xplore[0:m-1]: xplore[i] gives the location of the next (dfs)
         unexplored neighbor of i in lsub[*]

      segrep[0:nseg-1]: contains the list of supernodal representatives
         in topological order of the dfs. A supernode representative is the
         last column of a supernode.
         The maximum size of segrep[] is n.

      repfnz[0:W*m-1]: for a nonzero segment U[*,j] that ends at a
         supernodal representative r, repfnz[r] is the location of the first
         nonzero in this segment.  It is also used during the dfs: repfnz[r]>0
         indicates the supernode r has been explored.
         NOTE: There are W of them, each used for one column of a panel.

      panel_lsub[0:W*m-1]: temporary for the nonzeros row indices below
         the panel diagonal. These are filled in during dpanel_dfs(), and are
         used later in the inner LU factorization within the panel.
         panel_lsub[]/dense[] pair forms the SPA data structure.
         NOTE: There are W of them.

      dense[0:W*m-1]: sparse accumulating (SPA) vector for intermediate values;
               NOTE: there are W of them.

      tempv[0:*]: real temporary used for dense numeric kernels;
         The size of this array is defined by NUM_TEMPV() in slu_util.h.
         It is also used by the dropping routine ilu_ddrop_row().

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void dgsrfs | ( | trans_t | trans,
|  | SuperMatrix * | A,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | int * | perm_c,
|  | int * | perm_r,
|  | char * | equed,
|  | double * | R,
|  | double * | C,
|  | SuperMatrix * | B,
|  | SuperMatrix * | X,
|  | double * | ferr,
|  | double * | berr,
|  | SuperLUStat_t * | stat,
|  | int * | info
| ) |  |

#

    Purpose

      DGSRFS 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_D, 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_D, Mtype = SLU_TRLU.

      U       (input) SuperMatrix*
              The factor U from the factorization Pr*A*Pc=L*U as computed by
              dgstrf(). Use column-wise storage scheme,
              i.e., U has types: Stype = SLU_NC, Dtype = SLU_D, 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) double*, 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) double*, 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_D, 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_D, Mtype = SLU_GE.
              On entry, the solution matrix X, as computed by dgstrs().
              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) double*, 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) double*, 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.

Here is the call graph for this function:

void dgssv | ( | superlu_options_t * | options,
|  | SuperMatrix * | A,
|  | int * | perm_c,
|  | int * | perm_r,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | SuperMatrix * | B,
|  | SuperLUStat_t * | stat,
|  | int * | info
| ) |  |

#

    Purpose

    DGSSV solves the system of linear equations A*X=B, using the
    LU factorization from DGSTRF. It performs the following steps:

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

         1.1. Permute the columns of A, forming A*Pc, where Pc
              is a permutation matrix. For more details of this step,
              see sp_preorder.c.

         1.2. Factor A as Pr*A*Pc=L*U with the permutation Pr determined
              by Gaussian elimination with partial pivoting.
              L is unit lower triangular with offdiagonal entries
              bounded by 1 in magnitude, and U is upper triangular.

         1.3. Solve the system of equations A*X=B using the factored
              form of A.

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

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

         2.2. Factor A as Pr*transpose(A)*Pc=L*U with the permutation Pr
              determined by Gaussian elimination with partial pivoting.
              L is unit lower triangular with offdiagonal entries
              bounded by 1 in magnitude, and U is upper triangular.

         2.3. Solve the system of equations A*X=B using the factored
              form of A.

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

    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.
            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.

            If options->ColPerm = MY_PERMC or options->Fact = SamePattern or
               options->Fact = SamePattern_SameRowPerm, it is an input argument.
               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.
            Otherwise, it is an output argument.

    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->RowPerm = MY_PERMR or
               options->Fact = SamePattern_SameRowPerm, perm_r is an
               input argument.
            otherwise it is an output argument.

    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_D, 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_D, Mtype = SLU_TRU.

    B       (input/output) SuperMatrix*
            B has types: Stype = SLU_DN, Dtype = SLU_D, Mtype = SLU_GE.
            On entry, the right hand side matrix.
            On exit, the solution matrix if info = 0;

    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, and i is
                <= A->ncol: U(i,i) is exactly zero. The factorization has
                   been completed, but the factor U is exactly singular,
                   so the solution could not be computed.
                > A->ncol: number of bytes allocated when memory allocation
                   failure occurred, plus A->ncol.

Here is the call graph for this function:

void dgssvx | ( | 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,
|  | double * | ferr,
|  | double * | berr,
|  | GlobalLU_t * | Glu,
|  | mem_usage_t * | mem_usage,
|  | SuperLUStat_t * | stat,
|  | int * | info
| ) |  |

#

    Purpose

    DGSSVX solves the system of linear equations A*X=B or A'*X=B, using
    the LU factorization from dgstrf(). Error bounds on the solution and
    a condition estimate are also provided. It performs the following steps:

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

         1.1. If options->Equil = YES, 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 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 = A->ncol+1 is returned as a warning, but the
              routine still goes on to solve for X and computes error bounds
              as described below.

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

         1.7. If options->IterRefine != NOREFINE, iterative refinement is
              applied to improve the computed solution matrix and calculate
              error bounds and backward error estimates for it.

         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.

      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, 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 returns with info = i. Otherwise, the factored form
              of transpose(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 = A->nrow+1 is returned as
              a warning, but the routine still goes on to solve for X and
              computes error bounds as described below.

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

         2.7. If options->IterRefine != NOREFINE, iterative refinement is
              applied to improve the computed solution matrix and calculate
              error bounds and backward error estimates for it.

         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_D, 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.
            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).

    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_D, 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_D, 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_D, 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.

    X       (output) SuperMatrix*
            X has types: Stype = SLU_DN, Dtype = SLU_D, 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.

    FERR    (output) double*, 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.
            If options->IterRefine = NOREFINE, ferr = 1.0.

    BERR    (output) double*, 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).
            If options->IterRefine = NOREFINE, berr = 1.0.

    Glu      (input/output) GlobalLU_t *
             If options->Fact == SamePattern_SameRowPerm, it is an input;
                 The matrix A will be factorized assuming that a
                 factorization of a matrix with the same sparsity pattern
                 and similar numerical values was performed prior to this one.
                 Therefore, this factorization will reuse both row and column
            scaling factors R and C, both row and column permutation
            vectors perm_r and perm_c, and the L & U data structures
            set up from the previous factorization.
             Otherwise, it is an output.

    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: 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.
                 = 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.

Here is the call graph for this function:

void dgstrf | ( | superlu_options_t * | options,
|  | SuperMatrix * | A,
|  | int | relax,
|  | int | panel_size,
|  | int * | etree,
|  | void * | work,
|  | int | lwork,
|  | int * | perm_c,
|  | int * | perm_r,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | GlobalLU_t * | Glu,
|  | SuperLUStat_t * | stat,
|  | int * | info
| ) |  |

#

    Purpose

    DGSTRF computes an LU factorization of a general sparse m-by-n
    matrix A using partial pivoting with row interchanges.
    The factorization has the form
        Pr * A = L * U
    where Pr is a row permutation matrix, L is lower triangular with unit
    diagonal elements (lower trapezoidal if A->nrow > A->ncol), and U is upper
    triangular (upper trapezoidal if A->nrow < A->ncol).

    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.

    A        (input) SuperMatrix*
             Original matrix A, permuted by columns, of dimension
             (A->nrow, A->ncol). The type of A can be:
             Stype = SLU_NCP; Dtype = SLU_D; Mtype = SLU_GE.

    relax    (input) int
             To control degree of relaxing supernodes. If the number
             of nodes (columns) in a subtree of the elimination tree is less
             than relax, this subtree is considered as one supernode,
             regardless of the row structures of those columns.

    panel_size (input) int
             A panel consists of at most panel_size consecutive columns.

    etree    (input) int*, dimension (A->ncol)
             Elimination tree of A'*A.
             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.
             On input, the columns of A should be permuted so that the
             etree is in a certain postorder.

    work     (input/output) void*, size (lwork) (in bytes)
             User-supplied work space and space for the output data structures.
             Not referenced if lwork = 0;

    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
                  *info; no other side effects.

    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.
             When searching for diagonal, perm_c[*] is applied to the
             row subscripts of A, so that diagonal threshold pivoting
             can find the diagonal of A, rather than that of A*Pc.

    perm_r   (input/output) 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.
             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;

    L        (output) SuperMatrix*
             The factor L from the factorization Pr*A=L*U; use compressed row
             subscripts storage for supernodes, i.e., L has type:
             Stype = SLU_SC, Dtype = SLU_D, Mtype = SLU_TRLU.

    U        (output) SuperMatrix*
             The factor U from the factorization Pr*A*Pc=L*U. Use column-wise
             storage scheme, i.e., U has types: Stype = SLU_NC,
             Dtype = SLU_D, Mtype = SLU_TRU.

    Glu      (input/output) GlobalLU_t *
             If options->Fact == SamePattern_SameRowPerm, it is an input;
                 The matrix A will be factorized assuming that a
                 factorization of a matrix with the same sparsity pattern
                 and similar numerical values was performed prior to this one.
                 Therefore, this factorization will reuse both row and column
            scaling factors R and C, both row and column permutation
            vectors perm_r and perm_c, and the L & U data structures
            set up from the previous factorization.
             Otherwise, it is an output.

    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: U(i,i) is exactly zero. The factorization has
                   been completed, but the factor U is exactly singular,
                   and division by zero will occur if it is used to solve a
                   system of equations.
                > A->ncol: number of bytes allocated when memory allocation
                   failure occurred, plus A->ncol. If lwork = -1, it is
                   the estimated amount of space needed, plus A->ncol.

#

    Local Working Arrays:

      m = number of rows in the matrix
      n = number of columns in the matrix

      xprune[0:n-1]: xprune[*] points to locations in subscript
         vector lsub[*]. For column i, xprune[i] denotes the point where
         structural pruning begins. I.e. only xlsub[i],..,xprune[i]-1 need
         to be traversed for symbolic factorization.

      marker[0:3*m-1]: marker[i] = j means that node i has been
         reached when working on column j.
         Storage: relative to original row subscripts
         NOTE: There are 3 of them: marker/marker1 are used for panel dfs,
               see dpanel_dfs.c; marker2 is used for inner-factorization,
               see dcolumn_dfs.c.

      parent[0:m-1]: parent vector used during dfs
         Storage: relative to new row subscripts

      xplore[0:m-1]: xplore[i] gives the location of the next (dfs)
         unexplored neighbor of i in lsub[*]

      segrep[0:nseg-1]: contains the list of supernodal representatives
         in topological order of the dfs. A supernode representative is the
         last column of a supernode.
         The maximum size of segrep[] is n.

      repfnz[0:W*m-1]: for a nonzero segment U[*,j] that ends at a
         supernodal representative r, repfnz[r] is the location of the first
         nonzero in this segment.  It is also used during the dfs: repfnz[r]>0
         indicates the supernode r has been explored.
         NOTE: There are W of them, each used for one column of a panel.

      panel_lsub[0:W*m-1]: temporary for the nonzeros row indices below
         the panel diagonal. These are filled in during dpanel_dfs(), and are
         used later in the inner LU factorization within the panel.
         panel_lsub[]/dense[] pair forms the SPA data structure.
         NOTE: There are W of them.

      dense[0:W*m-1]: sparse accumulating (SPA) vector for intermediate values;
                       NOTE: there are W of them.

      tempv[0:*]: real temporary used for dense numeric kernels;
         The size of this array is defined by NUM_TEMPV() in slu_ddefs.h.

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void dgstrs | ( | trans_t | trans,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | int * | perm_c,
|  | int * | perm_r,
|  | SuperMatrix * | B,
|  | SuperLUStat_t * | stat,
|  | int * | info
| ) |  |

#

    Purpose

    DGSTRS solves a system of linear equations A*X=B or A'*X=B
    with A sparse and B dense, using the LU factorization computed by
    DGSTRF.

    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)

    L       (input) SuperMatrix*
            The factor L from the factorization Pr*A*Pc=L*U as computed by
            dgstrf(). Use compressed row subscripts storage for supernodes,
            i.e., L has types: Stype = SLU_SC, Dtype = SLU_D, Mtype = SLU_TRLU.

    U       (input) SuperMatrix*
            The factor U from the factorization Pr*A*Pc=L*U as computed by
            dgstrf(). Use column-wise storage scheme, i.e., U has types:
            Stype = SLU_NC, Dtype = SLU_D, Mtype = SLU_TRU.

    perm_c  (input) int*, dimension (L->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 (L->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.

    B       (input/output) SuperMatrix*
            B has types: Stype = SLU_DN, Dtype = SLU_D, Mtype = SLU_GE.
            On entry, the right hand side matrix.
            On exit, the solution matrix if info = 0;

    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

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void dinf_norm_error | ( | int | ,
|  | SuperMatrix * | ,
|  | double * |
| ) |  |

void dlaqgs | ( | SuperMatrix * | A,
|  | double * | r,
|  | double * | c,
|  | double | rowcnd,
|  | double | colcnd,
|  | double | amax,
|  | char * | equed
| ) |  |

#

    Purpose

      DLAQGS equilibrates a general sparse M by N matrix A using the row and
      scaling factors in the vectors R and C.

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

#

    Arguments

      A       (input/output) SuperMatrix*
              On exit, the equilibrated matrix.  See EQUED for the form of
              the equilibrated matrix. The type of A can be:
             Stype = NC; Dtype = SLU_D; Mtype = GE.

      R       (input) double*, dimension (A->nrow)
              The row scale factors for A.

      C       (input) double*, dimension (A->ncol)
              The column scale factors for A.

      ROWCND  (input) double
              Ratio of the smallest R(i) to the largest R(i).

      COLCND  (input) double
              Ratio of the smallest C(i) to the largest C(i).

      AMAX    (input) double
              Absolute value of largest matrix entry.

      EQUED   (output) char*
              Specifies the form of equilibration that was done.
              = 'N':  No equilibration
              = '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).

#

    Internal Parameters

      THRESH is a threshold value used to decide if row or column scaling
      should be done based on the ratio of the row or column scaling
      factors.  If ROWCND < THRESH, row scaling is done, and if
      COLCND < THRESH, column scaling is done.

      LARGE and SMALL are threshold values used to decide if row scaling
      should be done based on the absolute size of the largest matrix
      element.  If AMAX > LARGE or AMAX < SMALL, row scaling is done.

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int dldperm | ( | int | ,
|  | int | ,
|  | int | ,
|  | int | [],
|  | int | [],
|  | double | [],
|  | int | [],
|  | double | [],
|  | double | []
| ) |  |

int dLUMemInit | ( | fact_t | fact,
|  | void * | work,
|  | int | lwork,
|  | int | m,
|  | int | n,
|  | int | annz,
|  | int | panel_size,
|  | double | fill_ratio,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | GlobalLU_t * | Glu,
|  | int ** | iwork,
|  | double ** | dwork
| ) |  |

Memory-related.

    For those unpredictable size, estimate as fill_ratio * nnz(A).
    Return value:
        If lwork = -1, return the estimated amount of space required, plus n;
        otherwise, return the amount of space actually allocated when
        memory allocation failure occurred.

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int dLUMemXpand | ( | int | jcol,
|  | int | next,
|  | MemType | mem_type,
|  | int * | maxlen,
|  | GlobalLU_t * | Glu
| ) |  |

    Return value:   0 - successful return
                  > 0 - number of bytes allocated when run out of space

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void dLUWorkFree | ( | int * | ,
|  | double * | ,
|  | GlobalLU_t * |
| ) |  |

double dmach | ( | char * |  | ) |

int dmemory_usage | ( | const int | ,
|  | const int | ,
|  | const int | ,
|  | const int |
| ) |  |

double* doubleCalloc | ( | int |  | ) |

double* doubleMalloc | ( | int |  | ) |

void dpanel_bmod | ( | const int | m,
|  | const int | w,
|  | const int | jcol,
|  | const int | nseg,
|  | double * | dense,
|  | double * | tempv,
|  | int * | segrep,
|  | int * | repfnz,
|  | GlobalLU_t * | Glu,
|  | SuperLUStat_t * | stat
| ) |  |

#

    Purpose

       Performs numeric block updates (sup-panel) in topological order.
       It features: col-col, 2cols-col, 3cols-col, and sup-col updates.
       Special processing on the supernodal portion of L[*,j]

       Before entering this routine, the original nonzeros in the panel
       were already copied into the spa[m,w].

       Updated/Output parameters-
       dense[0:m-1,w]: L[*,j:j+w-1] and U[*,j:j+w-1] are returned
       collectively in the m-by-w vector dense[*].

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void dpanel_dfs | ( | const int | m,
|  | const int | w,
|  | const int | jcol,
|  | SuperMatrix * | A,
|  | int * | perm_r,
|  | int * | nseg,
|  | double * | dense,
|  | int * | panel_lsub,
|  | int * | segrep,
|  | int * | repfnz,
|  | int * | xprune,
|  | int * | marker,
|  | int * | parent,
|  | int * | xplore,
|  | GlobalLU_t * | Glu
| ) |  |

#

    Purpose

      Performs a symbolic factorization on a panel of columns [jcol, jcol+w).

      A supernode representative is the last column of a supernode.
      The nonzeros in U[*,j] are segments that end at supernodal
      representatives.

      The routine returns one list of the supernodal representatives
      in topological order of the dfs that generates them. This list is
      a superset of the topological order of each individual column within
      the panel.
      The location of the first nonzero in each supernodal segment
      (supernodal entry location) is also returned. Each column has a
      separate list for this purpose.

      Two marker arrays are used for dfs:
        marker[i] == jj, if i was visited during dfs of current column jj;
        marker1[i] >= jcol, if i was visited by earlier columns in this panel;

      marker: A-row –> A-row/col (0/1)
      repfnz: SuperA-col –> PA-row
      parent: SuperA-col –> SuperA-col
      xplore: SuperA-col –> index to L-structure

double dPivotGrowth | ( | int | ncols,
|  | SuperMatrix * | A,
|  | int * | perm_c,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U
| ) |  |

#

    Purpose

    Compute the reciprocal pivot growth factor of the leading ncols columns
    of the matrix, using the formula:
        min_j ( max_i(abs(A_ij)) / max_i(abs(U_ij)) )

#

    Arguments

    ncols    (input) int
             The number of columns of matrices A, L and U.

    A        (input) SuperMatrix*
             Original matrix A, permuted by columns, of dimension
             (A->nrow, A->ncol). The type of A can be:
             Stype = NC; Dtype = SLU_D; Mtype = GE.

    L        (output) SuperMatrix*
             The factor L from the factorization Pr*A=L*U; use compressed row
             subscripts storage for supernodes, i.e., L has type:
             Stype = SC; Dtype = SLU_D; Mtype = TRLU.

    U        (output) SuperMatrix*
             The factor U from the factorization Pr*A*Pc=L*U. Use column-wise
             storage scheme, i.e., U has types: Stype = NC;
             Dtype = SLU_D; Mtype = TRU.

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int dpivotL | ( | const int | jcol,
|  | const double | u,
|  | int * | usepr,
|  | int * | perm_r,
|  | int * | iperm_r,
|  | int * | iperm_c,
|  | int * | pivrow,
|  | GlobalLU_t * | Glu,
|  | SuperLUStat_t * | stat
| ) |  |

#

    Purpose

      Performs the numerical pivoting on the current column of L,
      and the CDIV operation.

      Pivot policy:
      (1) Compute thresh = u * max_(i>=j) abs(A_ij);
      (2) IF user specifies pivot row k and abs(A_kj) >= thresh THEN
              pivot row = k;
          ELSE IF abs(A_jj) >= thresh THEN
              pivot row = j;
          ELSE
              pivot row = m;

      Note: If you absolutely want to use a given pivot order, then set u=0.0.

      Return value: 0      success;
                    i > 0  U(i,i) is exactly zero.

void dPrint_CompCol_Matrix | ( | char * | ,
|  | SuperMatrix * |
| ) |  |

void dPrint_Dense_Matrix | ( | char * | ,
|  | SuperMatrix * |
| ) |  |

void dprint_lu_col | ( | char * | ,
|  | int | ,
|  | int | ,
|  | int * | ,
|  | GlobalLU_t * |
| ) |  |

void dPrint_SuperNode_Matrix | ( | char * | ,
|  | SuperMatrix * |
| ) |  |

void dpruneL | ( | const int | jcol,
|  | const int * | perm_r,
|  | const int | pivrow,
|  | const int | nseg,
|  | const int * | segrep,
|  | const int * | repfnz,
|  | int * | xprune,
|  | GlobalLU_t * | Glu
| ) |  |

#

    Purpose

      Prunes the L-structure of supernodes whose L-structure
      contains the current pivot row "pivrow"

double dqselect | ( | int | ,
|  | double * | ,
|  | int |
| ) |  |

int dQuerySpace | ( | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | mem_usage_t * | mem_usage
| ) |  |

    mem_usage consists of the following fields:

    for_lu (float)
         The amount of space used in bytes for the L data structures.
    total_needed (float)
         The amount of space needed in bytes to perform factorization.

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void dreadhb | ( | FILE * | ,
|  | int * | ,
|  | int * | ,
|  | int * | ,
|  | double ** | ,
|  | int ** | ,
|  | int ** |
| ) |  |

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void dreadMM | ( | FILE * | fp,
|  | int * | m,
|  | int * | n,
|  | int * | nonz,
|  | double ** | nzval,
|  | int ** | rowind,
|  | int ** | colptr
| ) |  |

brief

#

    Output parameters

      (nzval, rowind, colptr): (*rowind)[*] contains the row subscripts of
         nonzeros in columns of matrix A; (*nzval)[*] the numerical values;
         column i of A is given by (*nzval)[k], k = (*rowind)[i],...,
         (*rowind)[i+1]-1.

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void dreadmt | ( | int * | ,
|  | int * | ,
|  | int * | ,
|  | double ** | ,
|  | int ** | ,
|  | int ** |
| ) |  |

void dreadrb | ( | int * | ,
|  | int * | ,
|  | int * | ,
|  | double ** | ,
|  | int ** | ,
|  | int ** |
| ) |  |

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void dreadtriple | ( | int * | ,
|  | int * | ,
|  | int * | ,
|  | double ** | ,
|  | int ** | ,
|  | int ** |
| ) |  |

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void dSetRWork | ( | int | ,
|  | int | ,
|  | double * | ,
|  | double ** | ,
|  | double ** |
| ) |  |

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int dsnode_bmod | ( | const int | ,
|  | const int | ,
|  | const int | ,
|  | double * | ,
|  | double * | ,
|  | GlobalLU_t * | ,
|  | SuperLUStat_t * |
| ) |  |

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int dsnode_dfs | ( | const int | jcol,
|  | const int | kcol,
|  | const int * | asub,
|  | const int * | xa_begin,
|  | const int * | xa_end,
|  | int * | xprune,
|  | int * | marker,
|  | GlobalLU_t * | Glu
| ) |  |

#

    Purpose

       dsnode_dfs() - Determine the union of the row structures of those
       columns within the relaxed snode.
       Note: The relaxed snodes are leaves of the supernodal etree, therefore,
       the portion outside the rectangular supernode must be zero.

#

    Return value

        0   success;
       >0   number of bytes allocated when run out of memory.

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int dtrsm_ | ( | char * | ,
|  | char * | ,
|  | char * | ,
|  | char * | ,
|  | int * | ,
|  | int * | ,
|  | double * | ,
|  | double * | ,
|  | int * | ,
|  | double * | ,
|  | int * |
| ) |  |

int dtrsv_ | ( | char * | ,
|  | char * | ,
|  | char * | ,
|  | int * | ,
|  | double * | ,
|  | int * | ,
|  | double * | ,
|  | int * |
| ) |  |

void fixupL | ( | const int | ,
|  | const int * | ,
|  | GlobalLU_t * |
| ) |  |

void ilu_countnz | ( | const int | ,
|  | int * | ,
|  | int * | ,
|  | GlobalLU_t * |
| ) |  |

int ilu_dcolumn_dfs | ( | const int | m,
|  | const int | jcol,
|  | int * | perm_r,
|  | int * | nseg,
|  | int * | lsub_col,
|  | int * | segrep,
|  | int * | repfnz,
|  | int * | marker,
|  | int * | parent,
|  | int * | xplore,
|  | GlobalLU_t * | Glu
| ) |  |

#

    Purpose

      ILU_DCOLUMN_DFS performs a symbolic factorization on column jcol, and
      decide the supernode boundary.

      This routine does not use numeric values, but only use the RHS
      row indices to start the dfs.

      A supernode representative is the last column of a supernode.
      The nonzeros in U[*,j] are segments that end at supernodal
      representatives. The routine returns a list of such supernodal
      representatives in topological order of the dfs that generates them.
      The location of the first nonzero in each such supernodal segment
      (supernodal entry location) is also returned.

#

    Local parameters

      nseg: no of segments in current U[*,j]
      jsuper: jsuper=EMPTY if column j does not belong to the same
         supernode as j-1. Otherwise, jsuper=nsuper.

      marker2: A-row –> A-row/col (0/1)
      repfnz: SuperA-col –> PA-row
      parent: SuperA-col –> SuperA-col
      xplore: SuperA-col –> index to L-structure

#

    Return value

        0  success;

    0  number of bytes allocated when run out of space.

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int ilu_dcopy_to_ucol | ( | int | ,
|  | int | ,
|  | int * | ,
|  | int * | ,
|  | int * | ,
|  | double * | ,
|  | int | ,
|  | milu_t | ,
|  | double | ,
|  | int | ,
|  | double * | ,
|  | int * | ,
|  | GlobalLU_t * | ,
|  | double * |
| ) |  |

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int ilu_ddrop_row | ( | superlu_options_t * | ,
|  | int | ,
|  | int | ,
|  | double | ,
|  | int | ,
|  | int * | ,
|  | double * | ,
|  | GlobalLU_t * | ,
|  | double * | ,
|  | double * | ,
|  | int |
| ) |  |

void ilu_dpanel_dfs | ( | const int | m,
|  | const int | w,
|  | const int | jcol,
|  | SuperMatrix * | A,
|  | int * | perm_r,
|  | int * | nseg,
|  | double * | dense,
|  | double * | amax,
|  | int * | panel_lsub,
|  | int * | segrep,
|  | int * | repfnz,
|  | int * | marker,
|  | int * | parent,
|  | int * | xplore,
|  | GlobalLU_t * | Glu
| ) |  |

#

    Purpose

      Performs a symbolic factorization on a panel of columns [jcol, jcol+w).

      A supernode representative is the last column of a supernode.
      The nonzeros in U[*,j] are segments that end at supernodal
      representatives.

      The routine returns one list of the supernodal representatives
      in topological order of the dfs that generates them. This list is
      a superset of the topological order of each individual column within
      the panel.
      The location of the first nonzero in each supernodal segment
      (supernodal entry location) is also returned. Each column has a
      separate list for this purpose.

      Two marker arrays are used for dfs:
        marker[i] == jj, if i was visited during dfs of current column jj;
        marker1[i] >= jcol, if i was visited by earlier columns in this panel;

      marker: A-row –> A-row/col (0/1)
      repfnz: SuperA-col –> PA-row
      parent: SuperA-col –> SuperA-col
      xplore: SuperA-col –> index to L-structure

int ilu_dpivotL | ( | const int | jcol,
|  | const double | u,
|  | int * | usepr,
|  | int * | perm_r,
|  | int | diagind,
|  | int * | swap,
|  | int * | iswap,
|  | int * | marker,
|  | int * | pivrow,
|  | double | fill_tol,
|  | milu_t | milu,
|  | double | drop_sum,
|  | GlobalLU_t * | Glu,
|  | SuperLUStat_t * | stat
| ) |  |

#

    Purpose

      Performs the numerical pivoting on the current column of L,
      and the CDIV operation.

      Pivot policy:
      (1) Compute thresh = u * max_(i>=j) abs(A_ij);
      (2) IF user specifies pivot row k and abs(A_kj) >= thresh THEN
              pivot row = k;
          ELSE IF abs(A_jj) >= thresh THEN
              pivot row = j;
          ELSE
              pivot row = m;

      Note: If you absolutely want to use a given pivot order, then set u=0.0.

      Return value: 0         success;
               i > 0  U(i,i) is exactly zero.

int ilu_dQuerySpace | ( | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | mem_usage_t * | mem_usage
| ) |  |

    mem_usage consists of the following fields:

    for_lu (float)
         The amount of space used in bytes for the L data structures.
    total_needed (float)
         The amount of space needed in bytes to perform factorization.

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int ilu_dsnode_dfs | ( | const int | jcol,
|  | const int | kcol,
|  | const int * | asub,
|  | const int * | xa_begin,
|  | const int * | xa_end,
|  | int * | marker,
|  | GlobalLU_t * | Glu
| ) |  |

#

    Purpose

       ilu_dsnode_dfs() - Determine the union of the row structures of those
       columns within the relaxed snode.
       Note: The relaxed snodes are leaves of the supernodal etree, therefore,
       the portion outside the rectangular supernode must be zero.

#

    Return value

        0   success;
       >0   number of bytes allocated when run out of memory.

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int print_double_vec | ( | char * | ,
|  | int | ,
|  | double * |
| ) |  |

int sp_dgemm | ( | char * | transa,
|  | char * | transb,
|  | int | m,
|  | int | n,
|  | int | k,
|  | double | alpha,
|  | SuperMatrix * | A,
|  | double * | b,
|  | int | ldb,
|  | double | beta,
|  | double * | c,
|  | int | ldc
| ) |  |

#

    Purpose

      sp_d performs one of the matrix-matrix operations

         C := alpha*op( A )*op( B ) + beta*C,

      where  op( X ) is one of

         op( X ) = X   or   op( X ) = X'   or   op( X ) = conjg( X' ),

      alpha and beta are scalars, and A, B and C are matrices, with op( A )
      an m by k matrix,  op( B )  a  k by n matrix and  C an m by n matrix.

#

    Parameters

      TRANSA - (input) char*
               On entry, TRANSA specifies the form of op( A ) to be used in
               the matrix multiplication as follows:
                  TRANSA = 'N' or 'n',  op( A ) = A.
                  TRANSA = 'T' or 't',  op( A ) = A'.
                  TRANSA = 'C' or 'c',  op( A ) = conjg( A' ).
               Unchanged on exit.

      TRANSB - (input) char*
               On entry, TRANSB specifies the form of op( B ) to be used in
               the matrix multiplication as follows:
                  TRANSB = 'N' or 'n',  op( B ) = B.
                  TRANSB = 'T' or 't',  op( B ) = B'.
                  TRANSB = 'C' or 'c',  op( B ) = conjg( B' ).
               Unchanged on exit.

      M      - (input) int
               On entry,  M  specifies  the number of rows of the matrix
              op( A ) and of the matrix C.  M must be at least zero.
              Unchanged on exit.

      N      - (input) int
               On entry,  N specifies the number of columns of the matrix
              op( B ) and the number of columns of the matrix C. N must be
              at least zero.
              Unchanged on exit.

      K      - (input) int
               On entry, K specifies the number of columns of the matrix
              op( A ) and the number of rows of the matrix op( B ). K must
              be at least  zero.
              Unchanged on exit.

      ALPHA  - (input) double
               On entry, ALPHA specifies the scalar alpha.

      A      - (input) SuperMatrix*
               Matrix A with a sparse format, of dimension (A->nrow, A->ncol).
               Currently, the type of A can be:
                   Stype = NC or NCP; Dtype = SLU_D; Mtype = GE.
               In the future, more general A can be handled.

      B      - DOUBLE PRECISION array of DIMENSION ( LDB, kb ), where kb is
               n when TRANSB = 'N' or 'n',  and is  k otherwise.
               Before entry with  TRANSB = 'N' or 'n',  the leading k by n
               part of the array B must contain the matrix B, otherwise
               the leading n by k part of the array B must contain the
               matrix B.
               Unchanged on exit.

      LDB    - (input) int
               On entry, LDB specifies the first dimension of B as declared
               in the calling (sub) program. LDB must be at least max( 1, n ).
               Unchanged on exit.

      BETA   - (input) double
               On entry, BETA specifies the scalar beta. When BETA is
               supplied as zero then C need not be set on input.

      C      - DOUBLE PRECISION array of DIMENSION ( LDC, n ).
               Before entry, the leading m by n part of the array C must
               contain the matrix C,  except when beta is zero, in which
               case C need not be set on entry.
               On exit, the array C is overwritten by the m by n matrix
              ( alpha*op( A )*B + beta*C ).

      LDC    - (input) int
               On entry, LDC specifies the first dimension of C as declared
               in the calling (sub)program. LDC must be at least max(1,m).
               Unchanged on exit.

      ==== Sparse Level 3 Blas routine.

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int sp_dgemv | ( | char * | trans,
|  | double | alpha,
|  | SuperMatrix * | A,
|  | double * | x,
|  | int | incx,
|  | double | beta,
|  | double * | y,
|  | int | incy
| ) |  |

#

    Purpose

      sp_dgemv()  performs one of the matrix-vector operations
         y := alpha*A*x + beta*y,   or   y := alpha*A'*x + beta*y,
      where alpha and beta are scalars, x and y are vectors and A is a
      sparse A->nrow by A->ncol matrix.

#

    Parameters

      TRANS  - (input) char*
               On entry, TRANS specifies the operation to be performed as
               follows:
                  TRANS = 'N' or 'n'   y := alpha*A*x + beta*y.
                  TRANS = 'T' or 't'   y := alpha*A'*x + beta*y.
                  TRANS = 'C' or 'c'   y := alpha*A'*x + beta*y.

      ALPHA  - (input) double
               On entry, ALPHA specifies the scalar alpha.

      A      - (input) SuperMatrix*
               Matrix A with a sparse format, of dimension (A->nrow, A->ncol).
               Currently, the type of A can be:
                   Stype = NC or NCP; Dtype = SLU_D; Mtype = GE.
               In the future, more general A can be handled.

      X      - (input) double*, array of DIMENSION at least
               ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
               and at least
               ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
               Before entry, the incremented array X must contain the
               vector x.

      INCX   - (input) int
               On entry, INCX specifies the increment for the elements of
               X. INCX must not be zero.

      BETA   - (input) double
               On entry, BETA specifies the scalar beta. When BETA is
               supplied as zero then Y need not be set on input.

      Y      - (output) double*,  array of DIMENSION at least
               ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
               and at least
               ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
               Before entry with BETA non-zero, the incremented array Y
               must contain the vector y. On exit, Y is overwritten by the
               updated vector y.

      INCY   - (input) int
               On entry, INCY specifies the increment for the elements of
               Y. INCY must not be zero.

      ==== Sparse Level 2 Blas routine.

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int sp_dtrsv | ( | char * | uplo,
|  | char * | trans,
|  | char * | diag,
|  | SuperMatrix * | L,
|  | SuperMatrix * | U,
|  | double * | x,
|  | SuperLUStat_t * | stat,
|  | int * | info
| ) |  |

#

    Purpose

      sp_dtrsv() solves one of the systems of equations
          A*x = b,   or   A'*x = b,
      where b and x are n element vectors and A is a sparse unit , or
      non-unit, upper or lower triangular matrix.
      No test for singularity or near-singularity is included in this
      routine. Such tests must be performed before calling this routine.

#

    Parameters

      uplo   - (input) char*
               On entry, uplo specifies whether the matrix is an upper or
                lower triangular matrix as follows:
                   uplo = 'U' or 'u'   A is an upper triangular matrix.
                   uplo = 'L' or 'l'   A is a lower triangular matrix.

      trans  - (input) char*
                On entry, trans specifies the equations to be solved as
                follows:
                   trans = 'N' or 'n'   A*x = b.
                   trans = 'T' or 't'   A'*x = b.
                   trans = 'C' or 'c'   A'*x = b.

      diag   - (input) char*
                On entry, diag specifies whether or not A is unit
                triangular as follows:
                   diag = 'U' or 'u'   A is assumed to be unit triangular.
                   diag = 'N' or 'n'   A is not assumed to be unit
                                       triangular.

      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 = SC, Dtype = SLU_D, Mtype = TRLU.

      U       - (input) SuperMatrix*
                 The factor U from the factorization Pr*A*Pc=L*U.
                 U has types: Stype = NC, Dtype = SLU_D, Mtype = TRU.

      x       - (input/output) double*
                Before entry, the incremented array X must contain the n
                element right-hand side vector b. On exit, X is overwritten
                with the solution vector x.

      info    - (output) int*
                If *info = -i, the i-th argument had an illegal value.

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