<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html><head><meta http-equiv="Content-Type" content="text/html;charset=UTF-8"> <title>SuperLU: SRC/dgsisx.c File Reference</title> <link href="doxygen.css" rel="stylesheet" type="text/css"> <link href="tabs.css" rel="stylesheet" type="text/css"> </head><body> <!-- Generated by Doxygen 1.5.5 --> <div class="navigation" id="top"> <div class="tabs"> <ul> <li><a href="index.html"><span>Main Page</span></a></li> <li><a href="annotated.html"><span>Data Structures</span></a></li> <li class="current"><a href="files.html"><span>Files</span></a></li> </ul> </div> </div> <div class="contents"> <h1>SRC/dgsisx.c File Reference</h1>Computes an approximate solutions of linear equations A*X=B or A'*X=B. <a href="#_details">More...</a> <p> <code>#include "<a class="el" href="slu__ddefs_8h-source.html">slu_ddefs.h</a>"</code><br> <table border="0" cellpadding="0" cellspacing="0"> <tr><td></td></tr> <tr><td colspan="2"><br><h2>Functions</h2></td></tr> <tr><td class="memItemLeft" nowrap align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="dgsisx_8c.html#bcbb363b5bc23f538d1210f39bb1d672">dgsisx</a> (<a class="el" href="structsuperlu__options__t.html">superlu_options_t</a> *options, <a class="el" href="structSuperMatrix.html">SuperMatrix</a> *<a class="el" href="ilu__zdrop__row_8c.html#c900805a486cbb8489e3c176ed6e0d8e">A</a>, int *perm_c, int *perm_r, int *etree, char *equed, double *R, double *C, <a class="el" href="structSuperMatrix.html">SuperMatrix</a> *L, <a class="el" href="structSuperMatrix.html">SuperMatrix</a> *U, void *work, int lwork, <a class="el" href="structSuperMatrix.html">SuperMatrix</a> *B, <a class="el" href="structSuperMatrix.html">SuperMatrix</a> *X, double *recip_pivot_growth, double *rcond, <a class="el" href="structmem__usage__t.html">mem_usage_t</a> *mem_usage, <a class="el" href="structSuperLUStat__t.html">SuperLUStat_t</a> *stat, int *info)</td></tr> </table> <hr><a name="_details"></a><h2>Detailed Description</h2> <pre> -- SuperLU routine (version 4.2) -- Lawrence Berkeley National Laboratory. November, 2010 August, 2011 </pre> <hr><h2>Function Documentation</h2> <a class="anchor" name="bcbb363b5bc23f538d1210f39bb1d672"></a><!-- doxytag: member="dgsisx.c::dgsisx" ref="bcbb363b5bc23f538d1210f39bb1d672" args="(superlu_options_t *options, SuperMatrix *A, int *perm_c, int *perm_r, int *etree, char *equed, double *R, double *C, SuperMatrix *L, SuperMatrix *U, void *work, int lwork, SuperMatrix *B, SuperMatrix *X, double *recip_pivot_growth, double *rcond, mem_usage_t *mem_usage, SuperLUStat_t *stat, int *info)" --> <div class="memitem"> <div class="memproto"> <table class="memname"> <tr> <td class="memname">void dgsisx </td> <td>(</td> <td class="paramtype"><a class="el" href="structsuperlu__options__t.html">superlu_options_t</a> * </td> <td class="paramname"> <em>options</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype"><a class="el" href="structSuperMatrix.html">SuperMatrix</a> * </td> <td class="paramname"> <em>A</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">int * </td> <td class="paramname"> <em>perm_c</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">int * </td> <td class="paramname"> <em>perm_r</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">int * </td> <td class="paramname"> <em>etree</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">char * </td> <td class="paramname"> <em>equed</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">double * </td> <td class="paramname"> <em>R</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">double * </td> <td class="paramname"> <em>C</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype"><a class="el" href="structSuperMatrix.html">SuperMatrix</a> * </td> <td class="paramname"> <em>L</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype"><a class="el" href="structSuperMatrix.html">SuperMatrix</a> * </td> <td class="paramname"> <em>U</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">void * </td> <td class="paramname"> <em>work</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">int </td> <td class="paramname"> <em>lwork</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype"><a class="el" href="structSuperMatrix.html">SuperMatrix</a> * </td> <td class="paramname"> <em>B</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype"><a class="el" href="structSuperMatrix.html">SuperMatrix</a> * </td> <td class="paramname"> <em>X</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">double * </td> <td class="paramname"> <em>recip_pivot_growth</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">double * </td> <td class="paramname"> <em>rcond</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype"><a class="el" href="structmem__usage__t.html">mem_usage_t</a> * </td> <td class="paramname"> <em>mem_usage</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype"><a class="el" href="structSuperLUStat__t.html">SuperLUStat_t</a> * </td> <td class="paramname"> <em>stat</em>, </td> </tr> <tr> <td class="paramkey"></td> <td></td> <td class="paramtype">int * </td> <td class="paramname"> <em>info</em></td><td> </td> </tr> <tr> <td></td> <td>)</td> <td></td><td></td><td width="100%"></td> </tr> </table> </div> <div class="memdoc"> <p> <pre> Purpose =======</pre><p> <pre> DGSISX computes an approximate solutions of linear equations A*X=B or A'*X=B, using the ILU factorization from <a class="el" href="dgsitrf_8c.html#ffb50badd1ff77dbe19f6df360f98e50">dgsitrf()</a>. An estimation of the condition number is provided. The routine performs the following steps:</pre><p> <pre> 1. If A is stored column-wise (A->Stype = SLU_NC):</pre><p> <pre> 1.1. If options->Equil = YES or options->RowPerm = LargeDiag, scaling factors are computed to equilibrate the system: options->Trans = NOTRANS: diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B options->Trans = TRANS: (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B options->Trans = CONJ: (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if options->Trans=NOTRANS) or diag(C)*B (if options->Trans = TRANS or CONJ).</pre><p> <pre> 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 <a class="el" href="sp__preorder_8c.html" title="Permute and performs functions on columns of orginal matrix.">sp_preorder.c</a>.</pre><p> <pre> 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.</pre><p> <pre> 1.4. Compute the reciprocal pivot growth factor.</pre><p> <pre> 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.</pre><p> <pre> 1.6. The system of equations is solved for X using the factored form of A.</pre><p> <pre> 1.7. options->IterRefine is not used</pre><p> <pre> 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.</pre><p> <pre> 1.9. options for ILU only 1) If options->RowPerm = LargeDiag, MC64 is used to scale and permute the matrix to an I-matrix, that is Pr*Dr*A*Dc has entries of modulus 1 on the diagonal and off-diagonal entries of modulus at most 1. If MC64 fails, <a class="el" href="dgsequ_8c.html#af22b247cc134fb0ba90285e84ccebb4" title="Driver related.">dgsequ()</a> is used to equilibrate the system. ( Default: LargeDiag ) 2) options->ILU_DropTol = tau is the threshold for dropping. For L, it is used directly (for the whole row in a supernode); For U, ||A(:,i)||_oo * tau is used as the threshold for the i-th column. If a secondary dropping rule is required, tau will also be used to compute the second threshold. ( Default: 1e-4 ) 3) options->ILU_FillFactor = gamma, used as the initial guess of memory growth. If a secondary dropping rule is required, it will also be used as an upper bound of the memory. ( Default: 10 ) 4) options->ILU_DropRule specifies the dropping rule. Option Meaning ====== =========== DROP_BASIC: Basic dropping rule, supernodal based ILUTP(tau). DROP_PROWS: Supernodal based ILUTP(p,tau), p = gamma*nnz(A)/n. DROP_COLUMN: Variant of ILUTP(p,tau), for j-th column, p = gamma * nnz(A(:,j)). DROP_AREA: Variation of ILUTP, for j-th column, use nnz(F(:,1:j)) / nnz(A(:,1:j)) to control memory. DROP_DYNAMIC: Modify the threshold tau during factorizaion: If nnz(L(:,1:j)) / nnz(A(:,1:j)) > gamma tau_L(j) := MIN(tau_0, tau_L(j-1) * 2); Otherwise tau_L(j) := MAX(tau_0, tau_L(j-1) / 2); tau_U(j) uses the similar rule. NOTE: the thresholds used by L and U are separate. DROP_INTERP: Compute the second dropping threshold by interpolation instead of sorting (default). In this case, the actual fill ratio is not guaranteed smaller than gamma. DROP_PROWS, DROP_COLUMN and DROP_AREA are mutually exclusive. ( Default: DROP_BASIC | DROP_AREA ) 5) options->ILU_Norm is the criterion of measuring the magnitude of a row in a supernode of L. ( Default is INF_NORM ) options->ILU_Norm RowSize(x[1:n]) ================= =============== ONE_NORM ||x||_1 / n TWO_NORM ||x||_2 / sqrt(n) INF_NORM max{|x[i]|} 6) options->ILU_MILU specifies the type of MILU's variation. = SILU: do not perform Modified ILU; = SMILU_1 (not recommended): U(i,i) := U(i,i) + sum(dropped entries); = SMILU_2: U(i,i) := U(i,i) + SGN(U(i,i)) * sum(dropped entries); = SMILU_3: U(i,i) := U(i,i) + SGN(U(i,i)) * sum(|dropped entries|); NOTE: Even SMILU_1 does not preserve the column sum because of late dropping. ( Default: SILU ) 7) options->ILU_FillTol is used as the perturbation when encountering zero pivots. If some U(i,i) = 0, so that U is exactly singular, then U(i,i) := ||A(:,i)|| * options->ILU_FillTol ** (1 - i / n). ( Default: 1e-2 )</pre><p> <pre> 2. If A is stored row-wise (A->Stype = SLU_NR), apply the above algorithm to the transpose of A:</pre><p> <pre> 2.1. If options->Equil = YES or options->RowPerm = LargeDiag, scaling factors are computed to equilibrate the system: options->Trans = NOTRANS: diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B options->Trans = TRANS: (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B options->Trans = CONJ: (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A' is overwritten by diag(R)*A'*diag(C) and B by diag(R)*B (if trans='N') or diag(C)*B (if trans = 'T' or 'C').</pre><p> <pre> 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 <a class="el" href="sp__preorder_8c.html" title="Permute and performs functions on columns of orginal matrix.">sp_preorder.c</a>.</pre><p> <pre> 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.</pre><p> <pre> 2.4. Compute the reciprocal pivot growth factor.</pre><p> <pre> 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.</pre><p> <pre> 2.6. The system of equations is solved for X using the factored form of transpose(A).</pre><p> <pre> 2.7. If options->IterRefine is not used.</pre><p> <pre> 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.</pre><p> <pre> See <a class="el" href="supermatrix_8h.html" title="Defines matrix types.">supermatrix.h</a> for the definition of 'SuperMatrix' structure.</pre><p> <pre> Arguments =========</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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).</pre><p> <pre> If options->RowPerm = LargeDiag, MC64 is used to scale and permute the matrix to an I-matrix, that is A is modified as follows: P*Dr*A*Dc has entries of modulus 1 on the diagonal and off-diagonal entries of modulus at most 1. P is a permutation obtained from MC64. If MC64 fails, <a class="el" href="dgsequ_8c.html#af22b247cc134fb0ba90285e84ccebb4" title="Driver related.">dgsequ()</a> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> See argument 'mem_usage' for memory usage statistics.</pre><p> <pre> 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.</pre><p> <pre> 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'.</pre><p> <pre> 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.</pre><p> <pre> 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.</pre><p> <pre> mem_usage (output) mem_usage_t* Record the memory usage statistics, consisting of following fields:<ul> <li>for_lu (float) The amount of space used in bytes for L data structures.</li><li>total_needed (float) The amount of space needed in bytes to perform factorization.</li><li>expansions (int) The number of memory expansions during the LU factorization.</li></ul> </pre><p> <pre> stat (output) SuperLUStat_t* Record the statistics on runtime and floating-point operation count. See <a class="el" href="slu__util_8h.html" title="Utility header file.">slu_util.h</a> for the definition of 'SuperLUStat_t'.</pre><p> <pre> 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. </pre> </div> </div><p> </div> <hr size="1"><address style="text-align: right;"><small>Generated on Thu Aug 25 13:43:49 2011 for SuperLU by <a href="http://www.doxygen.org/index.html"> <img src="doxygen.png" alt="doxygen" align="middle" border="0"></a> 1.5.5 </small></address> </body> </html>