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/* ========================================================================== */ /* === ldl.c: sparse LDL' factorization and solve package =================== */ /* ========================================================================== */ /* LDL: a simple set of routines for sparse LDL' factorization. These routines * are not terrifically fast (they do not use dense matrix kernels), but the * code is very short. The purpose is to illustrate the algorithms in a very * concise manner, primarily for educational purposes. Although the code is * very concise, this package is slightly faster than the built-in sparse * Cholesky factorization in MATLAB 6.5 (chol), when using the same input * permutation. * * The routines compute the LDL' factorization of a real sparse symmetric * matrix A (or PAP' if a permutation P is supplied), and solve upper * and lower triangular systems with the resulting L and D factors. If A is * positive definite then the factorization will be accurate. A can be * indefinite (with negative values on the diagonal D), but in this case no * guarantee of accuracy is provided, since no numeric pivoting is performed. * * The n-by-n sparse matrix A is in compressed-column form. The nonzero values * in column j are stored in Ax [Ap [j] ... Ap [j+1]-1], with corresponding row * indices in Ai [Ap [j] ... Ap [j+1]-1]. Ap [0] = 0 is required, and thus * nz = Ap [n] is the number of nonzeros in A. Ap is an int array of size n+1. * The int array Ai and the double array Ax are of size nz. This data structure * is identical to the one used by MATLAB, except for the following * generalizations. The row indices in each column of A need not be in any * particular order, although they must be in the range 0 to n-1. Duplicate * entries can be present; any duplicates are summed. That is, if row index i * appears twice in a column j, then the value of A (i,j) is the sum of the two * entries. The data structure used here for the input matrix A is more * flexible than MATLAB's, which requires sorted columns with no duplicate * entries. * * Only the diagonal and upper triangular part of A (or PAP' if a permutation * P is provided) is accessed. The lower triangular parts of the matrix A or * PAP' can be present, but they are ignored. * * The optional input permutation is provided as an array P of length n. If * P [k] = j, the row and column j of A is the kth row and column of PAP'. * If P is present then the factorization is LDL' = PAP' or L*D*L' = A(P,P) in * 0-based MATLAB notation. If P is not present (a null pointer) then no * permutation is performed, and the factorization is LDL' = A. * * The lower triangular matrix L is stored in the same compressed-column * form (an int Lp array of size n+1, an int Li array of size Lp [n], and a * double array Lx of the same size as Li). It has a unit diagonal, which is * not stored. The row indices in each column of L are always returned in * ascending order, with no duplicate entries. This format is compatible with * MATLAB, except that it would be more convenient for MATLAB to include the * unit diagonal of L. Doing so here would add additional complexity to the * code, and is thus omitted in the interest of keeping this code short and * readable. * * The elimination tree is held in the Parent [0..n-1] array. It is normally * not required by the user, but it is required by ldl_numeric. The diagonal * matrix D is held as an array D [0..n-1] of size n. * * -------------------- * C-callable routines: * -------------------- * * ldl_symbolic: Given the pattern of A, computes the Lp and Parent arrays * required by ldl_numeric. Takes time proportional to the number of * nonzeros in L. Computes the inverse Pinv of P if P is provided. * Also returns Lnz, the count of nonzeros in each column of L below * the diagonal (this is not required by ldl_numeric). * ldl_numeric: Given the pattern and numerical values of A, the Lp array, * the Parent array, and P and Pinv if applicable, computes the * pattern and numerical values of L and D. * ldl_lsolve: Solves Lx=b for a dense vector b. * ldl_dsolve: Solves Dx=b for a dense vector b. * ldl_ltsolve: Solves L'x=b for a dense vector b. * ldl_perm: Computes x=Pb for a dense vector b. * ldl_permt: Computes x=P'b for a dense vector b. * ldl_valid_perm: checks the validity of a permutation vector * ldl_valid_matrix: checks the validity of the sparse matrix A * * ---------------------------- * Limitations of this package: * ---------------------------- * * In the interest of keeping this code simple and readable, ldl_symbolic and * ldl_numeric assume their inputs are valid. You can check your own inputs * prior to calling these routines with the ldl_valid_perm and ldl_valid_matrix * routines. Except for the two ldl_valid_* routines, no routine checks to see * if the array arguments are present (non-NULL). Like all C routines, no * routine can determine if the arrays are long enough and don't overlap. * * The ldl_numeric does check the numerical factorization, however. It returns * n if the factorization is successful. If D (k,k) is zero, then k is * returned, and L is only partially computed. * * No pivoting to control fill-in is performed, which is often critical for * obtaining good performance. I recommend that you compute the permutation P * using AMD or SYMAMD (approximate minimum degree ordering routines), or an * appropriate graph-partitioning based ordering. See the ldldemo.m routine for * an example in MATLAB, and the ldlmain.c stand-alone C program for examples of * how to find P. Routines for manipulating compressed-column matrices are * available in UMFPACK. AMD, SYMAMD, UMFPACK, and this LDL package are all * available at http://www.cise.ufl.edu/research/sparse. * * ------------------------- * Possible simplifications: * ------------------------- * * These routines could be made even simpler with a few additional assumptions. * If no input permutation were performed, the caller would have to permute the * matrix first, but the computation of Pinv, and the use of P and Pinv could be * removed. If only the diagonal and upper triangular part of A or PAP' are * present, then the tests in the "if (i < k)" statement in ldl_symbolic and * "if (i <= k)" in ldl_numeric, are always true, and could be removed (i can * equal k in ldl_symbolic, but then the body of the if statement would * correctly do no work since Flag [k] == k). If we could assume that no * duplicate entries are present, then the statement Y [i] += Ax [p] could be * replaced with Y [i] = Ax [p] in ldl_numeric. * * -------------------------- * Description of the method: * -------------------------- * * LDL computes the symbolic factorization by finding the pattern of L one row * at a time. It does this based on the following theory. Consider a sparse * system Lx=b, where L, x, and b, are all sparse, and where L comes from a * Cholesky (or LDL') factorization. The elimination tree (etree) of L is * defined as follows. The parent of node j is the smallest k > j such that * L (k,j) is nonzero. Node j has no parent if column j of L is completely zero * below the diagonal (j is a root of the etree in this case). The nonzero * pattern of x is the union of the paths from each node i to the root, for * each nonzero b (i). To compute the numerical solution to Lx=b, we can * traverse the columns of L corresponding to nonzero values of x. This * traversal does not need to be done in the order 0 to n-1. It can be done in * any "topological" order, such that x (i) is computed before x (j) if i is a * descendant of j in the elimination tree. * * The row-form of the LDL' factorization is shown in the MATLAB function * ldlrow.m in this LDL package. Note that row k of L is found via a sparse * triangular solve of L (1:k-1, 1:k-1) \ A (1:k-1, k), to use 1-based MATLAB * notation. Thus, we can start with the nonzero pattern of the kth column of * A (above the diagonal), follow the paths up to the root of the etree of the * (k-1)-by-(k-1) leading submatrix of L, and obtain the pattern of the kth row * of L. Note that we only need the leading (k-1)-by-(k-1) submatrix of L to * do this. The elimination tree can be constructed as we go. * * The symbolic factorization does the same thing, except that it discards the * pattern of L as it is computed. It simply counts the number of nonzeros in * each column of L and then constructs the Lp index array when it's done. The * symbolic factorization does not need to do this in topological order. * Compare ldl_symbolic with the first part of ldl_numeric, and note that the * while (len > 0) loop is not present in ldl_symbolic. * * LDL Version 1.0 (Dec. 31, 2003), Copyright (c) 2003 by Timothy A Davis, * University of Florida. All Rights Reserved. Developed while on sabbatical * at Stanford University and Lawrence Berkeley National Laboratory. Refer to * the README file for the License. Available at * http://www.cise.ufl.edu/research/sparse. */ #include "ldl.h" /* ========================================================================== */ /* === ldl_symbolic ========================================================= */ /* ========================================================================== */ /* The input to this routine is a sparse matrix A, stored in column form, and * an optional permutation P. The output is the elimination tree * and the number of nonzeros in each column of L. Parent [i] = k if k is the * parent of i in the tree. The Parent array is required by ldl_numeric. * Lnz [k] gives the number of nonzeros in the kth column of L, excluding the * diagonal. * * One workspace vector (Flag) of size n is required. * * If P is NULL, then it is ignored. The factorization will be LDL' = A. * Pinv is not computed. In this case, neither P nor Pinv are required by * ldl_numeric. * * If P is not NULL, then it is assumed to be a valid permutation. If * row and column j of A is the kth pivot, the P [k] = j. The factorization * will be LDL' = PAP', or A (p,p) in MATLAB notation. The inverse permutation * Pinv is computed, where Pinv [j] = k if P [k] = j. In this case, both P * and Pinv are required as inputs to ldl_numeric. * * The floating-point operation count of the subsequent call to ldl_numeric * is not returned, but could be computed after ldl_symbolic is done. It is * the sum of (Lnz [k]) * (Lnz [k] + 2) for k = 0 to n-1. */ void ldl_symbolic ( int n, /* A and L are n-by-n, where n >= 0 */ int Ap [ ], /* input of size n+1, not modified */ int Ai [ ], /* input of size nz=Ap[n], not modified */ int Lp [ ], /* output of size n+1, not defined on input */ int Parent [ ], /* output of size n, not defined on input */ int Lnz [ ], /* output of size n, not defined on input */ int Flag [ ], /* workspace of size n, not defn. on input or output */ int P [ ], /* optional input of size n */ int Pinv [ ] /* optional output of size n (used if P is not NULL) */ ) { int i, k, p, kk, p2 ; if (P) { /* If P is present then compute Pinv, the inverse of P */ for (k = 0 ; k < n ; k++) { Pinv [P [k]] = k ; } } for (k = 0 ; k < n ; k++) { /* L(k,:) pattern: all nodes reachable in etree from nz in A(0:k-1,k) */ Parent [k] = -1 ; /* parent of k is not yet known */ Flag [k] = k ; /* mark node k as visited */ Lnz [k] = 0 ; /* count of nonzeros in column k of L */ kk = (P) ? (P [k]) : (k) ; /* kth original, or permuted, column */ p2 = Ap [kk+1] ; for (p = Ap [kk] ; p < p2 ; p++) { /* A (i,k) is nonzero (original or permuted A) */ i = (Pinv) ? (Pinv [Ai [p]]) : (Ai [p]) ; if (i < k) { /* follow path from i to root of etree, stop at flagged node */ for ( ; Flag [i] != k ; i = Parent [i]) { /* find parent of i if not yet determined */ if (Parent [i] == -1) { Parent [i] = k ; } Lnz [i]++ ; /* L (k,i) is nonzero */ Flag [i] = k ; /* mark i as visited */ } } } } /* construct Lp index array from Lnz column counts */ Lp [0] = 0 ; for (k = 0 ; k < n ; k++) { Lp [k+1] = Lp [k] + Lnz [k] ; } } /* ========================================================================== */ /* === ldl_numeric ========================================================== */ /* ========================================================================== */ /* Given a sparse matrix A (the arguments n, Ap, Ai, and Ax) and its symbolic * analysis (Lp and Parent, and optionally P and Pinv), compute the numeric LDL' * factorization of A or PAP'. The outputs of this routine are arguments Li, * Lx, and D. It also requires three size-n workspaces (Y, Pattern, and Flag). */ int ldl_numeric /* returns n if successful, k if D (k,k) is zero */ ( int n, /* A and L are n-by-n, where n >= 0 */ int Ap [ ], /* input of size n+1, not modified */ int Ai [ ], /* input of size nz=Ap[n], not modified */ double Ax [ ], /* input of size nz=Ap[n], not modified */ int Lp [ ], /* input of size n+1, not modified */ int Parent [ ], /* input of size n, not modified */ int Lnz [ ], /* output of size n, not defn. on input */ int Li [ ], /* output of size lnz=Lp[n], not defined on input */ double Lx [ ], /* output of size lnz=Lp[n], not defined on input */ double D [ ], /* output of size n, not defined on input */ double Y [ ], /* workspace of size n, not defn. on input or output */ int Pattern [ ], /* workspace of size n, not defn. on input or output */ int Flag [ ], /* workspace of size n, not defn. on input or output */ int P [ ], /* optional input of size n */ int Pinv [ ] /* optional input of size n */ ) { double yi, l_ki ; int i, k, p, kk, p2, len, top ; for (k = 0 ; k < n ; k++) { /* compute nonzero Pattern of kth row of L, in topological order */ Y [k] = 0.0 ; /* Y (0:k) is now all zero */ top = n ; /* stack for pattern is empty */ Flag [k] = k ; /* mark node k as visited */ Lnz [k] = 0 ; /* count of nonzeros in column k of L */ kk = (P) ? (P [k]) : (k) ; /* kth original, or permuted, column */ p2 = Ap [kk+1] ; for (p = Ap [kk] ; p < p2 ; p++) { i = (Pinv) ? (Pinv [Ai [p]]) : (Ai [p]) ; /* get A(i,k) */ if (i <= k) { Y [i] += Ax [p] ; /* scatter A(i,k) into Y (sum duplicates) */ /* follow path from i to root of etree, stop at flagged node */ for (len = 0 ; Flag [i] != k ; i = Parent [i]) { Pattern [len++] = i ; /* L (k,i) is nonzero */ Flag [i] = k ; /* mark i as visited */ } while (len > 0) /* push path on top of stack */ { Pattern [--top] = Pattern [--len] ; } } } /* Pattern [top ... n-1] now contains nonzero pattern of L (:,k) */ /* compute numerical values kth row of L (a sparse triangular solve) */ D [k] = Y [k] ; /* get D (k,k) and clear Y (k) */ Y [k] = 0.0 ; for ( ; top < n ; top++) { i = Pattern [top] ; yi = Y [i] ; /* get and clear Y (i) */ Y [i] = 0.0 ; p2 = Lp [i] + Lnz [i] ; for (p = Lp [i] ; p < p2 ; p++) { Y [Li [p]] -= Lx [p] * yi ; } l_ki = yi / D [i] ; /* the nonzero entry L (k,i) */ D [k] -= l_ki * yi ; Li [p] = k ; /* store L(k,k )in column form of L */ Lx [p] = l_ki ; Lnz [i]++ ; /* increment count of nonzeros in col i */ } if (D [k] == 0.0) { return (k) ; /* failure, D (k,k) is zero */ } } return (n) ; /* success, diagonal of D is all nonzero */ } /* ========================================================================== */ /* === ldl_lsolve: solve Lx=b ============================================== */ /* ========================================================================== */ void ldl_lsolve ( int n, /* L is n-by-n, where n >= 0 */ double X [ ], /* size n. right-hand-side on input, soln. on output */ int Lp [ ], /* input of size n+1, not modified */ int Li [ ], /* input of size lnz=Lp[n], not modified */ double Lx [ ] /* input of size lnz=Lp[n], not modified */ ) { int j, p, p2 ; for (j = 0 ; j < n ; j++) { p2 = Lp [j+1] ; for (p = Lp [j] ; p < p2 ; p++) { X [Li [p]] -= Lx [p] * X [j] ; } } } /* ========================================================================== */ /* === ldl_dsolve: solve Dx=b ============================================== */ /* ========================================================================== */ void ldl_dsolve ( int n, /* D is n-by-n, where n >= 0 */ double X [ ], /* size n. right-hand-side on input, soln. on output */ double D [ ] /* input of size n, not modified */ ) { int j ; for (j = 0 ; j < n ; j++) { X [j] /= D [j] ; } } /* ========================================================================== */ /* === ldl_ltsolve: solve L'x=b ============================================ */ /* ========================================================================== */ void ldl_ltsolve ( int n, /* L is n-by-n, where n >= 0 */ double X [ ], /* size n. right-hand-side on input, soln. on output */ int Lp [ ], /* input of size n+1, not modified */ int Li [ ], /* input of size lnz=Lp[n], not modified */ double Lx [ ] /* input of size lnz=Lp[n], not modified */ ) { int j, p, p2 ; for (j = n-1 ; j >= 0 ; j--) { p2 = Lp [j+1] ; for (p = Lp [j] ; p < p2 ; p++) { X [j] -= Lx [p] * X [Li [p]] ; } } } /* ========================================================================== */ /* === ldl_perm: permute a vector, x=Pb ===================================== */ /* ========================================================================== */ void ldl_perm ( int n, /* size of X, B, and P */ double X [ ], /* output of size n. */ double B [ ], /* input of size n. */ int P [ ] /* input permutation array of size n. */ ) { int j ; for (j = 0 ; j < n ; j++) { X [j] = B [P [j]] ; } } /* ========================================================================== */ /* === ldl_permt: permute a vector, x=P'b =================================== */ /* ========================================================================== */ void ldl_permt ( int n, /* size of X, B, and P */ double X [ ], /* output of size n. */ double B [ ], /* input of size n. */ int P [ ] /* input permutation array of size n. */ ) { int j ; for (j = 0 ; j < n ; j++) { X [P [j]] = B [j] ; } } /* ========================================================================== */ /* === ldl_valid_perm: check if a permutation vector is valid =============== */ /* ========================================================================== */ int ldl_valid_perm /* returns 1 if valid, 0 otherwise */ ( int n, int P [ ], /* input of size n, a permutation of 0:n-1 */ int Flag [ ] /* workspace of size n */ ) { int j, k ; if (n < 0 || !Flag) { return (0) ; /* n must be >= 0, and Flag must be present */ } if (!P) { return (1) ; /* If NULL, P is assumed to be the identity perm. */ } for (j = 0 ; j < n ; j++) { Flag [j] = 0 ; /* clear the Flag array */ } for (k = 0 ; k < n ; k++) { j = P [k] ; if (j < 0 || j >= n || Flag [j] != 0) { return (0) ; /* P is not valid */ } Flag [j] = 1 ; } return (1) ; /* P is valid */ } /* ========================================================================== */ /* === ldl_valid_matrix: check if a sparse matrix is valid ================== */ /* ========================================================================== */ /* This routine checks to see if a sparse matrix A is valid for input to * ldl_symbolic and ldl_numeric. It returns 1 if the matrix is valid, 0 * otherwise. A is in sparse column form. The numerical values in column j * are stored in Ax [Ap [j] ... Ap [j+1]-1], with row indices in * Ai [Ap [j] ... Ap [j+1]-1]. The Ax array is not checked. */ int ldl_valid_matrix ( int n, int Ap [ ], int Ai [ ] ) { int j, p ; if (n < 0 || !Ap || !Ai || Ap [0] != 0) { return (0) ; /* n must be >= 0, and Ap and Ai must be present */ } for (j = 0 ; j < n ; j++) { if (Ap [j] > Ap [j+1]) { return (0) ; /* Ap must be monotonically nondecreasing */ } } for (p = 0 ; p < Ap [n] ; p++) { if (Ai [p] < 0 || Ai [p] >= n) { return (0) ; /* row indices must be in the range 0 to n-1 */ } } return (1) ; /* matrix is valid */ }

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