508 lines
20 KiB
C++
508 lines
20 KiB
C++
/* ========================================================================== */
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/* === ldl.c: sparse LDL' factorization and solve package =================== */
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/* ========================================================================== */
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/* LDL: a simple set of routines for sparse LDL' factorization. These routines
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* are not terrifically fast (they do not use dense matrix kernels), but the
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* code is very short. The purpose is to illustrate the algorithms in a very
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* concise manner, primarily for educational purposes. Although the code is
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* very concise, this package is slightly faster than the built-in sparse
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* Cholesky factorization in MATLAB 7.0 (chol), when using the same input
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* permutation.
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*
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* The routines compute the LDL' factorization of a real sparse symmetric
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* matrix A (or PAP' if a permutation P is supplied), and solve upper
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* and lower triangular systems with the resulting L and D factors. If A is
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* positive definite then the factorization will be accurate. A can be
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* indefinite (with negative values on the diagonal D), but in this case no
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* guarantee of accuracy is provided, since no numeric pivoting is performed.
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*
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* The n-by-n sparse matrix A is in compressed-column form. The nonzero values
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* in column j are stored in Ax [Ap [j] ... Ap [j+1]-1], with corresponding row
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* indices in Ai [Ap [j] ... Ap [j+1]-1]. Ap [0] = 0 is required, and thus
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* nz = Ap [n] is the number of nonzeros in A. Ap is an int array of size n+1.
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* The int array Ai and the double array Ax are of size nz. This data structure
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* is identical to the one used by MATLAB, except for the following
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* generalizations. The row indices in each column of A need not be in any
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* particular order, although they must be in the range 0 to n-1. Duplicate
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* entries can be present; any duplicates are summed. That is, if row index i
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* appears twice in a column j, then the value of A (i,j) is the sum of the two
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* entries. The data structure used here for the input matrix A is more
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* flexible than MATLAB's, which requires sorted columns with no duplicate
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* entries.
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*
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* Only the diagonal and upper triangular part of A (or PAP' if a permutation
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* P is provided) is accessed. The lower triangular parts of the matrix A or
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* PAP' can be present, but they are ignored.
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*
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* The optional input permutation is provided as an array P of length n. If
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* P [k] = j, the row and column j of A is the kth row and column of PAP'.
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* If P is present then the factorization is LDL' = PAP' or L*D*L' = A(P,P) in
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* 0-based MATLAB notation. If P is not present (a null pointer) then no
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* permutation is performed, and the factorization is LDL' = A.
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*
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* The lower triangular matrix L is stored in the same compressed-column
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* form (an int Lp array of size n+1, an int Li array of size Lp [n], and a
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* double array Lx of the same size as Li). It has a unit diagonal, which is
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* not stored. The row indices in each column of L are always returned in
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* ascending order, with no duplicate entries. This format is compatible with
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* MATLAB, except that it would be more convenient for MATLAB to include the
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* unit diagonal of L. Doing so here would add additional complexity to the
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* code, and is thus omitted in the interest of keeping this code short and
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* readable.
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*
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* The elimination tree is held in the Parent [0..n-1] array. It is normally
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* not required by the user, but it is required by ldl_numeric. The diagonal
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* matrix D is held as an array D [0..n-1] of size n.
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*
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* --------------------
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* C-callable routines:
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* --------------------
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*
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* ldl_symbolic: Given the pattern of A, computes the Lp and Parent arrays
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* required by ldl_numeric. Takes time proportional to the number of
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* nonzeros in L. Computes the inverse Pinv of P if P is provided.
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* Also returns Lnz, the count of nonzeros in each column of L below
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* the diagonal (this is not required by ldl_numeric).
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* ldl_numeric: Given the pattern and numerical values of A, the Lp array,
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* the Parent array, and P and Pinv if applicable, computes the
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* pattern and numerical values of L and D.
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* ldl_lsolve: Solves Lx=b for a dense vector b.
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* ldl_dsolve: Solves Dx=b for a dense vector b.
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* ldl_ltsolve: Solves L'x=b for a dense vector b.
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* ldl_perm: Computes x=Pb for a dense vector b.
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* ldl_permt: Computes x=P'b for a dense vector b.
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* ldl_valid_perm: checks the validity of a permutation vector
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* ldl_valid_matrix: checks the validity of the sparse matrix A
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*
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* ----------------------------
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* Limitations of this package:
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* ----------------------------
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*
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* In the interest of keeping this code simple and readable, ldl_symbolic and
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* ldl_numeric assume their inputs are valid. You can check your own inputs
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* prior to calling these routines with the ldl_valid_perm and ldl_valid_matrix
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* routines. Except for the two ldl_valid_* routines, no routine checks to see
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* if the array arguments are present (non-NULL). Like all C routines, no
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* routine can determine if the arrays are long enough and don't overlap.
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*
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* The ldl_numeric does check the numerical factorization, however. It returns
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* n if the factorization is successful. If D (k,k) is zero, then k is
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* returned, and L is only partially computed.
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*
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* No pivoting to control fill-in is performed, which is often critical for
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* obtaining good performance. I recommend that you compute the permutation P
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* using AMD or SYMAMD (approximate minimum degree ordering routines), or an
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* appropriate graph-partitioning based ordering. See the ldldemo.m routine for
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* an example in MATLAB, and the ldlmain.c stand-alone C program for examples of
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* how to find P. Routines for manipulating compressed-column matrices are
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* available in UMFPACK. AMD, SYMAMD, UMFPACK, and this LDL package are all
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* available at http://www.cise.ufl.edu/research/sparse.
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*
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* -------------------------
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* Possible simplifications:
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* -------------------------
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*
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* These routines could be made even simpler with a few additional assumptions.
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* If no input permutation were performed, the caller would have to permute the
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* matrix first, but the computation of Pinv, and the use of P and Pinv could be
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* removed. If only the diagonal and upper triangular part of A or PAP' are
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* present, then the tests in the "if (i < k)" statement in ldl_symbolic and
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* "if (i <= k)" in ldl_numeric, are always true, and could be removed (i can
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* equal k in ldl_symbolic, but then the body of the if statement would
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* correctly do no work since Flag [k] == k). If we could assume that no
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* duplicate entries are present, then the statement Y [i] += Ax [p] could be
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* replaced with Y [i] = Ax [p] in ldl_numeric.
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*
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* --------------------------
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* Description of the method:
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* --------------------------
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*
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* LDL computes the symbolic factorization by finding the pattern of L one row
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* at a time. It does this based on the following theory. Consider a sparse
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* system Lx=b, where L, x, and b, are all sparse, and where L comes from a
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* Cholesky (or LDL') factorization. The elimination tree (etree) of L is
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* defined as follows. The parent of node j is the smallest k > j such that
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* L (k,j) is nonzero. Node j has no parent if column j of L is completely zero
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* below the diagonal (j is a root of the etree in this case). The nonzero
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* pattern of x is the union of the paths from each node i to the root, for
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* each nonzero b (i). To compute the numerical solution to Lx=b, we can
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* traverse the columns of L corresponding to nonzero values of x. This
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* traversal does not need to be done in the order 0 to n-1. It can be done in
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* any "topological" order, such that x (i) is computed before x (j) if i is a
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* descendant of j in the elimination tree.
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*
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* The row-form of the LDL' factorization is shown in the MATLAB function
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* ldlrow.m in this LDL package. Note that row k of L is found via a sparse
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* triangular solve of L (1:k-1, 1:k-1) \ A (1:k-1, k), to use 1-based MATLAB
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* notation. Thus, we can start with the nonzero pattern of the kth column of
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* A (above the diagonal), follow the paths up to the root of the etree of the
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* (k-1)-by-(k-1) leading submatrix of L, and obtain the pattern of the kth row
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* of L. Note that we only need the leading (k-1)-by-(k-1) submatrix of L to
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* do this. The elimination tree can be constructed as we go.
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*
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* The symbolic factorization does the same thing, except that it discards the
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* pattern of L as it is computed. It simply counts the number of nonzeros in
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* each column of L and then constructs the Lp index array when it's done. The
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* symbolic factorization does not need to do this in topological order.
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* Compare ldl_symbolic with the first part of ldl_numeric, and note that the
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* while (len > 0) loop is not present in ldl_symbolic.
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*
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* LDL Version 1.3, Copyright (c) 2006 by Timothy A Davis,
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* University of Florida. All Rights Reserved. Developed while on sabbatical
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* at Stanford University and Lawrence Berkeley National Laboratory. Refer to
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* the README file for the License. Available at
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* http://www.cise.ufl.edu/research/sparse.
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*/
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#include "ldl.h"
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/* ========================================================================== */
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/* === ldl_symbolic ========================================================= */
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/* ========================================================================== */
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/* The input to this routine is a sparse matrix A, stored in column form, and
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* an optional permutation P. The output is the elimination tree
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* and the number of nonzeros in each column of L. Parent [i] = k if k is the
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* parent of i in the tree. The Parent array is required by ldl_numeric.
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* Lnz [k] gives the number of nonzeros in the kth column of L, excluding the
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* diagonal.
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*
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* One workspace vector (Flag) of size n is required.
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*
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* If P is NULL, then it is ignored. The factorization will be LDL' = A.
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* Pinv is not computed. In this case, neither P nor Pinv are required by
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* ldl_numeric.
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*
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* If P is not NULL, then it is assumed to be a valid permutation. If
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* row and column j of A is the kth pivot, the P [k] = j. The factorization
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* will be LDL' = PAP', or A (p,p) in MATLAB notation. The inverse permutation
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* Pinv is computed, where Pinv [j] = k if P [k] = j. In this case, both P
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* and Pinv are required as inputs to ldl_numeric.
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*
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* The floating-point operation count of the subsequent call to ldl_numeric
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* is not returned, but could be computed after ldl_symbolic is done. It is
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* the sum of (Lnz [k]) * (Lnz [k] + 2) for k = 0 to n-1.
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*/
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void LDL_symbolic
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(
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LDL_int n, /* A and L are n-by-n, where n >= 0 */
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LDL_int Ap [ ], /* input of size n+1, not modified */
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LDL_int Ai [ ], /* input of size nz=Ap[n], not modified */
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LDL_int Lp [ ], /* output of size n+1, not defined on input */
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LDL_int Parent [ ], /* output of size n, not defined on input */
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LDL_int Lnz [ ], /* output of size n, not defined on input */
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LDL_int Flag [ ], /* workspace of size n, not defn. on input or output */
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LDL_int P [ ], /* optional input of size n */
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LDL_int Pinv [ ] /* optional output of size n (used if P is not NULL) */
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)
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{
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LDL_int i, k, p, kk, p2 ;
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if (P)
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{
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/* If P is present then compute Pinv, the inverse of P */
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for (k = 0 ; k < n ; k++)
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{
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Pinv [P [k]] = k ;
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}
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}
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for (k = 0 ; k < n ; k++)
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{
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/* L(k,:) pattern: all nodes reachable in etree from nz in A(0:k-1,k) */
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Parent [k] = -1 ; /* parent of k is not yet known */
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Flag [k] = k ; /* mark node k as visited */
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Lnz [k] = 0 ; /* count of nonzeros in column k of L */
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kk = (P) ? (P [k]) : (k) ; /* kth original, or permuted, column */
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p2 = Ap [kk+1] ;
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for (p = Ap [kk] ; p < p2 ; p++)
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{
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/* A (i,k) is nonzero (original or permuted A) */
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i = (Pinv) ? (Pinv [Ai [p]]) : (Ai [p]) ;
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if (i < k)
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{
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/* follow path from i to root of etree, stop at flagged node */
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for ( ; Flag [i] != k ; i = Parent [i])
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{
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/* find parent of i if not yet determined */
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if (Parent [i] == -1) Parent [i] = k ;
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Lnz [i]++ ; /* L (k,i) is nonzero */
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Flag [i] = k ; /* mark i as visited */
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}
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}
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}
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}
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/* construct Lp index array from Lnz column counts */
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Lp [0] = 0 ;
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for (k = 0 ; k < n ; k++)
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{
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Lp [k+1] = Lp [k] + Lnz [k] ;
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}
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}
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/* ========================================================================== */
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/* === ldl_numeric ========================================================== */
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/* ========================================================================== */
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/* Given a sparse matrix A (the arguments n, Ap, Ai, and Ax) and its symbolic
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* analysis (Lp and Parent, and optionally P and Pinv), compute the numeric LDL'
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* factorization of A or PAP'. The outputs of this routine are arguments Li,
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* Lx, and D. It also requires three size-n workspaces (Y, Pattern, and Flag).
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*/
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LDL_int LDL_numeric /* returns n if successful, k if D (k,k) is zero */
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(
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LDL_int n, /* A and L are n-by-n, where n >= 0 */
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LDL_int Ap [ ], /* input of size n+1, not modified */
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LDL_int Ai [ ], /* input of size nz=Ap[n], not modified */
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double Ax [ ], /* input of size nz=Ap[n], not modified */
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LDL_int Lp [ ], /* input of size n+1, not modified */
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LDL_int Parent [ ], /* input of size n, not modified */
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LDL_int Lnz [ ], /* output of size n, not defn. on input */
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LDL_int Li [ ], /* output of size lnz=Lp[n], not defined on input */
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double Lx [ ], /* output of size lnz=Lp[n], not defined on input */
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double D [ ], /* output of size n, not defined on input */
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double Y [ ], /* workspace of size n, not defn. on input or output */
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LDL_int Pattern [ ],/* workspace of size n, not defn. on input or output */
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LDL_int Flag [ ], /* workspace of size n, not defn. on input or output */
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LDL_int P [ ], /* optional input of size n */
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LDL_int Pinv [ ] /* optional input of size n */
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)
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{
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double yi, l_ki ;
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LDL_int i, k, p, kk, p2, len, top ;
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for (k = 0 ; k < n ; k++)
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{
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/* compute nonzero Pattern of kth row of L, in topological order */
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Y [k] = 0.0 ; /* Y(0:k) is now all zero */
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top = n ; /* stack for pattern is empty */
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Flag [k] = k ; /* mark node k as visited */
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Lnz [k] = 0 ; /* count of nonzeros in column k of L */
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kk = (P) ? (P [k]) : (k) ; /* kth original, or permuted, column */
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p2 = Ap [kk+1] ;
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for (p = Ap [kk] ; p < p2 ; p++)
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{
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i = (Pinv) ? (Pinv [Ai [p]]) : (Ai [p]) ; /* get A(i,k) */
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if (i <= k)
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{
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Y [i] += Ax [p] ; /* scatter A(i,k) into Y (sum duplicates) */
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for (len = 0 ; Flag [i] != k ; i = Parent [i])
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{
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Pattern [len++] = i ; /* L(k,i) is nonzero */
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Flag [i] = k ; /* mark i as visited */
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}
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while (len > 0) Pattern [--top] = Pattern [--len] ;
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}
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}
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/* compute numerical values kth row of L (a sparse triangular solve) */
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D [k] = Y [k] ; /* get D(k,k) and clear Y(k) */
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Y [k] = 0.0 ;
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for ( ; top < n ; top++)
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{
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i = Pattern [top] ; /* Pattern [top:n-1] is pattern of L(:,k) */
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yi = Y [i] ; /* get and clear Y(i) */
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Y [i] = 0.0 ;
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p2 = Lp [i] + Lnz [i] ;
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for (p = Lp [i] ; p < p2 ; p++)
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{
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Y [Li [p]] -= Lx [p] * yi ;
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}
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l_ki = yi / D [i] ; /* the nonzero entry L(k,i) */
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D [k] -= l_ki * yi ;
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Li [p] = k ; /* store L(k,i) in column form of L */
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Lx [p] = l_ki ;
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Lnz [i]++ ; /* increment count of nonzeros in col i */
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}
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if (D [k] == 0.0) return (k) ; /* failure, D(k,k) is zero */
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}
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return (n) ; /* success, diagonal of D is all nonzero */
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}
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/* ========================================================================== */
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/* === ldl_lsolve: solve Lx=b ============================================== */
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/* ========================================================================== */
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void LDL_lsolve
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(
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LDL_int n, /* L is n-by-n, where n >= 0 */
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double X [ ], /* size n. right-hand-side on input, soln. on output */
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LDL_int Lp [ ], /* input of size n+1, not modified */
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LDL_int Li [ ], /* input of size lnz=Lp[n], not modified */
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double Lx [ ] /* input of size lnz=Lp[n], not modified */
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)
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{
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LDL_int j, p, p2 ;
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for (j = 0 ; j < n ; j++)
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{
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p2 = Lp [j+1] ;
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for (p = Lp [j] ; p < p2 ; p++)
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{
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X [Li [p]] -= Lx [p] * X [j] ;
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}
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}
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}
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/* ========================================================================== */
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/* === ldl_dsolve: solve Dx=b ============================================== */
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/* ========================================================================== */
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void LDL_dsolve
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(
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LDL_int n, /* D is n-by-n, where n >= 0 */
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double X [ ], /* size n. right-hand-side on input, soln. on output */
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double D [ ] /* input of size n, not modified */
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)
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{
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LDL_int j ;
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for (j = 0 ; j < n ; j++)
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{
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X [j] /= D [j] ;
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}
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}
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/* ========================================================================== */
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/* === ldl_ltsolve: solve L'x=b ============================================ */
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/* ========================================================================== */
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void LDL_ltsolve
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(
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LDL_int n, /* L is n-by-n, where n >= 0 */
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double X [ ], /* size n. right-hand-side on input, soln. on output */
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LDL_int Lp [ ], /* input of size n+1, not modified */
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LDL_int Li [ ], /* input of size lnz=Lp[n], not modified */
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double Lx [ ] /* input of size lnz=Lp[n], not modified */
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)
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{
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int j, p, p2 ;
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for (j = n-1 ; j >= 0 ; j--)
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{
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p2 = Lp [j+1] ;
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for (p = Lp [j] ; p < p2 ; p++)
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{
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X [j] -= Lx [p] * X [Li [p]] ;
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}
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}
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}
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/* ========================================================================== */
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/* === ldl_perm: permute a vector, x=Pb ===================================== */
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/* ========================================================================== */
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void LDL_perm
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(
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LDL_int n, /* size of X, B, and P */
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double X [ ], /* output of size n. */
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double B [ ], /* input of size n. */
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LDL_int P [ ] /* input permutation array of size n. */
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)
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{
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LDL_int j ;
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for (j = 0 ; j < n ; j++)
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{
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X [j] = B [P [j]] ;
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}
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}
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/* ========================================================================== */
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/* === ldl_permt: permute a vector, x=P'b =================================== */
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/* ========================================================================== */
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void LDL_permt
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(
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LDL_int n, /* size of X, B, and P */
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double X [ ], /* output of size n. */
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double B [ ], /* input of size n. */
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LDL_int P [ ] /* input permutation array of size n. */
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)
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{
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LDL_int j ;
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for (j = 0 ; j < n ; j++)
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{
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X [P [j]] = B [j] ;
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}
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}
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/* ========================================================================== */
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/* === ldl_valid_perm: check if a permutation vector is valid =============== */
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/* ========================================================================== */
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LDL_int LDL_valid_perm /* returns 1 if valid, 0 otherwise */
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(
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LDL_int n,
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LDL_int P [ ], /* input of size n, a permutation of 0:n-1 */
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LDL_int Flag [ ] /* workspace of size n */
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)
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{
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LDL_int j, k ;
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if (n < 0 || !Flag)
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{
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return (0) ; /* n must be >= 0, and Flag must be present */
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}
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if (!P)
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{
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return (1) ; /* If NULL, P is assumed to be the identity perm. */
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}
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for (j = 0 ; j < n ; j++)
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{
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Flag [j] = 0 ; /* clear the Flag array */
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}
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for (k = 0 ; k < n ; k++)
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{
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j = P [k] ;
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if (j < 0 || j >= n || Flag [j] != 0)
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{
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return (0) ; /* P is not valid */
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}
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Flag [j] = 1 ;
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}
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return (1) ; /* P is valid */
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}
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/* ========================================================================== */
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/* === ldl_valid_matrix: check if a sparse matrix is valid ================== */
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/* ========================================================================== */
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/* This routine checks to see if a sparse matrix A is valid for input to
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* ldl_symbolic and ldl_numeric. It returns 1 if the matrix is valid, 0
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* otherwise. A is in sparse column form. The numerical values in column j
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* are stored in Ax [Ap [j] ... Ap [j+1]-1], with row indices in
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* Ai [Ap [j] ... Ap [j+1]-1]. The Ax array is not checked.
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*/
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LDL_int LDL_valid_matrix
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(
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LDL_int n,
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LDL_int Ap [ ],
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LDL_int Ai [ ]
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)
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{
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LDL_int j, p ;
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if (n < 0 || !Ap || !Ai || Ap [0] != 0)
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{
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return (0) ; /* n must be >= 0, and Ap and Ai must be present */
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}
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for (j = 0 ; j < n ; j++)
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{
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if (Ap [j] > Ap [j+1])
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{
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return (0) ; /* Ap must be monotonically nondecreasing */
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}
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}
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for (p = 0 ; p < Ap [n] ; p++)
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{
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if (Ai [p] < 0 || Ai [p] >= n)
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{
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return (0) ; /* row indices must be in the range 0 to n-1 */
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}
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}
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return (1) ; /* matrix is valid */
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}
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