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/* ========================================================================== *//* === Include/cholmod_core.h =============================================== *//* ========================================================================== *//* -----------------------------------------------------------------------------* CHOLMOD/Include/cholmod_core.h. Version 0.6.* Copyright (C) 2005, Univ. of Florida. Author: Timothy A. Davis* CHOLMOD/Include/cholmod_core.h is licensed under Version 2.1 of the GNU* Lesser General Public License. See lesser.txt for a text of the license.* CHOLMOD is also available under other licenses; contact authors for details.* http://www.cise.ufl.edu/research/sparse* -------------------------------------------------------------------------- *//* CHOLMOD Core module: basic CHOLMOD objects and routines.* Required by all CHOLMOD modules. Requires no other module or package.** The CHOLMOD modules are:** Core basic data structures and definitions* Check check/print the 5 CHOLMOD objects, & 3 types of integer vectors* Cholesky sparse Cholesky factorization* Modify sparse Cholesky update/downdate/row-add/row-delete* MatrixOps sparse matrix functions (add, multiply, norm, ...)* Supernodal supernodal sparse Cholesky factorization* Partition graph-partitioning based orderings** The CHOLMOD objects:* --------------------** cholmod_common parameters, statistics, and workspace* cholmod_sparse a sparse matrix in compressed column form* cholmod_factor an LL' or LDL' factorization* cholmod_dense a dense matrix* cholmod_triplet a sparse matrix in "triplet" form** The Core module described here defines the CHOLMOD data structures, and* basic operations on them. To create and solve a sparse linear system Ax=b,* the user must create A and b, populate them with values, and then pass them* to the routines in the CHOLMOD Cholesky module. There are two primary* methods for creating A: (1) allocate space for a column-oriented sparse* matrix and fill it with pattern and values, or (2) create a triplet form* matrix and convert it to a sparse matrix. The latter option is simpler.** The matrices b and x are typically dense matrices, but can also be sparse.* You can allocate and free them as dense matrices with the* cholmod_allocate_dense and cholmod_free_dense routines.** The cholmod_factor object contains the symbolic and numeric LL' or LDL'* factorization of sparse symmetric matrix. The matrix must be positive* definite for an LL' factorization. It need only be symmetric and have well-* conditioned leading submatrices for it to have an LDL' factorization* (CHOLMOD does not pivot for numerical stability). It is typically created* with the cholmod_factorize routine in the Cholesky module, but can also* be initialized to L=D=I in the Core module and then modified by the Modify* module. It must be freed with cholmod_free_factor, defined below.** The Core routines for each object are described below. Each list is split* into two parts: the primary routines and secondary routines.** ============================================================================* === cholmod_common =========================================================* ============================================================================** The Common object contains control parameters, statistics, and* You must call cholmod_start before calling any other CHOLMOD routine, and* must call cholmod_finish as your last call to CHOLMOD, with two exceptions:* you may call cholmod_print_common and cholmod_check_common in the Check* module after calling cholmod_finish.** cholmod_start first call to CHOLMOD* cholmod_finish last call to CHOLMOD* -----------------------------* cholmod_defaults restore default parameters* cholmod_maxrank maximum rank for update/downdate* cholmod_allocate_work allocate workspace in Common* cholmod_free_work free workspace in Common* cholmod_clear_flag clear Flag workspace in Common* cholmod_error called when CHOLMOD encounters an error* cholmod_dbound for internal use in CHOLMOD only* cholmod_hypot compute sqrt (x*x + y*y) accurately* cholmod_divcomplex complex division, c = a/b** ============================================================================* === cholmod_sparse =========================================================* ============================================================================** A sparse matrix is held in compressed column form. In the basic type* ("packed", which corresponds to a MATLAB sparse matrix), an n-by-n matrix* with nz entries is held in three arrays: p of size n+1, i of size nz, and x* of size nz. Row indices of column j are held in i [p [j] ... p [j+1]-1] and* in the same locations in x. There may be no duplicate entries in a column.* Row indices in each column may be sorted or unsorted (CHOLMOD keeps track).* A->stype determines the storage mode: 0 if both upper/lower parts are stored,* -1 if A is symmetric and just tril(A) is stored, +1 if symmetric and triu(A)* is stored.** cholmod_allocate_sparse allocate a sparse matrix* cholmod_free_sparse free a sparse matrix* -----------------------------* cholmod_reallocate_sparse change the size (# entries) of sparse matrix* cholmod_nnz number of nonzeros in a sparse matrix* cholmod_speye sparse identity matrix* cholmod_spzeros sparse zero matrix* cholmod_transpose transpose a sparse matrix* cholmod_ptranspose transpose/permute a sparse matrix* cholmod_transpose_unsym transpose/permute an unsymmetric sparse matrix* cholmod_transpose_sym transpose/permute a symmetric sparse matrix* cholmod_sort sort row indices in each column of sparse matrix* cholmod_band C = tril (triu (A,k1), k2)* cholmod_band_inplace A = tril (triu (A,k1), k2)* cholmod_aat C = A*A'* cholmod_copy_sparse C = A, create an exact copy of a sparse matrix* cholmod_copy C = A, with possible change of stype* cholmod_add C = alpha*A + beta*B* cholmod_sparse_xtype change the xtype of a sparse matrix** ============================================================================* === cholmod_factor =========================================================* ============================================================================** The data structure for an LL' or LDL' factorization is too complex to* describe in one sentence. This object can hold the symbolic analysis alone,* or in combination with a "simplicial" (similar to a sparse matrix) or* "supernodal" form of the numerical factorization. Only the routine to free* a factor is primary, since a factor object is created by the factorization* routine (cholmod_factorize). It must be freed with cholmod_free_factor.** cholmod_free_factor free a factor* -----------------------------* cholmod_allocate_factor allocate a factor (LL' or LDL')* cholmod_reallocate_factor change the # entries in a factor* cholmod_change_factor change the type of factor (e.g., LDL' to LL')* cholmod_pack_factor pack the columns of a factor* cholmod_reallocate_column resize a single column of a factor* cholmod_factor_to_sparse create a sparse matrix copy of a factor* cholmod_copy_factor create a copy of a factor* cholmod_factor_xtype change the xtype of a factor** Note that there is no cholmod_sparse_to_factor routine to create a factor* as a copy of a sparse matrix. It could be done, after a fashion, but a* lower triangular sparse matrix would not necessarily have a chordal graph,* which would break the many CHOLMOD routines that rely on this property.** ============================================================================* === cholmod_dense ==========================================================* ============================================================================** The solve routines and some of the MatrixOps and Modify routines use dense* matrices as inputs. These are held in column-major order. With a leading* dimension of d, the entry in row i and column j is held in x [i+j*d].** cholmod_allocate_dense allocate a dense matrix* cholmod_free_dense free a dense matrix* -----------------------------* cholmod_zeros allocate a dense matrix of all zeros* cholmod_ones allocate a dense matrix of all ones* cholmod_eye allocate a dense identity matrix* cholmod_sparse_to_dense create a dense matrix copy of a sparse matrix* cholmod_dense_to_sparse create a sparse matrix copy of a dense matrix* cholmod_copy_dense create a copy of a dense matrix* cholmod_copy_dense2 copy a dense matrix (pre-allocated)* cholmod_dense_xtype change the xtype of a dense matrix** ============================================================================* === cholmod_triplet ========================================================* ============================================================================** A sparse matrix held in triplet form is the simplest one for a user to* create. It consists of a list of nz entries in arbitrary order, held in* three arrays: i, j, and x, each of length nk. The kth entry is in row i[k],* column j[k], with value x[k]. There may be duplicate values; if A(i,j)* appears more than once, its value is the sum of the entries with those row* and column indices.** cholmod_allocate_triplet allocate a triplet matrix* cholmod_triplet_to_sparse create a sparse matrix copy of a triplet matrix* cholmod_free_triplet free a triplet matrix* -----------------------------* cholmod_reallocate_triplet change the # of entries in a triplet matrix* cholmod_sparse_to_triplet create a triplet matrix copy of a sparse matrix* cholmod_copy_triplet create a copy of a triplet matrix* cholmod_triplet_xtype change the xtype of a triplet matrix** ============================================================================* === memory management ======================================================* ============================================================================** cholmod_malloc malloc wrapper* cholmod_calloc calloc wrapper* cholmod_free free wrapper* cholmod_realloc realloc wrapper* cholmod_realloc_multiple realloc wrapper for multiple objects** ============================================================================* === Core CHOLMOD prototypes ================================================* ============================================================================** All CHOLMOD routines (in all modules) use the following protocol for return* values, with one exception:** int TRUE (1) if successful, or FALSE (0) otherwise.* (exception: cholmod_divcomplex)* long a value >= 0 if successful, or -1 otherwise.* double a value >= 0 if successful, or -1 otherwise.* size_t a value > 0 if successful, or 0 otherwise.* void * a non-NULL pointer to newly allocated memory if* successful, or NULL otherwise.* cholmod_sparse * a non-NULL pointer to a newly allocated matrix* if successful, or NULL otherwise.* cholmod_factor * a non-NULL pointer to a newly allocated factor* if successful, or NULL otherwise.* cholmod_triplet * a non-NULL pointer to a newly allocated triplet* matrix if successful, or NULL otherwise.* cholmod_dense * a non-NULL pointer to a newly allocated triplet* matrix if successful, or NULL otherwise.** The last parameter to all routines is always a pointer to the CHOLMOD* Common object.** TRUE and FALSE are not defined here, since they may conflict with the user* program. A routine that described here returning TRUE or FALSE returns 1* or 0, respectively. Any TRUE/FALSE parameter is true if nonzero, false if* zero.*/#ifndef CHOLMOD_CORE_H#define CHOLMOD_CORE_H/* ========================================================================== *//* === CHOLMOD version ====================================================== *//* ========================================================================== *//* All versions of CHOLMOD will include the following definitions.* As an example, to test if the version you are using is 1.3 or later:** if (CHOLMOD_VERSION >= CHOLMOD_VER_CODE (1,3)) ...** This also works during compile-time:** #if CHOLMOD_VERSION >= CHOLMOD_VER_CODE (1,3)* printf ("This is version 1.3 or later\n") ;* #else* printf ("This is version is earlier than 1.3\n") ;* #endif*/#define CHOLMOD_DATE "Aug 30, 2005"#define CHOLMOD_VER_CODE(main,sub) ((main) * 1000 + (sub))#define CHOLMOD_MAIN_VERSION 0#define CHOLMOD_SUB_VERSION 6#define CHOLMOD_VERSION \CHOLMOD_VER_CODE(CHOLMOD_MAIN_VERSION,CHOLMOD_SUB_VERSION)/* ========================================================================== *//* === non-CHOLMOD include files ============================================ *//* ========================================================================== *//* This is the only non-CHOLMOD include file imposed on the user program.* It required for size_t definition used here. CHOLMOD itself includes other* ANSI C89 standard #include files, but does not expose them to the user.** CHOLMOD assumes that your C compiler is ANSI C89 compliant. It does not make* use of ANSI C99 features.*/#include <stddef.h>/* ========================================================================== *//* === CHOLMOD Common ======================================================= *//* ========================================================================== *//* itype defines the types of integer used: */#define CHOLMOD_INT 0 /* all integer arrays are int */#define CHOLMOD_INTLONG 1 /* most are int, some are long */#define CHOLMOD_LONG 2 /* all integer arrays are long *//* The itype of all parameters for all CHOLMOD routines must match.* FUTURE WORK: CHOLMOD_INTLONG is not yet supported.*//* dtype defines what the numerical type is (double or float): */#define CHOLMOD_DOUBLE 0 /* all numerical values are double */#define CHOLMOD_SINGLE 1 /* all numerical values are float *//* The dtype of all parameters for all CHOLMOD routines must match.** Scalar floating-point values are always passed as double arrays of size 2* (for the real and imaginary parts). They are typecast to float as needed.* FUTURE WORK: the float case is not supported yet.*//* xtype defines the kind of numerical values used: */#define CHOLMOD_PATTERN 0 /* pattern only, no numerical values */#define CHOLMOD_REAL 1 /* a real matrix */#define CHOLMOD_COMPLEX 2 /* a complex matrix (ANSI C99 compatible) */#define CHOLMOD_ZOMPLEX 3 /* a complex matrix (MATLAB compatible) *//* The xtype of all parameters for all CHOLMOD routines must match.** CHOLMOD_PATTERN: x and z are ignored.* CHOLMOD_DOUBLE: x is non-null of size nzmax, z is ignored.* CHOLMOD_COMPLEX: x is non-null of size 2*nzmax doubles, z is ignored.* CHOLMOD_ZOMPLEX: x and z are non-null of size nzmax** In the real case, z is ignored. The kth entry in the matrix is x [k].* There are two methods for the complex case. In the ANSI C99-compatible* CHOLMOD_COMPLEX case, the real and imaginary parts of the kth entry* are in x [2*k] and x [2*k+1], respectively. z is ignored. In the* MATLAB-compatible CHOLMOD_ZOMPLEX case, the real and imaginary* parts of the kth entry are in x [k] and z [k].** Scalar floating-point values are always passed as double arrays of size 2* (real and imaginary parts). The imaginary part of a scalar is ignored if* the routine operates on a real matrix.** These Modules support complex and zomplex matrices, with a few exceptions:** Check all routines* Cholesky all routines* Core all except cholmod_aat, add, band, copy* Demo all routines* Partition all routines* Supernodal all routines support any real, complex, or zomplex input.* There will never be a supernodal zomplex L; a complex* supernodal L is created if A is zomplex.* Tcov all routines* Valgrind all routines** These Modules provide partial support for complex and zomplex matrices:** MATLAB all routines support real and zomplex only, not complex,* with the exception of ldlupdate, which supports* real matrices only. This is a minor constraint since* MATLAB's matrices are all real or zomplex.* MatrixOps only norm_dense, norm_sparse, and sdmult support complex* and zomplex** These Modules do not support complex and zomplex matrices at all:** Modify all routines support real matrices only*//* Definitions for cholmod_common: */#define CHOLMOD_MAXMETHODS 9 /* maximum number of different methods that* cholmod_analyze can try. Must be >= 9. *//* Common->status values. zero means success, negative means a fatal error,* positive is a warning. */#define CHOLMOD_OK 0 /* success */#define CHOLMOD_NOT_INSTALLED (-1) /* failure: method not installed */#define CHOLMOD_OUT_OF_MEMORY (-2) /* failure: out of memory */#define CHOLMOD_TOO_LARGE (-3) /* failure: integer overflow occured */#define CHOLMOD_INVALID (-4) /* failure: invalid input */#define CHOLMOD_NOT_POSDEF (1) /* warning: matrix not pos. def. */#define CHOLMOD_DSMALL (2) /* warning: D for LDL' or diag(L) or* LL' has tiny absolute value *//* ordering method (also used for L->ordering) */#define CHOLMOD_NATURAL 0 /* use natural ordering */#define CHOLMOD_GIVEN 1 /* use given permutation */#define CHOLMOD_AMD 2 /* use minimum degree (AMD) */#define CHOLMOD_METIS 3 /* use METIS' nested dissection */#define CHOLMOD_NESDIS 4 /* use CHOLMOD's version of nested dissection:* node bisector applied recursively, followed* by constrained minimum degree (CSYMAMD or* CCOLAMD) */#define CHOLMOD_COLAMD 5 /* use AMD for A, COLAMD for A*A' *//* POSTORDERED is not a method, but a result of natural ordering followed by a* weighted postorder. It is used for L->ordering, not method [ ].ordering. */#define CHOLMOD_POSTORDERED 6 /* natural ordering, postordered. *//* supernodal strategy (for Common->supernodal) */#define CHOLMOD_SIMPLICIAL 0 /* always do simplicial */#define CHOLMOD_AUTO 1 /* select simpl/super depending on matrix */#define CHOLMOD_SUPERNODAL 2 /* always do supernodal */typedef struct cholmod_common_struct{/* ---------------------------------------------------------------------- *//* parameters for symbolic/numeric factorization and update/downdate *//* ---------------------------------------------------------------------- */double dbound ; /* Smallest absolute value of diagonal entries of D* for LDL' factorization and update/downdate/rowadd/* rowdel, or the diagonal of L for an LL' factorization.* Entries in the range 0 to dbound are replaced with dbound.* Entries in the range -dbound to 0 are replaced with -dbound. No* changes are made to the diagonal if dbound <= 0. Default: zero */double grow0 ; /* For a simplicial factorization, L->i and L->x can* grow if necessary. grow0 is the factor by which* it grows. For the initial space, L is of size MAX (1,grow0) times* the required space. If L runs out of space, the new size of L is* MAX(1.2,grow0) times the new required space. If you do not plan on* modifying the LDL' factorization in the Modify module, set grow0 to* zero (or set grow2 to 0, see below). Default: 1.2 */double grow1 ;size_t grow2 ; /* For a simplicial factorization, each column j of L* is initialized with space equal to* grow1*L->ColCount[j] + grow2. If grow0 < 1, grow1 < 1, or grow2 == 0,* then the space allocated is exactly equal to L->ColCount[j]. If the* column j runs out of space, it increases to grow1*need + grow2 in* size, where need is the total # of nonzeros in that column. If you do* not plan on modifying the factorization in the Modify module, set* grow2 to zero. Default: grow1 = 1.2, grow2 = 5. */size_t maxrank ; /* rank of maximum update/downdate. Valid values:* 2, 4, or 8. A value < 2 is set to 2, and a* value > 8 is set to 8. It is then rounded up to the next highest* power of 2, if not already a power of 2. Workspace (Xwork, below) of* size nrow-by-maxrank double's is allocated for the update/downdate.* If an update/downdate of rank-k is requested, with k > maxrank,* it is done in steps of maxrank. Default: 8, which is fastest.* Memory usage can be reduced by setting maxrank to 2 or 4.*/double supernodal_switch ; /* supernodal vs simplicial factorization */int supernodal ; /* If Common->supernodal <= CHOLMOD_SIMPLICIAL* (0) then cholmod_analyze performs a* simplicial analysis. If >= CHOLMOD_SUPERNODAL (2), then a supernodal* analysis is performed. If == CHOLMOD_AUTO (1) and* flop/nnz(L) < Common->supernodal_switch, then a simplicial analysis* is done. A supernodal analysis done otherwise.* Default: CHOLMOD_AUTO. Default supernodal_switch = 40 */int final_asis ; /* If TRUE, then ignore the other final_* parameters* (except for final_pack).* The factor is left as-is when done. Default: TRUE.*/int final_super ; /* If TRUE, leave a factor in supernodal form when* supernodal factorization is finished. If FALSE,* then convert to a simplicial factor when done.* Default: TRUE */int final_ll ; /* If TRUE, leave factor in LL' form when done.* Otherwise, leave in LDL' form. Default: FALSE */int final_pack ; /* If TRUE, pack the columns when done. If TRUE, and* cholmod_factorize is called with a symbolic L, L is* allocated with exactly the space required, using L->ColCount. If you* plan on modifying the factorization, set Common->final_pack to FALSE,* and each column will be given a little extra slack space for future* growth in fill-in due to updates. Default: TRUE */int final_monotonic ; /* If TRUE, ensure columns are monotonic when done.* Default: TRUE */int final_resymbol ;/* if cholmod_factorize performed a supernodal* factorization, final_resymbol is true, and* final_super is FALSE (convert a simplicial numeric factorization),* then numerically zero entries that resulted from relaxed supernodal* amalgamation are removed. This does not remove entries that are zero* due to exact numeric cancellation, since doing so would break the* update/downdate rowadd/rowdel routines. Default: FALSE. *//* supernodal relaxed amalgamation parameters: */double zrelax [3] ;size_t nrelax [3] ;/* Let ns be the total number of columns in two adjacent supernodes.* Let z be the fraction of zero entries in the two supernodes if they* are merged (z includes zero entries from prior amalgamations). The* two supernodes are merged if:* (ns <= nrelax [0]) || (no new zero entries added) ||* (ns <= nrelax [1] && z < zrelax [0]) ||* (ns <= nrelax [2] && z < zrelax [1]) || (z < zrelax [2])** Default parameters result in the following rule:* (ns <= 4) || (no new zero entries added) ||* (ns <= 16 && z < 0.8) || (ns <= 48 && z < 0.1) || (z < 0.05)*/int prefer_zomplex ; /* X = cholmod_solve (sys, L, B, Common) computes* x=A\b or solves a related system. If L and B are* both real, then X is real. Otherwise, X is returned as* CHOLMOD_COMPLEX if Common->prefer_zomplex is FALSE, or* CHOLMOD_ZOMPLEX if Common->prefer_zomplex is TRUE. This parameter* is needed because there is no supernodal zomplex L. Suppose the* caller wants all complex matrices to be stored in zomplex form* (MATLAB, for example). A supernodal L is returned in complex form* if A is zomplex. B can be real, and thus X = cholmod_solve (L,B)* should return X as zomplex. This cannot be inferred from the input* arguments L and B. Default: FALSE, since all data types are* supported in CHOLMOD_COMPLEX form and since this is the native type* of LAPACK and the BLAS. Note that the MATLAB/cholmod.c mexFunction* sets this parameter to TRUE, since MATLAB matrices are in* CHOLMOD_ZOMPLEX form.*/int prefer_upper ; /* cholmod_analyze and cholmod_factorize work* fastest when a symmetric matrix is stored in* upper triangular form when a fill-reducing ordering is used. In* MATLAB, this corresponds to how x=A\b works. When the matrix is* ordered as-is, they work fastest when a symmetric matrix is in lower* triangular form. In MATLAB, R=chol(A) does the opposite. This* parameter affects only how cholmod_read returns a symmetric matrix.* If TRUE (the default case), a symmetric matrix is always returned in* upper-triangular form (A->stype = 1). *//* ---------------------------------------------------------------------- *//* printing and error handling options *//* ---------------------------------------------------------------------- */int print ; /* print level. Default: 3 */int precise ; /* if TRUE, print 16 digits. Otherwise print 5 */int (*print_function) (const char *, ...) ; /* pointer to printf */int try_catch ; /* if TRUE, then ignore errors; CHOLMOD is in the middle* of a try/catch block. No error message is printed* and the Common->error_handler function is not called. */void (*error_handler) (int status, char *file, int line, char *message) ;/* Common->error_handler is the user's error handling routine. If not* NULL, this routine is called if an error occurs in CHOLMOD. status* can be CHOLMOD_OK (0), negative for a fatal error, and positive for* a warning. file is a string containing the name of the source code* file where the error occured, and line is the line number in that* file. message is a string describing the error in more detail. *//* ---------------------------------------------------------------------- *//* ordering options *//* ---------------------------------------------------------------------- *//* The cholmod_analyze routine can try many different orderings and select* the best one. It can also try one ordering method multiple times, with* different parameter settings. The default is to use three orderings,* the user's permutation (if provided), AMD which is the fastest ordering* and generally gives good fill-in, and METIS. CHOLMOD's nested dissection* (METIS with a constrained AMD) usually gives a better ordering than METIS* alone (by about 5% to 10%) but it takes more time.** If you know the method that is best for your matrix, set Common->nmethods* to 1 and set Common->method [0] to the set of parameters for that method.* If you set it to 1 and do not provide a permutation, then only AMD will* be called.** If METIS is not available, the default # of methods tried is 2 (the user* permutation, if any, and AMD).** To try other methods, set Common->nmethods to the number of methods you* want to try. The suite of default methods and their parameters is* described in the cholmod_defaults routine, and summarized here:** Common->method [i]:* i = 0: user-provided ordering (cholmod_analyze_p only)* i = 1: AMD (for both A and A*A')* i = 2: METIS* i = 3: CHOLMOD's nested dissection (NESDIS), default parameters* i = 4: natural* i = 5: NESDIS with nd_small = 20000* i = 6: NESDIS with nd_small = 4, no constrained minimum degree* i = 7: NESDIS with no dense node removal* i = 8: AMD for A, COLAMD for A*A'** You can modify the suite of methods you wish to try by modifying* Common.method [...] after calling cholmod_start or cholmod_defaults.** For example, to use AMD, followed by a weighted postordering:** Common->nmethods = 1 ;* Common->method [0].ordering = CHOLMOD_AMD ;* Common->postorder = TRUE ;** To use the natural ordering (with no postordering):** Common->nmethods = 1 ;* Common->method [0].ordering = CHOLMOD_NATURAL ;* Common->postorder = FALSE ;** If you are going to factorize hundreds or more matrices with the same* nonzero pattern, you may wish to spend a great deal of time finding a* good permutation. In this case, try setting Common->nmethods to 9.* The time spent in cholmod_analysis will be very high, but you need to* call it only once.** cholmod_analyze sets Common->current to a value between 0 and nmethods-1.* Each ordering method uses the set of options defined by this parameter.*/int nmethods ; /* The number of ordering methods to try. Default: 0.* nmethods = 0 is a special case. cholmod_analyze* will try the user-provided ordering (if given) and AMD. Let fl and* lnz be the flop count and nonzeros in L from AMD's ordering. Let* anz be the number of nonzeros in the upper or lower triangular part* of the symmetric matrix A. If fl/lnz < 500 or lnz/anz < 5, then this* is a good ordering, and METIS is not attempted. Otherwise, METIS is* tried. The best ordering found is used. If nmethods > 0, the* methods used are given in the method[ ] array, below. The first* three methods in the default suite of orderings is (1) use the given* permutation (if provided), (2) use AMD, and (3) use METIS. Maximum* allowed value is CHOLMOD_MAXMETHODS. */int current ; /* The current method being tried. Default: 0. Valid* range is 0 to nmethods-1. */int selected ; /* The best method found. *//* The suite of ordering methods and parameters: */struct cholmod_method_struct{/* statistics for this method */double lnz ; /* nnz(L) excl. zeros from supernodal amalgamation,* for a "pure" L */double fl ; /* flop count for a "pure", real simplicial LL'* factorization, with no extra work due to* amalgamation. Subtract n to get the LDL' flop count. Multiply* by about 4 if the matrix is complex or zomplex. *//* ordering method parameters */double prune_dense ;/* dense row/col control for AMD, SYMAMD, CSYMAMD,* and NESDIS (cholmod_nested_dissection). For a* symmetric n-by-n matrix, rows/columns with more than* MAX (16, prune_dense * sqrt (n)) entries are removed prior to* ordering. They appear at the end of the re-ordered matrix.** If prune_dense < 0, only completely dense rows/cols are removed.** This paramater is also the dense column control for COLAMD and* CCOLAMD. For an m-by-n matrix, columns with more than* MAX (16, prune_dense * sqrt (MIN (m,n))) entries are removed prior* to ordering. They appear at the end of the re-ordered matrix.* CHOLMOD factorizes A*A', so it calls COLAMD and CCOLAMD with A',* not A. Thus, this parameter affects the dense *row* control for* CHOLMOD's matrix, and the dense *column* control for COLAMD and* CCOLAMD.** Removing dense rows and columns improves the run-time of the* ordering methods. It has some impact on ordering quality* (usually minimal, sometimes good, sometimes bad).** Default: 10. */double prune_dense2 ;/* dense row control for COLAMD and CCOLAMD.* Rows with more than MAX (16, dense2 * sqrt (n))* for an m-by-n matrix are removed prior to ordering. CHOLMOD's* matrix is transposed before ordering it with COLAMD or CCOLAMD,* so this controls the dense *columns* of CHOLMOD's matrix, and* the dense *rows* of COLAMD's or CCOLAMD's matrix.** If prune_dense2 < 0, only completely dense rows/cols are removed.** Default: -1. Note that this is not the default for COLAMD and* CCOLAMD. -1 is best for Cholesky. 10 is best for LU. */int aggressive ; /* Aggresive absorption in AMD, COLAMD, SYMAMD,* CCOLAMD, and CSYMAMD. Default: TRUE */int order_for_lu ; /* CCOLAMD can be optimized to produce an ordering* for LU or Cholesky factorization. CHOLMOD only* performs a Cholesky factorization. However, you may wish to use* CHOLMOD as an interface for CCOLAMD but use it for your own LU* factorization. In this case, order_for_lu should be set to FALSE.* When factorizing in CHOLMOD itself, you should *** NEVER *** set* this parameter FALSE. Default: TRUE. */int nd_compress ; /* If TRUE, compress the graph and subgraphs before* partitioning them in NESDIS. Default: TRUE */size_t nd_small ; /* do not partition graphs with fewer nodes than* nd_small, in NESDIS. Default: 200 (same as* METIS) */int nd_camd ; /* If TRUE, follow the nested dissection ordering* with a constrained minimum degree ordering that* respects the partitioning just found. If you set nd_small very* small, you may not need this ordering, and can save time by* turning it off. Default: TRUE. *//* fill-reducing ordering to use */int ordering ;} method [CHOLMOD_MAXMETHODS + 1] ;int postorder ; /* If TRUE, cholmod_analyze follows the ordering with a* weighted postorder of the elimination tree. Improves* supernode amalgamation. Does not affect fundamental nnz(L) and* flop count. Default: TRUE. *//* ---------------------------------------------------------------------- *//* memory management routines *//* ---------------------------------------------------------------------- */void *(*malloc_memory) (size_t) ; /* pointer to malloc */void *(*realloc_memory) (void *, size_t) ; /* pointer to realloc */void (*free_memory) (void *) ; /* pointer to free */void *(*calloc_memory) (size_t, size_t) ; /* pointer to calloc *//* ---------------------------------------------------------------------- *//* routines for complex arithmetic *//* ---------------------------------------------------------------------- */int (*complex_divide) (double ax, double az, double bx, double bz,double *cx, double *cz) ;/* flag = complex_divide (ax, az, bx, bz, &cx, &cz) computes the complex* division c = a/b, where ax and az hold the real and imaginary part* of a, and b and c are stored similarly. flag is returned as 1 if* a divide-by-zero occurs, or 0 otherwise. By default, the function* pointer Common->complex_divide is set equal to cholmod_divcomplex.*/double (*hypotenuse) (double x, double y) ;/* s = hypotenuse (x,y) computes s = sqrt (x*x + y*y), but does so more* accurately. By default, the function pointer Common->hypotenuse is* set equal to cholmod_hypot. See also the hypot function in the C99* standard, which has an identical syntax and function. If you have* a C99-compliant compiler, you can set Common->hypotenuse = hypot. *//* ---------------------------------------------------------------------- *//* METIS workarounds *//* ---------------------------------------------------------------------- */double metis_memory ; /* This is a parameter for CHOLMOD's interface to* METIS, not a parameter to METIS itself. METIS* uses an amount of memory that is difficult to estimate precisely* beforehand. If it runs out of memory, it terminates your program.* All routines in CHOLMOD except for CHOLMOD's interface to METIS* return an error status and safely return to your program if they run* out of memory. To mitigate this problem, the CHOLMOD interface* can allocate a single block of memory equal in size to an empirical* upper bound of METIS's memory usage times the Common->metis_memory* parameter, and then immediately free it. It then calls METIS. If* this pre-allocation fails, it is possible that METIS will fail as* well, and so CHOLMOD returns with an out-of-memory condition without* calling METIS.** METIS_NodeND (used in the CHOLMOD_METIS ordering option) with its* default parameter settings typically uses about (4*nz+40n+4096)* times sizeof(int) memory, where nz is equal to the number of entries* in A for the symmetric case or AA' if an unsymmetric matrix is* being ordered (where nz includes both the upper and lower parts* of A or AA'). The observed "upper bound" (with 2 exceptions),* measured in an instrumented copy of METIS 4.0.1 on thousands of* matrices, is (10*nz+50*n+4096) * sizeof(int). Two large matrices* exceeded this bound, one by almost a factor of 2 (Gupta/gupta2).** If your program is terminated by METIS, try setting metis_memory to* 2.0, or even higher if needed. By default, CHOLMOD assumes that METIS* does not have this problem (so that CHOLMOD will work correctly when* this issue is fixed in METIS). Thus, the default value is zero.* This work-around is not guaranteed anyway.** If a matrix exceeds this predicted memory usage, AMD is attempted* instead. It, too, may run out of memory, but if it does so it will* not terminate your program.*/double metis_dswitch ; /* METIS_NodeND in METIS 4.0.1 gives a seg */size_t metis_nswitch ; /* fault with one matrix of order n = 3005 and* nz = 6,036,025. This is a very dense graph.* The workaround is to use AMD instead of METIS for matrices of dimension* greater than Common->metis_nswitch (default 3000) or more and with* density of Common->metis_dswitch (default 0.66) or more.* cholmod_nested_dissection has no problems with the same matrix, even* though it uses METIS_NodeComputeSeparator on this matrix. If this* seg fault does not affect you, set metis_nswitch to zero or less,* and CHOLMOD will not switch to AMD based just on the density of the* matrix (it will still switch to AMD if the metis_memory parameter* causes the switch).*//* ---------------------------------------------------------------------- *//* workspace *//* ---------------------------------------------------------------------- *//* CHOLMOD has several routines that take less time than the size of* workspace they require. Allocating and initializing the workspace would* dominate the run time, unless workspace is allocated and initialized* just once. CHOLMOD allocates this space when needed, and holds it here* between calls to CHOLMOD. cholmod_start sets these pointers to NULL* (which is why it must be the first routine called in CHOLMOD).* cholmod_finish frees the workspace (which is why it must be the last* call to CHOLMOD).*/size_t nrow ; /* size of Flag and Head */long mark ; /* mark value for Flag array */size_t iworksize ; /* size of Iwork. Upper bound: 6*nrow+ncol */size_t xworksize ; /* size of Xwork, in bytes.* maxrank*nrow*sizeof(double) for update/downdate.* 2*nrow*sizeof(double) otherwise *//* initialized workspace: contents needed between calls to CHOLMOD */void *Flag ; /* size nrow, an integer array. Kept cleared between* calls to cholmod rouines (Flag [i] < mark) */void *Head ; /* size nrow+1, an integer array. Kept cleared between* calls to cholmod routines (Head [i] = EMPTY) */void *Xwork ; /* a double array. Its size varies. It is nrow for* most routines (cholmod_rowfac, cholmod_add,* cholmod_aat, cholmod_norm, cholmod_ssmult) for the real case, twice* that when the input matrices are complex or zomplex. It is of size* 2*nrow for cholmod_rowadd and cholmod_rowdel. For cholmod_updown,* its size is maxrank*nrow where maxrank is 2, 4, or 8. Kept cleared* between calls to cholmod (set to zero). *//* uninitialized workspace, contents not needed between calls to CHOLMOD */void *Iwork ; /* size iworksize, 2*nrow+ncol for most routines,* up to 6*nrow+ncol for cholmod_analyze. */int itype ; /* If CHOLMOD_LONG, Flag, Head, and Iwork are long.* Otherwise all three arrays are int. */int dtype ; /* double or float *//* Common->itype and Common->dtype are used to define the types of all* sparse matrices, triplet matrices, dense matrices, and factors* created using this Common struct. The itypes and dtypes of all* parameters to all CHOLMOD routines must match. *//* ---------------------------------------------------------------------- *//* statistics *//* ---------------------------------------------------------------------- *//* fl and lnz are set only in cholmod_analyze and cholmod_rowcolcounts,* in the Cholesky modudle. modfl is set only in the Modify module. */int status ; /* error code */double fl ; /* LL' flop count from most recent analysis */double lnz ; /* fundamental nz in L */double anz ; /* nonzeros in tril(A) if A is symmetric/lower,* triu(A) if symmetric/upper, or tril(A*A') if* unsymmetric, in last call to cholmod_analyze. */double modfl ; /* flop count from most recent update/downdate/* rowadd/rowdel (excluding flops to modify the* solution to Lx=b, if computed) */size_t malloc_count ; /* # of objects malloc'ed minus the # free'd*/size_t memory_usage ; /* peak memory usage in bytes */size_t memory_inuse ; /* current memory usage in bytes */} cholmod_common ;/* -------------------------------------------------------------------------- *//* cholmod_start: first call to CHOLMOD *//* -------------------------------------------------------------------------- */int cholmod_start(cholmod_common *Common) ;int cholmod_l_start (cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_finish: last call to CHOLMOD *//* -------------------------------------------------------------------------- */int cholmod_finish(cholmod_common *Common) ;int cholmod_l_finish (cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_defaults: restore default parameters *//* -------------------------------------------------------------------------- */int cholmod_defaults(cholmod_common *Common) ;int cholmod_l_defaults (cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_maxrank: return valid maximum rank for update/downdate *//* -------------------------------------------------------------------------- */size_t cholmod_maxrank /* returns validated value of Common->maxrank */(/* ---- input ---- */size_t n, /* A and L will have n rows *//* --------------- */cholmod_common *Common) ;size_t cholmod_l_maxrank (size_t, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_allocate_work: allocate workspace in Common *//* -------------------------------------------------------------------------- */int cholmod_allocate_work(/* ---- input ---- */size_t nrow, /* size: Common->Flag (nrow), Common->Head (nrow+1) */size_t iworksize, /* size of Common->Iwork */size_t xworksize, /* size of Common->Xwork *//* --------------- */cholmod_common *Common) ;int cholmod_l_allocate_work (size_t, size_t, size_t, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_free_work: free workspace in Common *//* -------------------------------------------------------------------------- */int cholmod_free_work(cholmod_common *Common) ;int cholmod_l_free_work (cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_clear_flag: clear Flag workspace in Common *//* -------------------------------------------------------------------------- */long cholmod_clear_flag(cholmod_common *Common) ;long cholmod_l_clear_flag (cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_error: called when CHOLMOD encounters an error *//* -------------------------------------------------------------------------- */int cholmod_error(/* ---- input ---- */int status, /* error status */char *file, /* name of source code file where error occured */int line, /* line number in source code file where error occured*/char *message, /* error message *//* --------------- */cholmod_common *Common) ;int cholmod_l_error (int, char *, int, char *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_dbound: for internal use in CHOLMOD only *//* -------------------------------------------------------------------------- */double cholmod_dbound /* returns modified diagonal entry of D or L */(/* ---- input ---- */double dj, /* diagonal entry of D for LDL' or L for LL' *//* --------------- */cholmod_common *Common) ;double cholmod_l_dbound (double, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_hypot: compute sqrt (x*x + y*y) accurately *//* -------------------------------------------------------------------------- */double cholmod_hypot(/* ---- input ---- */double x, double y) ;double cholmod_l_hypot (double, double) ;/* -------------------------------------------------------------------------- *//* cholmod_divcomplex: complex division, c = a/b *//* -------------------------------------------------------------------------- */int cholmod_divcomplex /* return 1 if divide-by-zero, 0 otherise */(/* ---- input ---- */double ar, double ai, /* real and imaginary parts of a */double br, double bi, /* real and imaginary parts of b *//* ---- output --- */double *cr, double *ci /* real and imaginary parts of c */) ;int cholmod_l_divcomplex (double, double, double, double, double *, double *) ;/* ========================================================================== *//* === Core/cholmod_sparse ================================================== *//* ========================================================================== *//* A sparse matrix stored in compressed-column form. */typedef struct cholmod_sparse_struct{size_t nrow ; /* the matrix is nrow-by-ncol */size_t ncol ;size_t nzmax ; /* maximum number of entries in the matrix *//* pointers to int or long: */void *p ; /* p [0..ncol], the column pointers */void *i ; /* i [0..nzmax-1], the row indices *//* for unpacked matrices only: */void *nz ; /* nz [0..ncol-1], the # of nonzeros in each col. In* packed form, the nonzero pattern of column j is in* A->i [A->p [j] ... A->p [j+1]-1]. In unpacked form, column j is in* A->i [A->p [j] ... A->p [j]+A->nz[j]-1] instead. In both cases, the* numerical values (if present) are in the corresponding locations in* the array x (or z if A->xtype is CHOLMOD_ZOMPLEX). *//* pointers to double or float: */void *x ; /* size nzmax or 2*nzmax, if present */void *z ; /* size nzmax, if present */int stype ; /* Describes what parts of the matrix are considered:** 0: matrix is "unsymmetric": use both upper and lower triangular parts* (the matrix may actually be symmetric in pattern and value, but* both parts are explicitly stored and used). May be square or* rectangular.* >0: matrix is square and symmetric, use upper triangular part.* Entries in the lower triangular part are ignored.* <0: matrix is square and symmetric, use lower triangular part.* Entries in the upper triangular part are ignored.** Note that stype>0 and stype<0 are different for cholmod_sparse and* cholmod_triplet. See the cholmod_triplet data structure for more* details.*/int itype ; /* CHOLMOD_INT: p, i, and nz are int.* CHOLMOD_INTLONG: p is long, i and nz are int.* CHOLMOD_LONG: p, i, and nz are long. */int xtype ; /* pattern, real, complex, or zomplex */int dtype ; /* x and z are double or float */int sorted ; /* TRUE if columns are sorted, FALSE otherwise */int packed ; /* TRUE if packed (nz must present), FALSE otherwise* (nz is ignored) */} cholmod_sparse ;/* -------------------------------------------------------------------------- *//* cholmod_allocate_sparse: allocate a sparse matrix *//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_allocate_sparse(/* ---- input ---- */size_t nrow, /* # of rows of A */size_t ncol, /* # of columns of A */size_t nzmax, /* max # of nonzeros of A */int sorted, /* TRUE if columns of A sorted, FALSE otherwise */int packed, /* TRUE if A will be packed, FALSE otherwise */int stype, /* stype of A */int xtype, /* CHOLMOD_PATTERN, _REAL, _COMPLEX, or _ZOMPLEX *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_allocate_sparse (size_t, size_t, size_t, int, int,int, int, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_free_sparse: free a sparse matrix *//* -------------------------------------------------------------------------- */int cholmod_free_sparse(/* ---- in/out --- */cholmod_sparse **A, /* matrix to deallocate, NULL on output *//* --------------- */cholmod_common *Common) ;int cholmod_l_free_sparse (cholmod_sparse **, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_reallocate_sparse: change the size (# entries) of sparse matrix *//* -------------------------------------------------------------------------- */int cholmod_reallocate_sparse(/* ---- input ---- */size_t nznew, /* new # of entries in A *//* ---- in/out --- */cholmod_sparse *A, /* matrix to reallocate *//* --------------- */cholmod_common *Common) ;int cholmod_l_reallocate_sparse ( size_t, cholmod_sparse *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_nnz: return number of nonzeros in a sparse matrix *//* -------------------------------------------------------------------------- */long cholmod_nnz(/* ---- input ---- */cholmod_sparse *A,/* --------------- */cholmod_common *Common) ;long cholmod_l_nnz (cholmod_sparse *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_speye: sparse identity matrix *//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_speye(/* ---- input ---- */size_t nrow, /* # of rows of A */size_t ncol, /* # of columns of A */int xtype, /* CHOLMOD_PATTERN, _REAL, _COMPLEX, or _ZOMPLEX *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_speye (size_t, size_t, int, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_spzeros: sparse zero matrix *//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_spzeros(/* ---- input ---- */size_t nrow, /* # of rows of A */size_t ncol, /* # of columns of A */size_t nzmax, /* max # of nonzeros of A */int xtype, /* CHOLMOD_PATTERN, _REAL, _COMPLEX, or _ZOMPLEX *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_spzeros (size_t, size_t, size_t, int,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_transpose: transpose a sparse matrix *//* -------------------------------------------------------------------------- *//* Return A' or A.' The "values" parameter is 0, 1, or 2 to denote the pattern* transpose, the array transpose (A.'), and the complex conjugate transpose* (A').*/cholmod_sparse *cholmod_transpose(/* ---- input ---- */cholmod_sparse *A, /* matrix to transpose */int values, /* 0: pattern, 1: array transpose, 2: conj. transpose *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_transpose (cholmod_sparse *, int, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_transpose_unsym: transpose an unsymmetric sparse matrix *//* -------------------------------------------------------------------------- *//* Compute F = A', A (:,f)', or A (p,f)', where A is unsymmetric and F is* already allocated. See cholmod_transpose for a simpler routine. */int cholmod_transpose_unsym(/* ---- input ---- */cholmod_sparse *A, /* matrix to transpose */int values, /* 0: pattern, 1: array transpose, 2: conj. transpose */int *Perm, /* size nrow, if present (can be NULL) */int *fset, /* subset of 0:(A->ncol)-1 */size_t fsize, /* size of fset *//* ---- output --- */cholmod_sparse *F, /* F = A', A(:,f)', or A(p,f)' *//* --------------- */cholmod_common *Common) ;int cholmod_l_transpose_unsym (cholmod_sparse *, int, long *, long *, size_t,cholmod_sparse *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_transpose_sym: transpose a symmetric sparse matrix *//* -------------------------------------------------------------------------- *//* Compute F = A' or A (p,p)', where A is symmetric and F is already allocated.* See cholmod_transpose for a simpler routine. */int cholmod_transpose_sym(/* ---- input ---- */cholmod_sparse *A, /* matrix to transpose */int values, /* 0: pattern, 1: array transpose, 2: conj. transpose */int *Perm, /* size nrow, if present (can be NULL) *//* ---- output --- */cholmod_sparse *F, /* F = A' or A(p,p)' *//* --------------- */cholmod_common *Common) ;int cholmod_l_transpose_sym (cholmod_sparse *, int, long *, cholmod_sparse *,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_ptranspose: transpose a sparse matrix *//* -------------------------------------------------------------------------- *//* Return A' or A(p,p)' if A is symmetric. Return A', A(:,f)', or A(p,f)' if* A is unsymmetric. */cholmod_sparse *cholmod_ptranspose(/* ---- input ---- */cholmod_sparse *A, /* matrix to transpose */int values, /* 0: pattern, 1: array transpose, 2: conj. transpose */int *Perm, /* if non-NULL, F = A(p,f) or A(p,p) */int *fset, /* subset of 0:(A->ncol)-1 */size_t fsize, /* size of fset *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_ptranspose (cholmod_sparse *, int, long *, long *,size_t, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_sort: sort row indices in each column of sparse matrix *//* -------------------------------------------------------------------------- */int cholmod_sort(/* ---- in/out --- */cholmod_sparse *A, /* matrix to sort *//* --------------- */cholmod_common *Common) ;int cholmod_l_sort (cholmod_sparse *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_band: C = tril (triu (A,k1), k2) *//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_band(/* ---- input ---- */cholmod_sparse *A, /* matrix to extract band matrix from */long k1, /* ignore entries below the k1-st diagonal */long k2, /* ignore entries above the k2-nd diagonal */int mode, /* >0: numerical, 0: pattern, <0: pattern (no diag) *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_band (cholmod_sparse *, long, long, int,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_band_inplace: A = tril (triu (A,k1), k2) *//* -------------------------------------------------------------------------- */int cholmod_band_inplace(/* ---- input ---- */long k1, /* ignore entries below the k1-st diagonal */long k2, /* ignore entries above the k2-nd diagonal */int mode, /* >0: numerical, 0: pattern, <0: pattern (no diag) *//* ---- in/out --- */cholmod_sparse *A, /* matrix from which entries not in band are removed *//* --------------- */cholmod_common *Common) ;int cholmod_l_band_inplace (long, long, int, cholmod_sparse *,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_aat: C = A*A' or A(:,f)*A(:,f)' *//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_aat(/* ---- input ---- */cholmod_sparse *A, /* input matrix; C=A*A' is constructed */int *fset, /* subset of 0:(A->ncol)-1 */size_t fsize, /* size of fset */int mode, /* >0: numerical, 0: pattern, <0: pattern (no diag),* -2: pattern only, no diagonal, add 50%+n extra* space to C *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_aat (cholmod_sparse *, long *, size_t, int,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_copy_sparse: C = A, create an exact copy of a sparse matrix *//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_copy_sparse(/* ---- input ---- */cholmod_sparse *A, /* matrix to copy *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_copy_sparse (cholmod_sparse *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_copy: C = A, with possible change of stype *//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_copy(/* ---- input ---- */cholmod_sparse *A, /* matrix to copy */int stype, /* requested stype of C */int mode, /* >0: numerical, 0: pattern, <0: pattern (no diag) *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_copy (cholmod_sparse *, int, int, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_add: C = alpha*A + beta*B *//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_add(/* ---- input ---- */cholmod_sparse *A, /* matrix to add */cholmod_sparse *B, /* matrix to add */double alpha [2], /* scale factor for A */double beta [2], /* scale factor for B */int values, /* if TRUE compute the numerical values of C */int sorted, /* if TRUE, sort columns of C *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_add (cholmod_sparse *, cholmod_sparse *, double *,double *, int, int, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_sparse_xtype: change the xtype of a sparse matrix *//* -------------------------------------------------------------------------- */int cholmod_sparse_xtype(/* ---- input ---- */int to_xtype, /* requested xtype (pattern, real, complex, zomplex) *//* ---- in/out --- */cholmod_sparse *A, /* sparse matrix to change *//* --------------- */cholmod_common *Common) ;int cholmod_l_sparse_xtype (int, cholmod_sparse *, cholmod_common *) ;/* ========================================================================== *//* === Core/cholmod_factor ================================================== *//* ========================================================================== *//* A symbolic and numeric factorization, either simplicial or supernodal.* In all cases, the row indices in the columns of L are kept sorted. */typedef struct cholmod_factor_struct{/* ---------------------------------------------------------------------- *//* for both simplicial and supernodal factorizations *//* ---------------------------------------------------------------------- */size_t n ; /* L is n-by-n */size_t minor ; /* If the factorization failed, L->minor is the column* at which it failed (in the range 0 to n-1). A value* of n means the factorization was successful or* the matrix has not yet been factorized. *//* ---------------------------------------------------------------------- *//* symbolic ordering and analysis *//* ---------------------------------------------------------------------- */void *Perm ; /* size n, permutation used */void *ColCount ; /* size n, column counts for simplicial L *//* ---------------------------------------------------------------------- *//* simplicial factorization *//* ---------------------------------------------------------------------- */size_t nzmax ; /* size of i and x */void *p ; /* p [0..ncol], the column pointers */void *i ; /* i [0..nzmax-1], the row indices */void *x ; /* x [0..nzmax-1], the numerical values */void *z ;void *nz ; /* nz [0..ncol-1], the # of nonzeros in each column.* i [p [j] ... p [j]+nz[j]-1] contains the row indices,* and the numerical values are in the same locatins* in x. The value of i [p [k]] is always k. */void *next ; /* size ncol+2. next [j] is the next column in i/x */void *prev ; /* size ncol+2. prev [j] is the prior column in i/x.* head of the list is ncol+1, and the tail is ncol. *//* ---------------------------------------------------------------------- *//* supernodal factorization *//* ---------------------------------------------------------------------- *//* Note that L->x is shared with the simplicial data structure. L->x has* size L->nzmax for a simplicial factor, and size L->xsize for a supernodal* factor. */size_t nsuper ; /* number of supernodes */size_t ssize ; /* size of s, integer part of supernodes */size_t xsize ; /* size of x, real part of supernodes */size_t maxcsize ; /* size of largest update matrix */size_t maxesize ; /* max # of rows in supernodes, excl. triangular part */void *super ; /* size nsuper+1, first col in each supernode */void *pi ; /* size nsuper+1, pointers to integer patterns */void *px ; /* size nsuper+1, pointers to real parts */void *s ; /* size ssize, integer part of supernodes *//* ---------------------------------------------------------------------- *//* factorization type *//* ---------------------------------------------------------------------- */int ordering ; /* ordering method used */int is_ll ; /* TRUE if LL', FALSE if LDL' */int is_super ; /* TRUE if supernodal, FALSE if simplicial */int is_monotonic ; /* TRUE if columns of L appear in order 0..n-1.* Only applicable to simplicial numeric types. *//* There are 8 types of factor objects that cholmod_factor can represent* (only 6 are used):** Numeric types (xtype is not CHOLMOD_PATTERN)* --------------------------------------------** simplicial LDL': (is_ll FALSE, is_super FALSE). Stored in compressed* column form, using the simplicial components above (nzmax, p, i,* x, z, nz, next, and prev). The unit diagonal of L is not stored,* and D is stored in its place. There are no supernodes.** simplicial LL': (is_ll TRUE, is_super FALSE). Uses the same storage* scheme as the simplicial LDL', except that D does not appear.* The first entry of each column of L is the diagonal entry of* that column of L.** supernodal LDL': (is_ll FALSE, is_super TRUE). Not used.* FUTURE WORK: add support for supernodal LDL'** supernodal LL': (is_ll TRUE, is_super TRUE). A supernodal factor,* using the supernodal components described above (nsuper, ssize,* xsize, maxcsize, maxesize, super, pi, px, s, x, and z).*** Symbolic types (xtype is CHOLMOD_PATTERN)* -----------------------------------------** simplicial LDL': (is_ll FALSE, is_super FALSE). Nothing is present* except Perm and ColCount.** simplicial LL': (is_ll TRUE, is_super FALSE). Identical to the* simplicial LDL', except for the is_ll flag.** supernodal LDL': (is_ll FALSE, is_super TRUE). Not used.* FUTURE WORK: add support for supernodal LDL'** supernodal LL': (is_ll TRUE, is_super TRUE). A supernodal symbolic* factorization. The simplicial symbolic information is present* (Perm and ColCount), as is all of the supernodal factorization* except for the numerical values (x and z).*/int itype ; /* The integer arrays are Perm, ColCount, p, i, nz,* next, prev, super, pi, px, and s. If itype is* CHOLMOD_INT, all of these are int arrays.* CHOLMOD_INTLONG: p, pi, px are long, all others int.* CHOLMOD_LONG: all integer arrays are long. */int xtype ; /* pattern, real, complex, or zomplex */int dtype ; /* x and z double or float */} cholmod_factor ;/* -------------------------------------------------------------------------- *//* cholmod_allocate_factor: allocate a factor (symbolic LL' or LDL') *//* -------------------------------------------------------------------------- */cholmod_factor *cholmod_allocate_factor(/* ---- input ---- */size_t n, /* L is n-by-n *//* --------------- */cholmod_common *Common) ;cholmod_factor *cholmod_l_allocate_factor (size_t, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_free_factor: free a factor *//* -------------------------------------------------------------------------- */int cholmod_free_factor(/* ---- in/out --- */cholmod_factor **L, /* factor to free, NULL on output *//* --------------- */cholmod_common *Common) ;int cholmod_l_free_factor (cholmod_factor **, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_reallocate_factor: change the # entries in a factor *//* -------------------------------------------------------------------------- */int cholmod_reallocate_factor(/* ---- input ---- */size_t nznew, /* new # of entries in L *//* ---- in/out --- */cholmod_factor *L, /* factor to modify *//* --------------- */cholmod_common *Common) ;int cholmod_l_reallocate_factor (size_t, cholmod_factor *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_change_factor: change the type of factor (e.g., LDL' to LL') *//* -------------------------------------------------------------------------- */int cholmod_change_factor(/* ---- input ---- */int to_xtype, /* to CHOLMOD_PATTERN, _REAL, _COMPLEX, _ZOMPLEX */int to_ll, /* TRUE: convert to LL', FALSE: LDL' */int to_super, /* TRUE: convert to supernodal, FALSE: simplicial */int to_packed, /* TRUE: pack simplicial columns, FALSE: do not pack */int to_monotonic, /* TRUE: put simplicial columns in order, FALSE: not *//* ---- in/out --- */cholmod_factor *L, /* factor to modify *//* --------------- */cholmod_common *Common) ;int cholmod_l_change_factor ( int, int, int, int, int, cholmod_factor *,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_pack_factor: pack the columns of a factor *//* -------------------------------------------------------------------------- *//* Pack the columns of a simplicial factor. Unlike cholmod_change_factor,* it can pack the columns of a factor even if they are not stored in their* natural order (non-monotonic). */int cholmod_pack_factor(/* ---- in/out --- */cholmod_factor *L, /* factor to modify *//* --------------- */cholmod_common *Common) ;int cholmod_l_pack_factor (cholmod_factor *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_reallocate_column: resize a single column of a factor *//* -------------------------------------------------------------------------- */int cholmod_reallocate_column(/* ---- input ---- */size_t j, /* the column to reallocate */size_t need, /* required size of column j *//* ---- in/out --- */cholmod_factor *L, /* factor to modify *//* --------------- */cholmod_common *Common) ;int cholmod_l_reallocate_column (size_t, size_t, cholmod_factor *,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_factor_to_sparse: create a sparse matrix copy of a factor *//* -------------------------------------------------------------------------- *//* Only operates on numeric factors, not symbolic ones */cholmod_sparse *cholmod_factor_to_sparse(/* ---- in/out --- */cholmod_factor *L, /* factor to copy, converted to symbolic on output *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_factor_to_sparse (cholmod_factor *,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_copy_factor: create a copy of a factor *//* -------------------------------------------------------------------------- */cholmod_factor *cholmod_copy_factor(/* ---- input ---- */cholmod_factor *L, /* factor to copy *//* --------------- */cholmod_common *Common) ;cholmod_factor *cholmod_l_copy_factor (cholmod_factor *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_factor_xtype: change the xtype of a factor *//* -------------------------------------------------------------------------- */int cholmod_factor_xtype(/* ---- input ---- */int to_xtype, /* requested xtype (real, complex, or zomplex) *//* ---- in/out --- */cholmod_factor *L, /* factor to change *//* --------------- */cholmod_common *Common) ;int cholmod_l_factor_xtype (int, cholmod_factor *, cholmod_common *) ;/* ========================================================================== *//* === Core/cholmod_dense =================================================== *//* ========================================================================== *//* A dense matrix in column-oriented form. It has no itype since it contains* no integers. Entry in row i and column j is located in x [i+j*d].*/typedef struct cholmod_dense_struct{size_t nrow ; /* the matrix is nrow-by-ncol */size_t ncol ;size_t nzmax ; /* maximum number of entries in the matrix */size_t d ; /* leading dimension (d >= nrow must hold) */void *x ; /* size nzmax or 2*nzmax, if present */void *z ; /* size nzmax, if present */int xtype ; /* pattern, real, complex, or zomplex */int dtype ; /* x and z double or float */} cholmod_dense ;/* -------------------------------------------------------------------------- *//* cholmod_allocate_dense: allocate a dense matrix (contents uninitialized) *//* -------------------------------------------------------------------------- */cholmod_dense *cholmod_allocate_dense(/* ---- input ---- */size_t nrow, /* # of rows of matrix */size_t ncol, /* # of columns of matrix */size_t d, /* leading dimension */int xtype, /* CHOLMOD_REAL, _COMPLEX, or _ZOMPLEX *//* --------------- */cholmod_common *Common) ;cholmod_dense *cholmod_l_allocate_dense (size_t, size_t, size_t, int,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_zeros: allocate a dense matrix and set it to zero *//* -------------------------------------------------------------------------- */cholmod_dense *cholmod_zeros(/* ---- input ---- */size_t nrow, /* # of rows of matrix */size_t ncol, /* # of columns of matrix */int xtype, /* CHOLMOD_REAL, _COMPLEX, or _ZOMPLEX *//* --------------- */cholmod_common *Common) ;cholmod_dense *cholmod_l_zeros (size_t, size_t, int, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_ones: allocate a dense matrix and set it to all ones *//* -------------------------------------------------------------------------- */cholmod_dense *cholmod_ones(/* ---- input ---- */size_t nrow, /* # of rows of matrix */size_t ncol, /* # of columns of matrix */int xtype, /* CHOLMOD_REAL, _COMPLEX, or _ZOMPLEX *//* --------------- */cholmod_common *Common) ;cholmod_dense *cholmod_l_ones (size_t, size_t, int, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_eye: allocate a dense matrix and set it to the identity matrix *//* -------------------------------------------------------------------------- */cholmod_dense *cholmod_eye(/* ---- input ---- */size_t nrow, /* # of rows of matrix */size_t ncol, /* # of columns of matrix */int xtype, /* CHOLMOD_REAL, _COMPLEX, or _ZOMPLEX *//* --------------- */cholmod_common *Common) ;cholmod_dense *cholmod_l_eye (size_t, size_t, int, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_free_dense: free a dense matrix *//* -------------------------------------------------------------------------- */int cholmod_free_dense(/* ---- in/out --- */cholmod_dense **X, /* dense matrix to deallocate, NULL on output *//* --------------- */cholmod_common *Common) ;int cholmod_l_free_dense (cholmod_dense **, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_sparse_to_dense: create a dense matrix copy of a sparse matrix *//* -------------------------------------------------------------------------- */cholmod_dense *cholmod_sparse_to_dense(/* ---- input ---- */cholmod_sparse *A, /* matrix to copy *//* --------------- */cholmod_common *Common) ;cholmod_dense *cholmod_l_sparse_to_dense (cholmod_sparse *,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_dense_to_sparse: create a sparse matrix copy of a dense matrix *//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_dense_to_sparse(/* ---- input ---- */cholmod_dense *X, /* matrix to copy */int values, /* TRUE if values to be copied, FALSE otherwise *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_dense_to_sparse (cholmod_dense *, int,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_copy_dense: create a copy of a dense matrix *//* -------------------------------------------------------------------------- */cholmod_dense *cholmod_copy_dense(/* ---- input ---- */cholmod_dense *X, /* matrix to copy *//* --------------- */cholmod_common *Common) ;cholmod_dense *cholmod_l_copy_dense (cholmod_dense *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_copy_dense2: copy a dense matrix (pre-allocated) *//* -------------------------------------------------------------------------- */int cholmod_copy_dense2(/* ---- input ---- */cholmod_dense *X, /* matrix to copy *//* ---- output --- */cholmod_dense *Y, /* copy of matrix X *//* --------------- */cholmod_common *Common) ;int cholmod_l_copy_dense2 (cholmod_dense *, cholmod_dense *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_dense_xtype: change the xtype of a dense matrix *//* -------------------------------------------------------------------------- */int cholmod_dense_xtype(/* ---- input ---- */int to_xtype, /* requested xtype (real, complex,or zomplex) *//* ---- in/out --- */cholmod_dense *X, /* dense matrix to change *//* --------------- */cholmod_common *Common) ;int cholmod_l_dense_xtype (int, cholmod_dense *, cholmod_common *) ;/* ========================================================================== *//* === Core/cholmod_triplet ================================================= *//* ========================================================================== *//* A sparse matrix stored in triplet form. */typedef struct cholmod_triplet_struct{size_t nrow ; /* the matrix is nrow-by-ncol */size_t ncol ;size_t nzmax ; /* maximum number of entries in the matrix */size_t nnz ; /* number of nonzeros in the matrix */void *i ; /* i [0..nzmax-1], the row indices */void *j ; /* j [0..nzmax-1], the column indices */void *x ; /* size nzmax or 2*nzmax, if present */void *z ; /* size nzmax, if present */int stype ; /* Describes what parts of the matrix are considered:** 0: matrix is "unsymmetric": use both upper and lower triangular parts* (the matrix may actually be symmetric in pattern and value, but* both parts are explicitly stored and used). May be square or* rectangular.* >0: matrix is square and symmetric. Entries in the lower triangular* part are transposed and added to the upper triangular part when* the matrix is converted to cholmod_sparse form.* <0: matrix is square and symmetric. Entries in the upper triangular* part are transposed and added to the lower triangular part when* the matrix is converted to cholmod_sparse form.** Note that stype>0 and stype<0 are different for cholmod_sparse and* cholmod_triplet. The reason is simple. You can permute a symmetric* triplet matrix by simply replacing a row and column index with their* new row and column indices, via an inverse permutation. Suppose* P = L->Perm is your permutation, and Pinv is an array of size n.* Suppose a symmetric matrix A is represent by a triplet matrix T, with* entries only in the upper triangular part. Then the following code:** Ti = T->i ;* Tj = T->j ;* for (k = 0 ; k < n ; k++) Pinv [P [k]] = k ;* for (k = 0 ; k < nz ; k++) Ti [k] = Pinv [Ti [k]] ;* for (k = 0 ; k < nz ; k++) Tj [k] = Pinv [Tj [k]] ;** creates the triplet form of C=P*A*P'. However, if T initially* contains just the upper triangular entries (T->stype = 1), after* permutation it has entries in both the upper and lower triangular* parts. These entries should be transposed when constructing the* cholmod_sparse form of A, which is what cholmod_triplet_to_sparse* does. Thus:** C = cholmod_triplet_to_sparse (T, 0, &Common) ;** will return the matrix C = P*A*P'.** Since the triplet matrix T is so simple to generate, it's quite easy* to remove entries that you do not want, prior to converting T to the* cholmod_sparse form. So if you include these entries in T, CHOLMOD* assumes that there must be a reason (such as the one above). Thus,* no entry in a triplet matrix is ever ignored.*/int itype ; /* CHOLMOD_LONG: i and j are long. Otherwise int. */int xtype ; /* pattern, real, complex, or zomplex */int dtype ; /* x and z are double or float */} cholmod_triplet ;/* -------------------------------------------------------------------------- *//* cholmod_allocate_triplet: allocate a triplet matrix *//* -------------------------------------------------------------------------- */cholmod_triplet *cholmod_allocate_triplet(/* ---- input ---- */size_t nrow, /* # of rows of T */size_t ncol, /* # of columns of T */size_t nzmax, /* max # of nonzeros of T */int stype, /* stype of T */int xtype, /* CHOLMOD_PATTERN, _REAL, _COMPLEX, or _ZOMPLEX *//* --------------- */cholmod_common *Common) ;cholmod_triplet *cholmod_l_allocate_triplet (size_t, size_t, size_t, int, int,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_free_triplet: free a triplet matrix *//* -------------------------------------------------------------------------- */int cholmod_free_triplet(/* ---- in/out --- */cholmod_triplet **T, /* triplet matrix to deallocate, NULL on output *//* --------------- */cholmod_common *Common) ;int cholmod_l_free_triplet (cholmod_triplet **, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_reallocate_triplet: change the # of entries in a triplet matrix *//* -------------------------------------------------------------------------- */int cholmod_reallocate_triplet(/* ---- input ---- */size_t nznew, /* new # of entries in T *//* ---- in/out --- */cholmod_triplet *T, /* triplet matrix to modify *//* --------------- */cholmod_common *Common) ;int cholmod_l_reallocate_triplet (size_t, cholmod_triplet *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_sparse_to_triplet: create a triplet matrix copy of a sparse matrix*//* -------------------------------------------------------------------------- */cholmod_triplet *cholmod_sparse_to_triplet(/* ---- input ---- */cholmod_sparse *A, /* matrix to copy *//* --------------- */cholmod_common *Common) ;cholmod_triplet *cholmod_l_sparse_to_triplet (cholmod_sparse *,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_triplet_to_sparse: create a sparse matrix copy of a triplet matrix*//* -------------------------------------------------------------------------- */cholmod_sparse *cholmod_triplet_to_sparse(/* ---- input ---- */cholmod_triplet *T, /* matrix to copy */int nzmax, /* allocate at least this much space in output matrix *//* --------------- */cholmod_common *Common) ;cholmod_sparse *cholmod_l_triplet_to_sparse (cholmod_triplet *, int,cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_copy_triplet: create a copy of a triplet matrix *//* -------------------------------------------------------------------------- */cholmod_triplet *cholmod_copy_triplet(/* ---- input ---- */cholmod_triplet *T, /* matrix to copy *//* --------------- */cholmod_common *Common) ;cholmod_triplet *cholmod_l_copy_triplet (cholmod_triplet *, cholmod_common *) ;/* -------------------------------------------------------------------------- *//* cholmod_triplet_xtype: change the xtype of a triplet matrix *//* -------------------------------------------------------------------------- */int cholmod_triplet_xtype(/* ---- input ---- */int to_xtype, /* requested xtype (pattern, real, complex,or zomplex)*//* ---- in/out --- */cholmod_triplet *T, /* triplet matrix to change *//* --------------- */cholmod_common *Common) ;int cholmod_l_triplet_xtype (int, cholmod_triplet *, cholmod_common *) ;/* ========================================================================== *//* === Core/cholmod_memory ================================================== *//* ========================================================================== *//* The user may make use of these, just like malloc and free. You can even* malloc an object and safely free it with cholmod_free, and visa versa* (except that the memory usage statistics will be corrupted). These routines* do differ from malloc and free. If cholmod_free is given a NULL pointer,* for example, it does nothing (unlike the ANSI free). cholmod_realloc does* not return NULL if given a non-NULL pointer and a nonzero size, even if it* fails (it returns the original pointer and sets an error code in* Common->status instead).** CHOLMOD keeps track of the amount of memory it has allocated, and so the* cholmod_free routine also takes the size of the object being freed. This* is only used for statistics. If you, the user of CHOLMOD, pass the wrong* size, the only consequence is that the memory usage statistics will be* corrupted.*/void *cholmod_malloc /* returns pointer to the newly malloc'd block */(/* ---- input ---- */size_t n, /* number of items */size_t size, /* size of each item *//* --------------- */cholmod_common *Common) ;void *cholmod_l_malloc (size_t, size_t, cholmod_common *) ;void *cholmod_calloc /* returns pointer to the newly calloc'd block */(/* ---- input ---- */size_t n, /* number of items */size_t size, /* size of each item *//* --------------- */cholmod_common *Common) ;void *cholmod_l_calloc (size_t, size_t, cholmod_common *) ;void *cholmod_free /* always returns NULL */(/* ---- input ---- */size_t n, /* number of items */size_t size, /* size of each item *//* ---- in/out --- */void *p, /* block of memory to free *//* --------------- */cholmod_common *Common) ;void *cholmod_l_free (size_t, size_t, void *, cholmod_common *) ;void *cholmod_realloc /* returns pointer to reallocated block */(/* ---- input ---- */size_t nnew, /* requested # of items in reallocated block */size_t size, /* size of each item *//* ---- in/out --- */void *p, /* block of memory to realloc */size_t *n, /* current size on input, nnew on output if successful*//* --------------- */cholmod_common *Common) ;void *cholmod_l_realloc (size_t, size_t, void *, size_t *, cholmod_common *) ;int cholmod_realloc_multiple(/* ---- input ---- */size_t nnew, /* requested # of items in reallocated blocks */int nint, /* number of int/long blocks */int xtype, /* CHOLMOD_PATTERN, _REAL, _COMPLEX, or _ZOMPLEX *//* ---- in/out --- */void **I, /* int or long block */void **J, /* int or long block */void **X, /* complex, double, or float block */void **Z, /* zomplex case only: double or float block */size_t *n, /* current size of the I,J,X,Z blocks on input,* nnew on output if successful *//* --------------- */cholmod_common *Common) ;int cholmod_l_realloc_multiple (size_t, int, int, void **, void **, void **,void **, size_t *, cholmod_common *) ;/* ========================================================================== *//* === Numerical relop macros =============================================== *//* ========================================================================== *//* These macros correctly handle the NaN case.** CHOLMOD_IS_NAN(x):* True if x is NaN. False otherwise. The commonly-existing isnan(x)* function could be used, but it's not in Kernighan & Ritchie 2nd edition* (ANSI C89). It may appear in <math.h>, but I'm not certain about* portability. The expression x != x is true if and only if x is NaN,* according to the IEEE 754 floating-point standard.** CHOLMOD_IS_ZERO(x):* True if x is zero. False if x is nonzero, NaN, or +/- Inf.* This is (x == 0) if the compiler is IEEE 754 compliant.** CHOLMOD_IS_NONZERO(x):* True if x is nonzero, NaN, or +/- Inf. False if x zero.* This is (x != 0) if the compiler is IEEE 754 compliant.** CHOLMOD_IS_LT_ZERO(x):* True if x is < zero or -Inf. False if x is >= 0, NaN, or +Inf.* This is (x < 0) if the compiler is IEEE 754 compliant.** CHOLMOD_IS_GT_ZERO(x):* True if x is > zero or +Inf. False if x is <= 0, NaN, or -Inf.* This is (x > 0) if the compiler is IEEE 754 compliant.** CHOLMOD_IS_LE_ZERO(x):* True if x is <= zero or -Inf. False if x is > 0, NaN, or +Inf.* This is (x <= 0) if the compiler is IEEE 754 compliant.*/#ifdef CHOLMOD_WINDOWS/* Yes, this is exceedingly ugly. Blame Microsoft, which hopelessly *//* violates the IEEE 754 floating-point standard in a bizarre way. *//* If you're using an IEEE 754-compliant compiler, then x != x is true *//* iff x is NaN. For Microsoft, (x < x) is true iff x is NaN. *//* So either way, this macro safely detects a NaN. */#define CHOLMOD_IS_NAN(x) (((x) != (x)) || (((x) < (x))))#define CHOLMOD_IS_ZERO(x) (((x) == 0.) && !CHOLMOD_IS_NAN(x))#define CHOLMOD_IS_NONZERO(x) (((x) != 0.) || CHOLMOD_IS_NAN(x))#define CHOLMOD_IS_LT_ZERO(x) (((x) < 0.) && !CHOLMOD_IS_NAN(x))#define CHOLMOD_IS_GT_ZERO(x) (((x) > 0.) && !CHOLMOD_IS_NAN(x))#define CHOLMOD_IS_LE_ZERO(x) (((x) <= 0.) && !CHOLMOD_IS_NAN(x))#else/* These all work properly, according to the IEEE 754 standard ... except on *//* a PC with windows. Works fine in Linux on the same PC... */#define CHOLMOD_IS_NAN(x) ((x) != (x))#define CHOLMOD_IS_ZERO(x) ((x) == 0.)#define CHOLMOD_IS_NONZERO(x) ((x) != 0.)#define CHOLMOD_IS_LT_ZERO(x) ((x) < 0.)#define CHOLMOD_IS_GT_ZERO(x) ((x) > 0.)#define CHOLMOD_IS_LE_ZERO(x) ((x) <= 0.)#endif#endif