SPRAL SSIDS Sparse Symmetric Indefinite Direct Solver Fortran
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1. 2 Usage overview 2 1 Calling sequences Access to the package requires a USE statement use spral_ssids The following procedures are available to the user ssids_analyse accepts the matrix data in compressed sparse column format and optionally checks it for duplicates and out of range entries The user may supply an elimination order otherwise one is generated Using this elimination order ssids_analyse analyses the sparsity pattern of the matrix and prepares the data structures for the factorization ssids_analyse_coord is an alternative to ssids_analyse that may be used if the user has the matrix data in coordinate format Again the user may supply an elimination order otherwise one is generated ssids_analyse_coord checks the matrix data for duplicates and out of range entries stores it in compressed sparse column format and then proceeds in the same way as ssids_analyse ssids_factor uses the data structures set up by ssids_analyse to compute a sparse factorization More than one call to ssids_factor may follow a call to ssids_analyse allowing more than one matrix with the same sparsity pattern but different numerical values to be factorized without multiple calls to ssids_analyse An option exists to scale the matrix ssids_solve uses the computed factors generated by ssids_factor to solve systems AX B for one or more right hand sides B Multiple calls to ssids_solve may follow a call to ssids_2. if one and more diagonal entries of A is missing and out of range and or duplicated variable indices have been found 6 Returned by ssids_analyse and ssids_analyse_coord if A is found be structurally singular This will overwrite any of the above warnings 7 Returned by ssids_factor if optionsf action is set to true and the matrix is found to be structurally or numerically singular 8 Returned by ssids_factor if optionsjordering 2 i e a matching based ordering was used but the associated scaling was not i e options scaling 3 7 Method ssids_analyse and ssids_analyse_coord If check is set to true on the call to ssids_analyse or if ssids_analyse_coord is called the user supplied matrix data is checked for errors The cleaned integer matrix data duplicates are summed and out of range indices discarded is stored in akeep The use of checking is optional on a call to ssids_analyse as it incurs both time and memory overheads However it is recommended since the behaviour of the other routines in the package is unpredictable if duplicates and or out of range variable indices are entered If the user has supplied an elimination order it is checked for errors Otherwise an elimination order is generated by the package The elimination order is used to construct an assembly tree On exit from ssids_analyse and ssids_analyse_coord order is set so that order i holds the position of variable 7 in th
3. will be placed in d 2 i 7 1 2 n 1 and d 2 n will be set to zero 4 3 ssids_alter To alter D following an indefinite factorization call ssids_alter d akeep fkeep options inform Note that this routine is not compatabile with the option options presolve 1 d is an INTENTCIN array of package type with extents 2 and n The diagonal entries of D7 will be altered to d 1 i i 1 2 n and the off diagonal entries will be altered to d 2 i i 1 2 n 1 and PLD PL will no longer be a factorization of A akeep isan INTENTCIN scalar of type ssids_akeep that must be unchanged since the call to ssids_factor fkeep is an INTENT INOUT scalar of type ssids_fkeep that must be unchanged since the call to ssids_factor 5 Derived types 5 1 ssids_options The derived data type ssids_options is used to specify the options used within SSIDS The components that are automatically given default values in the definition of the type are http www numerical rl ac uk spral 7 17 March 2014 SPRAL SSIDS Version 1 0 0 5 DERIVED TYPES Printing options print_level is a scalar of type INTEGER that is used to control the level of printing The different levels are lt 0 No printing 0 Error and warning messages only 1 As 0 plus basic diagnostic printing gt 1 As 1 plus some additional diagnostic printing The default is print_level 0 unit_diagnostics is a scalar of type INTEGER that holds
4. 3 3 ssids_solve To solve the linear system AX B after a call to ssids_factor for a single right hand side call ssids_solve x1 akeep fkeep options inform job for one or more right hand sides call ssids_solve nrhs x ldx akeep fkeep options inform job Partial solutions may be performed by appropriately setting the optional parameter job x1 is an INTENTCINOUT array of package type and size n It must be set such that x1 i holds the component of the right hand side for variables i On exit x1 i holds the solution for variable i nrhs is an INTENT IN scalar of type INTEGER that holds the number of right hand sides x isan INTENTCINOUT array of package type with extents ldx and nrhs It must be set so that x i j holds the component of the right hand side for variable 7 to the j th system On exit x i j holds the solution for variable i to the j th system ldx is an INTENT IN scalar of type INTEGER that must be set to the first extent of array x akeep isan INTENTCIN scalar of type ssids_akeep that must be unchanged since the last call to ssids_factor fkeep is an INTENT IN scalar of type ssids_fkeep that must be unchanged since the last call to ssids_factor options is an INTENTCIN scalar of type ssids_options Its components specify the algorithmic options used by the subroutine as explained in Section 5 1 inform is an INTENT OQUT scalar of type ssids_inform Its components provide inf
5. ac uk spral 2 17 March 2014 SPRAL SSIDS Version 1 0 0 2 USAGE OVERVIEW Figure 1 Data format example matrix LI 2 2 3 3 2 2 4 4 4 4 5 5 6 6 3 3 7 7 8 8 6 6 8 8 9 9 2 3 Achieving bit compatibility Care has been taken to ensure bit compatibility is achieved using this solver That is consecutive runs with the same data on the same machine produces exactly the same solution 2 4 Optional arguments We use square brackets to indicate optional arguments In each call optional arguments follow the argument inform Since we reserve the right to add additional optional arguments in future releases of the code we strongly recommend that all optional arguments be called by keyword not by position 2 5 Integer real and package types INTEGER denotes default INTEGER and INTEGER long denotes INTEGER kind selected_int_kind 18 REAL denotes double precision real We also use the term package type to mean the same 2 6 Data formats 2 6 1 Compressed Sparse Column CSC Format This standard data format consists of the following data integer z n size of matrix integer size n 1 Sf ptr column pointers integer size ptr n 1 1 row row indices real size ptr n 1 1 val numerical values Non zero matrix entries are ordered by increasing column index and stored in the arrays row and val such that row k holds the row number and val k holds the value of the k th entry The ptr array stores column pointers suc
6. is a scalar of type INTEGER In the event of an allocation or deallocation error it holds the Fortran stat parameter if it is available and is set to 0 otherwise cublas_error is a scalar of type INTEGER In the event of an error return from the CUBLAS library it holds the error code returned and is 0 otherwise cuda_error is a scalar of type INTEGER In the event of a CUDA error it holds the error code returned and is O otherwise Note that due to the asynchronous nature of GPU execution the reported error may have a cause errors external to SSIDS 6 Return codes A successful return from a subroutine in the package is indicated by inform flag having the value zero A negative value is associated with an error message that by default will be output on unit optionsZunit_error Possible negative values are 1 An error has been made in the sequence of calls this includes calling a subroutine after an error that cannot be recovered from 2 Returned by ssids_analyse and ssids_analyse_coord ifn lt 0 Also returned by ssids_analyse_coord if ne lt 1 3 Returned by ssids_analyse if there is an error in ptr 4 Returned by ssids_analyse if all the variable indices in one or more columns are out of range Also returned by ssids_analyse_coord if all entries are out of range 5 Returned by ssids_factor if posdef false and options action false when the matrix is found to be singular The user may reset t
7. time The pivoting condition is chosen to ensure that all entries of L have absolute value less than u This limits the growth of the entries of the D factor and ensures that any solves will be backwards stable The details are described in 1 If a pivot candidate does not pass the pivot tests at a node it is delayed to its parent node where further elimination operations may make it acceptable Delaying pivots leads to additional fill in and floating point operations beyond that predicted by ssids_analyse and ssids_analyse_coord and may result in additional memory allocations being required The number of delayed pivots can often be reduced by using appropriate scaling At each non root node the majority of the floating point operations involve the formation of the generated element This is handled by a single dedicated kernel again see 1 for details At the end of the factorization data structures for use in future calls to ssids_solve are prepared If options presolve 1 the block of L corresponding to the eliminated variables is explicitly inverted to accelerate future calls to ssids_solve at the cost of making ssids_factor slower ssids_solve If options use_gpu_solve false data is moved to the CPU if required and the BLAS calls are used to perform a solve using the assembly tree and factors generated on previous calls Otherwise the solve is conducted on the GPU in a similar fashion If options presolve 0 custom G
8. used if nemin lt 1 Options used by ssids_factor scaling is a scalar of type default INTEGER that controls the use of scaling The available options are lt 0 No scaling if scale is not present or user supplied scaling if scale is present 1 Compute a scaling using a weighted bipartite matching via the Hungarian Algorithm MC64 algorithm 2 Compute a scaling using a weighted bipartite matching via the Auction Algorithm may be lower quality than that computed using the Hungarian Algorithm but can be considerably faster 3 A matching based ordering has been generated during the analyse phase using options ordering 2 Use the scaling generated as a side effect of this process The scaling will be the same as that generated with options scaling 1 if the matrix values have not changed This option will generate an error if a matching based ordering was not used gt 4 Compute a scaling using the norm equilibration algorithm of Ruiz The default is scaling 0 http www numerical rl ac uk spral 8 17 March 2014 SPRAL SSIDS Version 1 0 0 5 DERIVED TYPES Options used by ssids_factor with posdef false A indefinite action is a scalar of type default LOGICAL If the matrix is found to be singular has rank less than the number of non empty rows the computation continues after issuing a warning if action has the value true or terminates with an error if it has the value false The default is action
9. 3 6 8 8 9 row 1 ptr n 1 1 1 2 2 3 5 3 4 5 val 1 ptr mt 1 1 2 0 1 0 4 0 1 0 1 0 3 0 2 0 2 0 The right hand side with solution 1 0 2 0 3 0 4 0 5 0 x 1 n 4 0 17 0 19 0 6 0 12 0 Perform analyse and factorise with data checking check true call ssids_analyse check n ptr row akeep options inform if inform flag lt 0 go to 100 call ssids_factor posdef val akeep fkeep options inform if inform flag lt 0 go to 100 Solve call ssids_solve x akeep fkeep options inform if inform flag lt 0 go to 100 http www numerical rl ac uk spral 13 17 March 2014 SPRAL SSIDS Version 1 0 0 8 EXAMPLE write a 3es18 10 The computed solution is x 1 n Determine and print the pivot order call ssids_enquire_indef akeep fkeep options inform piv_order piv_order write a 10i5 Pivot order piv_order 1 n 100 continue call ssids_free akeep fkeep cuda_error end program ssids_example This produces the following output Warning from ssids_analyse Warning flag 4 one or more diagonal entries is missing The computed solution is 1 0000000000E 00 2 0000000000E 00 3 0000000000E 00 4 0000000000E 00 5 0000000000E 00 Pivot order 4 5 2 f 3 EPSRC Funded by EPSRC Engineering and Physical Sciences grant EP J010553 1 Research Council http www numerical rl ac uk spral 14 17 March 2014
10. EY science Technology SPRAL SPRAL_SSIDS Sparse Symmetric Indefinite Direct Solver Fortran User Guide This package solves one or more sets of n x n sparse symmetric equations AX B using a multifrontal method on an NVIDIA GPU The following cases are covered 1 A is indefinite SSIDS computes the sparse factorization A PLD PL where P is a permutation matrix L is unit lower triangular and D is block diagonal with blocks of size 1 x 1 and 2 x 2 2 Ais positive definite SSIDS computes the sparse Cholesky factorization A PL PL where P is a permutation matrix and L is lower triangular However as SSIDS is designed primarily for indefinite systems this may be slower than a dedicated Cholesky solver SSIDS returns bit compatible results An option exists to scale the matrix In this case the factorization of the scaled matrix A SAS is computed where S is a diagonal scaling matrix Jonathan Hogg STFC Rutherford Appleton Laboratory Evgueni Ovtchinnikov STFC Rutherford Appleton Laboratory Jennifer Scott STFC Rutherford Appleton Laboratory Major version history 2014 03 17 Version 1 0 0 Initial release 1 Installation Please see the SPRAL install documentation In particular note that e A CUDA compiler nvcc is required e A METIS library is required e A BLAS library is required in addition to CUBLAS http www numerical rl ac uk spral 1 17 March 2014 SPRAL SSIDS Version 1 0 0 2 USAGE OVERVIEW
11. PU implementations of _trsv and _gemv are used to handle multiple independent operations If multiple right hand sides are to be solved for the single right hand side solve is looped over If options presolve 1 _trsv can be replaced by the much more parallel and hence faster _gemv In this case multiple right hand sides are handled at the same time References 1 J D Hogg E Ovtchinnikov and J A Scott 2014 A sparse symmetric indefinite direct solver for GPU architectures RAL Technical Report RAL P 2014 0xx to appear 8 Example Suppose we wish to factorize the matrix 2 1 1 4 1 1 A Te sox 2 2 0 1 2 http www numerical rl ac uk spral 12 17 March 2014 SPRAL SSIDS Version 1 0 0 8 EXAMPLE and then solve for the right hand side 17 B 19 12 The following code may be used examples Fortran ssids f90 Example code for SPRAL_SSIDS package program ssids_example use spral_ssids implicit none Derived types type ssids_akeep akeep type ssids_fkeep fkeep type ssids_options options type ssids_inform inform Parameters integer parameter wp kind 0 0d0 Matrix data logical posdef integer n ptr 6 row 8 real wp val 8 Other variables integer piv_order 5 cuda_error real wp x 5 logical check Data for matrix 2 1 1 4 1 1 1 3 2 2 1 2 posdef false n 5 ptr 1 n 1 1
12. actorize the matrix call ssids_factor posdef val akeep fkeep options inform scale ptr row posdef is an INTENTCIN scalar of type LOGICAL that must be set to true if the matrix is positive definite and false if it is indefinite val is an INTENTCIN array of package type that must hold the numerical values of the entries of the matrix as described in Section 2 6 akeep isan INTENTCIN scalar of type ssids_akeep that must be unchanged since the call to ssids_analyse or ssids_analyse_coord fkeep is an INTENTCINOUT scalar of type ssids_fkeep It is used to hold data about the problem being solved and must be passed unchanged to the other subroutines options is an INTENTCIN scalar of type ssids_options Its components specify the algorithmic options used by the subroutine as explained in Section 5 1 inform is an INTENT QUT scalar of type ssids_inform Its components provide information about the execution of the subroutine as explained in Section 5 2 scale is an optional INTENTCINOUT array of type REAL and size n If present and options scaling 0 it must contain the diagonal entries of the scaling matrix S On exit scale i will contain the i th diagonal entry of the scaling matrix S ptr androw are optional INTENT IN arrays of type INTEGER They are only accessed if ssids_analyse was called with check set to false In this case they must both be present and must be unchanged since that call
13. algorithm details in Section 6 matrix_dup is a scalar of type INTEGER On exit from ssids_analyse with check set to true or from ssids_analyse_coord it holds the number of duplicate entries that were found and summed matrix_missing diag is a scalar of type INTEGER On exit from ssids_analyse with check set to true or from ssids_analyse_coord it holds the number of diagonal entries without an explicitly provided value matrix_outrange is a scalar of type INTEGER On exit from ssids_analyse with check set to true or from ssids_analyse_coord it holds the number of out of range entries that were found and discarded matrix_rank is scalar of type INTEGER On exit from ssids_analyse and ssids_analyse_coord it holds the structural rank of A if available otherwise it is set to n On exit from ssids_factor it holds the computed rank of the factorized matrix maxdepth is a scalar of type INTEGER On exit from ssids_analyse or ssids_analyse_coord it holds the maximum depth of the assembly tree maxfront is a scalar of type INTEGER On exit from ssids_analyse or ssids_analyse_coord it holds the maximum front size in the positive definite case or in the indefinite case with the same pivot sequence On exit from ssids_factor it holds the maximum front size num_delay is scalar of type INTEGER On exit from ssids_factor it holds the number of eliminations that were delayed that is the total number of f
14. all should be made to free memory and CUDA resources allocated by SSIDS and associated with the derived data types akeep and or fkeep using calls to ssids_free akeep is an INTENTC INOUT scalar of type ssids_akeep that must be passed unchanged On exit allocatable components will have been deallocated and CUDA resources released fkeep is an INTENT INOUT scalar of type ssids_fkeep that must be passed unchanged On exit allocatable components will have been deallocated and CUDA resources released cuda_error is an INTENT OUT scalar of type default integer On exit a non zero value gives a CUDA error code This may indicate either a failure to deallocate GPU memory or a pre existing CUDA error condition Note that due to the asynchronous nature of GPU execution the reported error may have a cause errors external to SSIDS 4 Advanced subroutines 4 1 ssids_enquire_posdef To obtain the matrix D following a positive definite factorization call ssids_enquire_posdef akeep fkeep options inform d akeep isan INTENTCIN scalar of type ssids_akeep that must be unchanged since the call to ssids_factor fkeep is an INTENT IN scalar of type ssids_fkeep that must be unchanged since the call to ssids_factor http www numerical rl ac uk spral 6 17 March 2014 SPRAL SSIDS Version 1 0 0 5 DERIVED TYPES options is an INTENTCIN scalar of type ssids_options Its components specify the algorithmic options used by the subroutin
15. e as explained in Section 5 1 inform is an INTENT OQUT scalar of type ssids_inform Its components provide information about the execution of the subroutine as explained in Section 5 2 d is an INTENT OUT array of type REAL and size n The i th diagonal entry of D will be placed in d i 4 2 ssids_enquire_indef To obtain the matrix D and or the pivot order following an indefinite factorization call ssids_enquire_indef akeep fkeep options inform piv_order d akeep isan INTENTCIN scalar of type ssids_akeep that must be unchanged since the call to ssids_factor fkeep is an INTENT IN scalar of type ssids_fkeep that must be unchanged since the call to ssids_factor options is an INTENTCIN scalar of type ssids_options Its components specify the algorithmic options used by the subroutine as explained in Section 5 1 inform is an INTENT OQUT scalar of type ssids_inform Its components provide information about the execution of the subroutine as explained in Section 5 2 piv_order is an optional INTENT OUT array of type INTEGER and size n If present on exit piv_order i gives the position of variable in the pivot order The sign will be positive if 7 is a 1 x 1 pivot and negative if i is part of a 2 x 2 pivot d is an optional INTENT OUT array of package type with extents 2 and n If present on exit diagonal entries of D will be placed in d 1 i i 1 2 n the off diagonal entries of D
16. e elimination order If an ordering was supplied by the user this order may differ but will be equivalent in terms of fill in http www numerical rl ac uk spral 11 17 March 2014 SPRAL SSIDS Version 1 0 0 8 EXAMPLE ssids_factor ssids_factor optionally computes a scaling and then performs the numerical factorization The user must specify whether or not the matrix is positive definite If posdef is set to true no pivoting is performed and the computation will terminate with an error if a non positive pivot is encountered The factorization uses the assembly tree that was set up by the analyse phase At each node entries from A and if it is not a leaf node the generated elements and any delayed pivots from its child nodes must be assembled Separate kernels handle each of these The kernel that performs the assembly from the child nodes considers one parent child assembly at a time Each generated element from a child is divided into a number of tiles and a thread block launched to assemble each tile into a dense submatrix using a simple mapping array to determine the destination row and column of each entry Bit compatibility is achieved by ensuring the child entries are always assembled in the same order A dense partial factorization of the fully summed columns is then performed The fully summed columns are split into a number of tiles that are each handled by an associated block Factorization proceeds one column of tiles at a
17. es Out of range entries are ignored To illustrate the coordinate format the following arrays describe the matrix shown in Figure 1 n 5 ne 9 row 1 9 1 2 3 4 3 5 4 5 5 col 1 9 1 1 2 1 3 3 4 4 5 val 1 9 1 1 2 2 4 4 3 3 5 5 6 6 7 7 8 8 9 9 3 Basic Subroutines 3 1 ssids_analyse and ssids_analyse_coord To analyse the sparsity pattern and prepare for the factorization for Compressed Sparse Column CSC format call ssids_analyse check n ptr row akeep options inform order val for Coordinate format call ssids_analyse_coord n ne row col akeep options inform order val Matrix data should be supplied as described in Section 2 6 As the package uses CSC format internally ssids_analyse with checking disabled provides the most efficient interface check is an INTENTCIN scalar of type LOGICAL If set to true the matrix data is checked for errors and the cleaned matrix duplicates are summed and out of range entries discarded is stored in akeep Otherwise for data in CSC format no checking of the matrix data is carried out and ptr and row must be passed unchanged to the factorization routines Checking is always performed when the coordinate format is used n ptr row are INTENT IN variables of type INTEGER specifying the lower triangular part of A in CSC format see Section 2 6 1 n ne row col are INTENTCIN variables of type INTEGER speci
18. factor An option is available to perform a partial solution ssids_free should be called after all other calls are complete for a problem including after an error return that does not allow the computation to continue It frees memory referenced by components of the derived types This routine also allows created by either ssids_analyse or ssids_factorize to be freed separately In addition the following routines may be called 2 2 ssids_enquire_posdef may be called in the positive definite case to obtain the pivots used ssids_enquire_indef may be called in the indefinite case to obtain the pivot sequence used by the factorization and the entries of D ssids_alter may be called in the indefinite case to alter the entries of D71 Note that this means that PLD PL is no longer a factorization of A Derived types For each problem the user must employ the derived types defined by the package to declare scalars of the types ssids_options ssids_inform ssids_akeep and ssids_fkeep The following pseudo code illustrates this use spral_ssids type ssids_options options type ssids_inform inform type ssids_akeep akeep type ssids_fkeep fkeep The components of ssids_options and ssids_inform are explained in Sections 5 1 and 5 2 The components of ssids_akeep and ssids_fkeep are used to pass data between the subroutines of the package and must not be altered by the user http www numerical rl
19. fying the lower or upper triangular part of A in coordinate format see Section 2 6 2 akeep is an INTENT OUT scalar of type ssids_akeep It is used to hold data about the problem being solved and must be passed unchanged to the other subroutines options is an INTENTCIN scalar of type ssids_options Its components specify the algorithmic options used by the subroutine as explained in Section 5 1 inform is an INTENT OQUT scalar of type ssids_inform Its components provide information about the execution of the subroutine as explained in Section 5 2 order isan optional INTENT CINOUT array of type INTEGER and size n If optionsfordering 0 order must be present and order i must hold the position of variable i in the elimination order On exit order contains the elimination order that ssids_factor will be given it is passed to these routines as part of akeep this order may give slightly more fill in than the user supplied order and in the indefinite case may be modified by ssids_factor to maintain numerical stability val is an optional INTENTCIN array of package type that must hold the numerical values of the entries of the matrix as described in Section 2 6 val must be present if a matching based elimination ordering is required optionsjordering 2 and is otherwise ignored http www numerical rl ac uk spral 4 17 March 2014 SPRAL SSIDS Version 1 0 0 3 BASIC SUBROUTINES 3 2 ssids_factor To f
20. h that ptr i is the position in row and val of the first entry in the i th column and ptr n 1 is one more than the total number of entries Entries that are zero including those on the diagonal need not be specified If this format is used SSIDS requires only the lower triangular entries of A and there should be no duplicate entries If the check argument to ssids_analyse is true out of range entries including those in the upper triangle will be discarded and any duplicates will be summed To illustrate the CSC format the following arrays describe the matrix shown in Figure 1 n 5 ptr 1 6 row 1 9 val 1 9 4 5 10 2 2 4 3 3 wl 2 3 4 5 ll AnA e O Bee 2 6 2 Coordinate Format This standard data format consists of the following data integer i on size of matrix integer ne number of non zero entries integer size ne row row indices integer size ne col column indices real size ne val numerical values http www numerical rl ac uk spral 3 17 March 2014 SPRAL SSIDS Version 1 0 0 3 BASIC SUBROUTINES The arrays should be set such that the k th entry is in row row k and column col k with value val k Entries that are zero including those on the diagonal need not be specified If this format is used SSIDS requires that each entry of A should be present only in the lower or upper triangular part Entries present in both will be summed as will any duplicate entri
21. he matrix values in val and recall ssids_factor 6 Returned by ssids_factor if posdef true and the matrix is found to be not positive definite This may be because the Hungarian scaling determined that the matrix was structurally singular The user may reset the matrix values in val and recall ssids_factor 7 Returned by ssids_factor if ssids_analyse was called with check set to false but ptr and or row is not present 8 Returned by ssids_analyse and ssids_analyse_coord if options ordering is out of range or optionsf ordering 0 and the user has either failed to provide an elimination order or an error has been found in the user supplied elimination order as supplied in order 9 Returned by ssids_analyse and ssids_analyse_coord if options ordering 2 but val was not supplied http www numerical rl ac uk spral 10 17 March 2014 SPRAL SSIDS Version 1 0 0 7 METHOD 10 Returned by ssids_solve if there is an error in the size of array x that is ldx lt n or nrhs lt 1 The user may reset 1dx and or nrhs and recall ssids_solve 11 Returned by ssids_solve if job is out of range The user may reset job and recall ssids_solve 12 Returned by spral_ssids_solve and spral_ssids_alter if the selected combination of options use_gpu_solve and options presolve are not compatible with the requested operation 13 Returned by ssids_enquire_posdef if posdef false o
22. n the last call to ssids_factor 14 Returned by ssids_enquire_indef if posdef true on the last call to ssids_factor 15 Returned by ssids_factor if options scaling 3 but a matching based ordering was not used during the call to ssids_analyse or ssids_analyse_coord i e was called with options ordering 2 50 Allocation error If available the stat parameter is returned in inform stat 51 CUDA error The CUDA error return value is returned in inform cuda_error 52 CUBLAS error The CUBLAS error return value is returned in inform cublas_error A positive value of inform flag is used to warn the user that the input matrix data may be faulty or that the subroutine cannot guarantee the solution obtained Possible values are 1 Returned by ssids_analyse and ssids_analyse_coord if out of range variable indices found Any such entries are ignored and the computation continues inform matrix_outrange is set to the number of such entries 2 Returned by ssids_analyse and ssids_analyse_coord if duplicated indices found Duplicates are recorded and the corresponding entries are summed inform matrix_dup is set to the number of such entries 3 Returned by ssids_analyse and ssids_analyse_coord if both out of range and duplicated variable indices found 4 Returned by ssids_analyse and ssids_analyse_coord if one and more diagonal entries of A is missing 5 Returned by ssids_analyse and ssids_analyse_coord
23. ormation about the execution of the subroutine as explained in Section 5 2 http www numerical rl ac uk spral 5 17 March 2014 SPRAL SSIDS Version 1 0 0 4 ADVANCED SUBROUTINES job is an optional INTENTCIN scalar of type INTEGER If absent AX B is solved In the positive definite case the Cholesky factorization that has been computed may be expressed in the form SAS PL PL where P is a permutation matrix and L is lower triangular In the indefinite case the factorization that has been computed may be expressed in the form SAS PL D PL where P is a permutation matrix L is unit lower triangular and D is block diagonal with blocks of order 1 and 2 S is a diagonal scaling matrix S is equal to the identity if options scaling 0 and scale is not present on the last call to ssids_factor A partial solution may be computed by setting job to have one of the following values 1 for solving PLX SB 2 for solving DX B indefinite case only 3 for solving PL S X B 4 for solving D PL S X B indefinite case only 3 4 ssids_free To free memory and resources allocated by ssids_analyse or ssids_analyse_coord call ssids_free akeep cuda_error allocated by ssids_factor call ssids_free fkeep cuda_error both at once call ssids_free akeep fkeep cuda_error Once all other calls are complete for a problem or after an error return that does not allow the computation to continue a c
24. the unit number for diagnostic printing Printing is suppressed if unit_diagnostics lt 0 The default is unit_diagnostics 6 unit_error is a scalar of type INTEGER that holds the unit number for error messages Printing of error messages is suppressed if unit_error lt 0 The default is unit_error 6 unit_warning is a scalar of type INTEGER that holds the unit number for warning messages Printing of warning messages is suppressed if unit_warning lt 0 The default is unit_warning 6 Options used by ssids_analyse and ssids_analyse_coord ordering is a scalar of type INTEGER If set to 0 the user must supply an elimination order in order otherwise an elimination order will be computed by ssids_analyse or ssids_analyse_coord The options are O User supplied ordering is used 1 METIS ordering with default settings is used 2 A matching based elimination ordering is computed the Hungarian algorithm is used to identify large off diagonal entries A restricted METIS ordering is then used that forces these on to the subdiagonal This option should only be chosen for indefinite systems A scaling is also computed that may be used in ssids_factor see options scaling below The default is ordering 1 Restriction ordering 0 1 2 nemin is a scalar of type INTEGER that controls node amalgamation Two neighbours in the elimination tree are merged if they both involve fewer than nemin eliminations The default is nemin 8 The default is
25. true u is a scalar of type REAL that holds the relative pivot tolerance u The default is u 0 01 Values outside the range 0 0 5 are treated as the closest value in that range Options used by ssids_factor and ssids_solve use_gpu_solve is a scalar of type LOGICAL that controls whether to use the CPU or GPU for the ssids_solve If true the GPU is used but some more advanced features are not available see the description of the job parameter in Section 3 3 Setting use_gpu_solve false is only compatible with options presolve 0 The default value is use_gpu_solve true presolve is a scalar of type INTEGER that controls the amount of extra work performed during factorization to accelerate the solve It can take the following values O Minimal work is performed during ssids_factor to prepare for the solve 1 The explicit inverse of the nelimxnelim block in each supernode is precalculated during ssids_factor where nelim is the number of variables eliminated at that supernode As the matrix L is overwritten the routine ssids_alter cannot be used This option is not compatible with options use_gpu_solve false The default option is presolve 0 5 2 ssids_inform The derived data type ssids_inform is used to hold parameters that give information about the progress and needs of the algorithm The components of ssids_inform in alphabetical order are flag is a scalar of type INTEGER that gives the exit status of the
26. ully summed variables that were passed to the father node because of stability considerations If a variable is passed further up the tree it will be counted again http www numerical rl ac uk spral 9 17 March 2014 SPRAL SSIDS Version 1 0 0 6 RETURN CODES num_factor is scalar of type INTEGER long On exit from ssids_analyse or ssids_analyse_coord it holds the number of entries that will be in the factor L in the positive definite case or in the indefinite case with the same pivot sequence On exit from ssids_factor it holds the actual number of entries in the factor L In the indefinite case 2n entries of D are also held num_flops is scalar of type INTEGER long On exit from ssids_analyse or ssids_analyse_coord it holds the number of floating point operations that will be needed to perform the factorization in the positive definite case or in the indefinite case with the same pivot sequence On exit from ssids_factor it holds the number of floating point operations performed num_neg is a scalar of type INTEGER On exit from ssids_factor it holds the number of negative eigenvalues of the matrix D num_sup is a scalar of type INTEGER On exit from ssids_analyse or ssids_analyse_coord it holds the number of supernodes in the problem num_two is scalar of type INTEGER On exit from ssids_factor it holds the number of 2 x 2 pivots used by the factorization that is the number of 2 x 2 blocks in D stat
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