CoinFactorization Class Reference

This deals with Factorization and Updates. More...

#include <CoinFactorization.hpp>

Collaboration diagram for CoinFactorization:

Collaboration graph
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List of all members.

used by factorization

void getAreas (int numberRows, int numberColumns, CoinBigIndex maximumL, CoinBigIndex maximumU)
 Gets space for a factorization, called by constructors.
void preProcess (int state, int possibleDuplicates=-1)
 PreProcesses raw triplet data.
int factor ()
 Does most of factorization.
int replaceColumnPFI (CoinIndexedVector *regionSparse, int pivotRow, double alpha)
 Replaces one Column to basis for PFI returns 0=OK, 1=Probably OK, 2=singular, 3=no room.
int factorSparse ()
 Does sparse phase of factorization return code is <0 error, 0= finished.
int factorSparseSmall ()
 Does sparse phase of factorization (for smaller problems) return code is <0 error, 0= finished.
int factorSparseLarge ()
 Does sparse phase of factorization (for larger problems) return code is <0 error, 0= finished.
int factorDense ()
 Does dense phase of factorization return code is <0 error, 0= finished.
bool pivotOneOtherRow (int pivotRow, int pivotColumn)
 Pivots when just one other row so faster?
bool pivotRowSingleton (int pivotRow, int pivotColumn)
 Does one pivot on Row Singleton in factorization.
bool pivotColumnSingleton (int pivotRow, int pivotColumn)
 Does one pivot on Column Singleton in factorization.
bool getColumnSpace (int iColumn, int extraNeeded)
 Gets space for one Column with given length, may have to do compression (returns True if successful), also moves existing vector, extraNeeded is over and above present.
bool getColumnSpaceIterateR (int iColumn, double value, int iRow)
 getColumnSpaceIterateR.
CoinBigIndex getColumnSpaceIterate (int iColumn, double value, int iRow)
 getColumnSpaceIterate.
bool getRowSpace (int iRow, int extraNeeded)
 Gets space for one Row with given length, may have to do compression (returns True if successful), also moves existing vector.
bool getRowSpaceIterate (int iRow, int extraNeeded)
 Gets space for one Row with given length while iterating, may have to do compression (returns True if successful), also moves existing vector.
void checkConsistency ()
 Checks that row and column copies look OK.
void addLink (int index, int count)
 Adds a link in chain of equal counts.
void deleteLink (int index)
 Deletes a link in chain of equal counts.
void separateLinks (int count, bool rowsFirst)
 Separate out links with same row/column count.
void cleanup ()
 Cleans up at end of factorization.
void updateColumnL (CoinIndexedVector *region, int *indexIn) const
 Updates part of column (FTRANL).
void updateColumnLDensish (CoinIndexedVector *region, int *indexIn) const
 Updates part of column (FTRANL) when densish.
void updateColumnLSparse (CoinIndexedVector *region, int *indexIn) const
 Updates part of column (FTRANL) when sparse.
void updateColumnLSparsish (CoinIndexedVector *region, int *indexIn) const
 Updates part of column (FTRANL) when sparsish.
void updateColumnR (CoinIndexedVector *region) const
 Updates part of column (FTRANR) without FT update.
void updateColumnRFT (CoinIndexedVector *region, int *indexIn)
 Updates part of column (FTRANR) with FT update.
void updateColumnU (CoinIndexedVector *region, int *indexIn) const
 Updates part of column (FTRANU).
void updateColumnUSparse (CoinIndexedVector *regionSparse, int *indexIn) const
 Updates part of column (FTRANU) when sparse.
void updateColumnUSparsish (CoinIndexedVector *regionSparse, int *indexIn) const
 Updates part of column (FTRANU) when sparsish.
int updateColumnUDensish (double *COIN_RESTRICT region, int *COIN_RESTRICT regionIndex) const
 Updates part of column (FTRANU).
void updateTwoColumnsUDensish (int &numberNonZero1, double *COIN_RESTRICT region1, int *COIN_RESTRICT index1, int &numberNonZero2, double *COIN_RESTRICT region2, int *COIN_RESTRICT index2) const
 Updates part of 2 columns (FTRANU) real work.
void updateColumnPFI (CoinIndexedVector *regionSparse) const
 Updates part of column PFI (FTRAN) (after rest).
void permuteBack (CoinIndexedVector *regionSparse, CoinIndexedVector *outVector) const
 Permutes back at end of updateColumn.
void updateColumnTransposePFI (CoinIndexedVector *region) const
 Updates part of column transpose PFI (BTRAN) (before rest).
void updateColumnTransposeU (CoinIndexedVector *region, int smallestIndex) const
 Updates part of column transpose (BTRANU), assumes index is sorted i.e.
void updateColumnTransposeUSparsish (CoinIndexedVector *region, int smallestIndex) const
 Updates part of column transpose (BTRANU) when sparsish, assumes index is sorted i.e.
void updateColumnTransposeUDensish (CoinIndexedVector *region, int smallestIndex) const
 Updates part of column transpose (BTRANU) when densish, assumes index is sorted i.e.
void updateColumnTransposeUSparse (CoinIndexedVector *region) const
 Updates part of column transpose (BTRANU) when sparse, assumes index is sorted i.e.
void updateColumnTransposeR (CoinIndexedVector *region) const
 Updates part of column transpose (BTRANR).
void updateColumnTransposeRDensish (CoinIndexedVector *region) const
 Updates part of column transpose (BTRANR) when dense.
void updateColumnTransposeRSparse (CoinIndexedVector *region) const
 Updates part of column transpose (BTRANR) when sparse.
void updateColumnTransposeL (CoinIndexedVector *region) const
 Updates part of column transpose (BTRANL).
void updateColumnTransposeLDensish (CoinIndexedVector *region) const
 Updates part of column transpose (BTRANL) when densish by column.
void updateColumnTransposeLByRow (CoinIndexedVector *region) const
 Updates part of column transpose (BTRANL) when densish by row.
void updateColumnTransposeLSparsish (CoinIndexedVector *region) const
 Updates part of column transpose (BTRANL) when sparsish by row.
void updateColumnTransposeLSparse (CoinIndexedVector *region) const
 Updates part of column transpose (BTRANL) when sparse (by Row).
int checkPivot (double saveFromU, double oldPivot) const
 Returns accuracy status of replaceColumn returns 0=OK, 1=Probably OK, 2=singular.
template<class T>
bool pivot (int pivotRow, int pivotColumn, CoinBigIndex pivotRowPosition, CoinBigIndex pivotColumnPosition, double work[], unsigned int workArea2[], int increment, int increment2, T markRow[], int largeInteger)
 Gets space for a factorization, called by constructors.

Public Member Functions

Constructors and destructor and copy
 CoinFactorization ()
 Default constructor.
 CoinFactorization (const CoinFactorization &other)
 Copy constructor.
 ~CoinFactorization ()
 Destructor.
void almostDestructor ()
 Delete all stuff (leaves as after CoinFactorization()).
void show_self () const
 Debug show object (shows one representation).
int saveFactorization (const char *file) const
 Debug - save on file - 0 if no error.
int restoreFactorization (const char *file, bool factor=false)
 Debug - restore from file - 0 if no error on file.
void sort () const
 Debug - sort so can compare.
CoinFactorizationoperator= (const CoinFactorization &other)
 = copy
Do factorization
int factorize (const CoinPackedMatrix &matrix, int rowIsBasic[], int columnIsBasic[], double areaFactor=0.0)
 When part of LP - given by basic variables.
int factorize (int numberRows, int numberColumns, CoinBigIndex numberElements, CoinBigIndex maximumL, CoinBigIndex maximumU, const int indicesRow[], const int indicesColumn[], const double elements[], int permutation[], double areaFactor=0.0)
 When given as triplets.
int factorizePart1 (int numberRows, int numberColumns, CoinBigIndex estimateNumberElements, int *indicesRow[], int *indicesColumn[], double *elements[], double areaFactor=0.0)
 Two part version for maximum flexibility This part creates arrays for user to fill.
int factorizePart2 (int permutation[], int exactNumberElements)
 This is part two of factorization Arrays belong to factorization and were returned by part 1 If status okay, permutation has pivot rows - this is only needed If status is singular, then basic variables have pivot row and ones thrown out have -1 returns 0 -okay, -1 singular, -99 memory.
double conditionNumber () const
 Condition number - product of pivots after factorization.
general stuff such as permutation or status
int status () const
 Returns status.
void setStatus (int value)
 Sets status.
int pivots () const
 Returns number of pivots since factorization.
void setPivots (int value)
 Sets number of pivots since factorization.
int * permute () const
 Returns address of permute region.
int * pivotColumn () const
 Returns address of pivotColumn region (also used for permuting).
double * pivotRegion () const
 Returns address of pivot region.
int * permuteBack () const
 Returns address of permuteBack region.
int * pivotColumnBack () const
 Returns address of pivotColumnBack region (also used for permuting).
CoinBigIndexstartRowL () const
 Start of each row in L.
CoinBigIndexstartColumnL () const
 Start of each column in L.
int * indexColumnL () const
 Index of column in row for L.
int * indexRowL () const
 Row indices of L.
double * elementByRowL () const
 Elements in L (row copy).
int numberRowsExtra () const
 Number of Rows after iterating.
void setNumberRows (int value)
 Set number of Rows after factorization.
int numberRows () const
 Number of Rows after factorization.
CoinBigIndex numberL () const
 Number in L.
CoinBigIndex baseL () const
 Base of L.
int maximumRowsExtra () const
 Maximum of Rows after iterating.
int numberColumns () const
 Total number of columns in factorization.
int numberElements () const
 Total number of elements in factorization.
int numberForrestTomlin () const
 Length of FT vector.
int numberGoodColumns () const
 Number of good columns in factorization.
double areaFactor () const
 Whether larger areas needed.
void areaFactor (double value)
 Returns status.
double adjustedAreaFactor () const
 Returns areaFactor but adjusted for dense.
void relaxAccuracyCheck (double value)
 Allows change of pivot accuracy check 1.0 == none >1.0 relaxed.
double getAccuracyCheck () const
 Returns status.
int messageLevel () const
 Level of detail of messages.
void messageLevel (int value)
 Returns status.
int maximumPivots () const
 Maximum number of pivots between factorizations.
void maximumPivots (int value)
 Returns status.
int denseThreshold () const
 Gets dense threshold.
void setDenseThreshold (int value)
 Sets dense threshold.
double pivotTolerance () const
 Pivot tolerance.
void pivotTolerance (double value)
 Returns status.
double zeroTolerance () const
 Zero tolerance.
void zeroTolerance (double value)
 Returns status.
double slackValue () const
 Whether slack value is +1 or -1.
void slackValue (double value)
 Returns status.
double maximumCoefficient () const
 Returns maximum absolute value in factorization.
bool forrestTomlin () const
 true if Forrest Tomlin update, false if PFI
void setForrestTomlin (bool value)
 Returns status.
bool spaceForForrestTomlin () const
 True if FT update and space.
some simple stuff
int numberDense () const
 Returns number of dense rows.
CoinBigIndex numberElementsU () const
 Returns number in U area.
void setNumberElementsU (CoinBigIndex value)
 Setss number in U area.
CoinBigIndex lengthAreaU () const
 Returns length of U area.
CoinBigIndex numberElementsL () const
 Returns number in L area.
CoinBigIndex lengthAreaL () const
 Returns length of L area.
CoinBigIndex numberElementsR () const
 Returns number in R area.
CoinBigIndex numberCompressions () const
 Number of compressions done.
int * numberInRow () const
 Number of entries in each row.
int * numberInColumn () const
 Number of entries in each column.
double * elementU () const
 Elements of U.
int * indexRowU () const
 Row indices of U.
CoinBigIndexstartColumnU () const
 Start of each column in U.
int maximumColumnsExtra ()
 Maximum number of Columns after iterating.
int biasLU () const
 L to U bias 0 - U bias, 1 - some U bias, 2 some L bias, 3 L bias.
void setBiasLU (int value)
 Returns number of dense rows.
int persistenceFlag () const
 Array persistence flag If 0 then as now (delete/new) 1 then only do arrays if bigger needed 2 as 1 but give a bit extra if bigger needed.
void setPersistenceFlag (int value)
 Returns number of dense rows.
rank one updates which do exist
int replaceColumn (CoinIndexedVector *regionSparse, int pivotRow, double pivotCheck, bool checkBeforeModifying=false)
 Replaces one Column to basis, returns 0=OK, 1=Probably OK, 2=singular, 3=no room If checkBeforeModifying is true will do all accuracy checks before modifying factorization.
various uses of factorization (return code number elements)
which user may want to know about

int updateColumnFT (CoinIndexedVector *regionSparse, CoinIndexedVector *regionSparse2)
 Updates one column (FTRAN) from regionSparse2 Tries to do FT update number returned is negative if no room regionSparse starts as zero and is zero at end.
int updateColumn (CoinIndexedVector *regionSparse, CoinIndexedVector *regionSparse2, bool noPermute=false) const
 This version has same effect as above with FTUpdate==false so number returned is always >=0.
int updateTwoColumnsFT (CoinIndexedVector *regionSparse1, CoinIndexedVector *regionSparse2, CoinIndexedVector *regionSparse3, bool noPermuteRegion3=false)
 Updates one column (FTRAN) from region2 Tries to do FT update number returned is negative if no room.
int updateColumnTranspose (CoinIndexedVector *regionSparse, CoinIndexedVector *regionSparse2) const
 Updates one column (BTRAN) from regionSparse2 regionSparse starts as zero and is zero at end Note - if regionSparse2 packed on input - will be packed on output.
void goSparse ()
 makes a row copy of L for speed and to allow very sparse problems
int sparseThreshold () const
 get sparse threshold
void sparseThreshold (int value)
 set sparse threshold
various uses of factorization (return code number elements)
which user may not want to know about (left over from my LP code)

void clearArrays ()
 Get rid of all memory.
various updates - none of which have been written!
int add (CoinBigIndex numberElements, int indicesRow[], int indicesColumn[], double elements[])
 Adds given elements to Basis and updates factorization, can increase size of basis.
int addColumn (CoinBigIndex numberElements, int indicesRow[], double elements[])
 Adds one Column to basis, can increase size of basis.
int addRow (CoinBigIndex numberElements, int indicesColumn[], double elements[])
 Adds one Row to basis, can increase size of basis.
int deleteColumn (int Row)
 Deletes one Column from basis, returns rank.
int deleteRow (int Row)
 Deletes one Row from basis, returns rank.
int replaceRow (int whichRow, int numberElements, const int indicesColumn[], const double elements[])
 Replaces one Row in basis, At present assumes just a singleton on row is in basis returns 0=OK, 1=Probably OK, 2=singular, 3 no space.
void emptyRows (int numberToEmpty, const int which[])
 Takes out all entries for given rows.
used by ClpFactorization
void checkSparse ()
 See if worth going sparse.
bool collectStatistics () const
 For statistics.
void setCollectStatistics (bool onOff) const
 For statistics.
void gutsOfDestructor (int type=1)
 The real work of constructors etc 0 just scalars, 1 bit normal.
void gutsOfInitialize (int type)
 1 bit - tolerances etc, 2 more, 4 dummy arrays
void gutsOfCopy (const CoinFactorization &other)
 See if worth going sparse.
void resetStatistics ()
 Reset all sparsity etc statistics.

Protected Attributes

data
double pivotTolerance_
 Pivot tolerance.
double zeroTolerance_
 Zero tolerance.
double slackValue_
 Whether slack value is +1 or -1.
double areaFactor_
 How much to multiply areas by.
double relaxCheck_
 Relax check on accuracy in replaceColumn.
int numberRows_
 Number of Rows in factorization.
int numberRowsExtra_
 Number of Rows after iterating.
int maximumRowsExtra_
 Maximum number of Rows after iterating.
int numberColumns_
 Number of Columns in factorization.
int numberColumnsExtra_
 Number of Columns after iterating.
int maximumColumnsExtra_
 Maximum number of Columns after iterating.
int numberGoodU_
 Number factorized in U (not row singletons).
int numberGoodL_
 Number factorized in L.
int maximumPivots_
 Maximum number of pivots before factorization.
int numberPivots_
 Number pivots since last factorization.
CoinBigIndex totalElements_
 Number of elements in U (to go) or while iterating total overall.
CoinBigIndex factorElements_
 Number of elements after factorization.
CoinIntArrayWithLength pivotColumn_
 Pivot order for each Column.
CoinIntArrayWithLength permute_
 Permutation vector for pivot row order.
CoinIntArrayWithLength permuteBack_
 DePermutation vector for pivot row order.
CoinIntArrayWithLength pivotColumnBack_
 Inverse Pivot order for each Column.
int status_
 Status of factorization.
int numberTrials_
 Number of trials before rejection.
CoinBigIndexArrayWithLength startRowU_
 Start of each Row as pointer.
CoinIntArrayWithLength numberInRow_
 Number in each Row.
CoinIntArrayWithLength numberInColumn_
 Number in each Column.
CoinIntArrayWithLength numberInColumnPlus_
 Number in each Column including pivoted.
CoinIntArrayWithLength firstCount_
 First Row/Column with count of k, can tell which by offset - Rows then Columns.
CoinIntArrayWithLength nextCount_
 Next Row/Column with count.
CoinIntArrayWithLength lastCount_
 Previous Row/Column with count.
CoinIntArrayWithLength nextColumn_
 Next Column in memory order.
CoinIntArrayWithLength lastColumn_
 Previous Column in memory order.
CoinIntArrayWithLength nextRow_
 Next Row in memory order.
CoinIntArrayWithLength lastRow_
 Previous Row in memory order.
CoinIntArrayWithLength saveColumn_
 Columns left to do in a single pivot.
CoinIntArrayWithLength markRow_
 Marks rows to be updated.
int messageLevel_
 Detail in messages.
int biggerDimension_
 Larger of row and column size.
CoinIntArrayWithLength indexColumnU_
 Base address for U (may change).
CoinIntArrayWithLength pivotRowL_
 Pivots for L.
CoinDoubleArrayWithLength pivotRegion_
 Inverses of pivot values.
int numberSlacks_
 Number of slacks at beginning of U.
int numberU_
 Number in U.
CoinBigIndex maximumU_
 Maximum space used in U.
CoinBigIndex lengthU_
 Length of U.
CoinBigIndex lengthAreaU_
 Length of area reserved for U.
CoinDoubleArrayWithLength elementU_
 Elements of U.
CoinIntArrayWithLength indexRowU_
 Row indices of U.
CoinBigIndexArrayWithLength startColumnU_
 Start of each column in U.
CoinBigIndexArrayWithLength convertRowToColumnU_
 Converts rows to columns in U.
CoinBigIndex numberL_
 Number in L.
CoinBigIndex baseL_
 Base of L.
CoinBigIndex lengthL_
 Length of L.
CoinBigIndex lengthAreaL_
 Length of area reserved for L.
CoinDoubleArrayWithLength elementL_
 Elements of L.
CoinIntArrayWithLength indexRowL_
 Row indices of L.
CoinBigIndexArrayWithLength startColumnL_
 Start of each column in L.
bool doForrestTomlin_
 true if Forrest Tomlin update, false if PFI
int numberR_
 Number in R.
CoinBigIndex lengthR_
 Length of R stuff.
CoinBigIndex lengthAreaR_
 length of area reserved for R
double * elementR_
 Elements of R.
int * indexRowR_
 Row indices for R.
CoinBigIndexArrayWithLength startColumnR_
 Start of columns for R.
double * denseArea_
 Dense area.
int * densePermute_
 Dense permutation.
int numberDense_
 Number of dense rows.
int denseThreshold_
 Dense threshold.
CoinDoubleArrayWithLength workArea_
 First work area.
CoinUnsignedIntArrayWithLength workArea2_
 Second work area.
CoinBigIndex numberCompressions_
 Number of compressions done.
double ftranCountInput_
 Below are all to collect.
double ftranCountAfterL_
 Pivot tolerance.
double ftranCountAfterR_
 Pivot tolerance.
double ftranCountAfterU_
 Pivot tolerance.
double btranCountInput_
 Pivot tolerance.
double btranCountAfterU_
 Pivot tolerance.
double btranCountAfterR_
 Pivot tolerance.
double btranCountAfterL_
 Pivot tolerance.
int numberFtranCounts_
 We can roll over factorizations.
int numberBtranCounts_
 Pivot tolerance.
double ftranAverageAfterL_
 While these are average ratios collected over last period.
double ftranAverageAfterR_
 Pivot tolerance.
double ftranAverageAfterU_
 Pivot tolerance.
double btranAverageAfterU_
 Pivot tolerance.
double btranAverageAfterR_
 Pivot tolerance.
double btranAverageAfterL_
 Pivot tolerance.
bool collectStatistics_
 For statistics.
int sparseThreshold_
 Below this use sparse technology - if 0 then no L row copy.
int sparseThreshold2_
 And one for "sparsish".
CoinBigIndexArrayWithLength startRowL_
 Start of each row in L.
CoinIntArrayWithLength indexColumnL_
 Index of column in row for L.
CoinDoubleArrayWithLength elementByRowL_
 Elements in L (row copy).
CoinIntArrayWithLength sparse_
 Sparse regions.
int biasLU_
 L to U bias 0 - U bias, 1 - some U bias, 2 some L bias, 3 L bias.
int persistenceFlag_
 Array persistence flag If 0 then as now (delete/new) 1 then only do arrays if bigger needed 2 as 1 but give a bit extra if bigger needed.

Friends

void CoinFactorizationUnitTest (const std::string &mpsDir)

Detailed Description

This deals with Factorization and Updates.

This class started with a parallel simplex code I was writing in the mid 90's. The need for parallelism led to many complications and I have simplified as much as I could to get back to this.

I was aiming at problems where I might get speed-up so I was looking at dense problems or ones with structure. This led to permuting input and output vectors and to increasing the number of rows each rank-one update. This is still in as a minor overhead.

I have also put in handling for hyper-sparsity. I have taken out all outer loop unrolling, dense matrix handling and most of the book-keeping for slacks. Also I always use FTRAN approach to updating even if factorization fairly dense. All these could improve performance.

I blame some of the coding peculiarities on the history of the code but mostly it is just because I can't do elegant code (or useful comments).

I am assuming that 32 bits is enough for number of rows or columns, but CoinBigIndex may be redefined to get 64 bits.

Definition at line 46 of file CoinFactorization.hpp.


Constructor & Destructor Documentation

CoinFactorization::CoinFactorization (  ) 

Default constructor.

CoinFactorization::CoinFactorization ( const CoinFactorization other  ) 

Copy constructor.

CoinFactorization::~CoinFactorization (  ) 

Destructor.


Member Function Documentation

void CoinFactorization::almostDestructor (  ) 

Delete all stuff (leaves as after CoinFactorization()).

void CoinFactorization::show_self (  )  const

Debug show object (shows one representation).

int CoinFactorization::saveFactorization ( const char *  file  )  const

Debug - save on file - 0 if no error.

int CoinFactorization::restoreFactorization ( const char *  file,
bool  factor = false 
)

Debug - restore from file - 0 if no error on file.

If factor true then factorizes as if called from ClpFactorization

void CoinFactorization::sort (  )  const

Debug - sort so can compare.

CoinFactorization& CoinFactorization::operator= ( const CoinFactorization other  ) 

= copy

int CoinFactorization::factorize ( const CoinPackedMatrix matrix,
int  rowIsBasic[],
int  columnIsBasic[],
double  areaFactor = 0.0 
)

When part of LP - given by basic variables.

Actually does factorization. Arrays passed in have non negative value to say basic. If status is okay, basic variables have pivot row - this is only needed If status is singular, then basic variables have pivot row and ones thrown out have -1 returns 0 -okay, -1 singular, -2 too many in basis, -99 memory

int CoinFactorization::factorize ( int  numberRows,
int  numberColumns,
CoinBigIndex  numberElements,
CoinBigIndex  maximumL,
CoinBigIndex  maximumU,
const int  indicesRow[],
const int  indicesColumn[],
const double  elements[],
int  permutation[],
double  areaFactor = 0.0 
)

When given as triplets.

Actually does factorization. maximumL is guessed maximum size of L part of final factorization, maximumU of U part. These are multiplied by areaFactor which can be computed by user or internally. Arrays are copied in. I could add flag to delete arrays to save a bit of memory. If status okay, permutation has pivot rows - this is only needed If status is singular, then basic variables have pivot row and ones thrown out have -1 returns 0 -okay, -1 singular, -99 memory

int CoinFactorization::factorizePart1 ( int  numberRows,
int  numberColumns,
CoinBigIndex  estimateNumberElements,
int *  indicesRow[],
int *  indicesColumn[],
double *  elements[],
double  areaFactor = 0.0 
)

Two part version for maximum flexibility This part creates arrays for user to fill.

estimateNumberElements is safe estimate of number returns 0 -okay, -99 memory

int CoinFactorization::factorizePart2 ( int  permutation[],
int  exactNumberElements 
)

This is part two of factorization Arrays belong to factorization and were returned by part 1 If status okay, permutation has pivot rows - this is only needed If status is singular, then basic variables have pivot row and ones thrown out have -1 returns 0 -okay, -1 singular, -99 memory.