/* ========================================================================== */
/* === KLU_kernel =========================================================== */
/* ========================================================================== */
/* Sparse left-looking LU factorization, with partial pivoting. Based on
* Gilbert & Peierl's method, with a non-recursive DFS and with Eisenstat &
* Liu's symmetric pruning. No user-callable routines are in this file.
*/
#include "klu_internal.h"
/* ========================================================================== */
/* === dfs ================================================================== */
/* ========================================================================== */
/* Does a depth-first-search, starting at node j. */
static Int dfs
(
/* input, not modified on output: */
Int j, /* node at which to start the DFS */
Int k, /* mark value, for the Flag array */
Int Pinv [ ], /* Pinv [i] = k if row i is kth pivot row, or EMPTY if
* row i is not yet pivotal. */
Int Llen [ ], /* size n, Llen [k] = # nonzeros in column k of L */
Int Lip [ ], /* size n, Lip [k] is position in LU of column k of L */
/* workspace, not defined on input or output */
Int Stack [ ], /* size n */
/* input/output: */
Int Flag [ ], /* Flag [i] == k means i is marked */
Int Lpend [ ], /* for symmetric pruning */
Int top, /* top of stack on input*/
Unit LU [],
Int *Lik, /* Li row index array of the kth column */
Int *plength,
/* other, not defined on input or output */
Int Ap_pos [ ] /* keeps track of position in adj list during DFS */
)
{
Int i, pos, jnew, head, l_length ;
Int *Li ;
l_length = *plength ;
head = 0 ;
Stack [0] = j ;
ASSERT (Flag [j] != k) ;
while (head >= 0)
{
j = Stack [head] ;
jnew = Pinv [j] ;
ASSERT (jnew >= 0 && jnew < k) ; /* j is pivotal */
if (Flag [j] != k) /* a node is not yet visited */
{
/* first time that j has been visited */
Flag [j] = k ;
PRINTF (("[ start dfs at %d : new %d\n", j, jnew)) ;
/* set Ap_pos [head] to one past the last entry in col j to scan */
Ap_pos [head] =
(Lpend [jnew] == EMPTY) ? Llen [jnew] : Lpend [jnew] ;
}
/* add the adjacent nodes to the recursive stack by iterating through
* until finding another non-visited pivotal node */
Li = (Int *) (LU + Lip [jnew]) ;
for (pos = --Ap_pos [head] ; pos >= 0 ; --pos)
{
i = Li [pos] ;
if (Flag [i] != k)
{
/* node i is not yet visited */
if (Pinv [i] >= 0)
{
/* keep track of where we left off in the scan of the
* adjacency list of node j so we can restart j where we
* left off. */
Ap_pos [head] = pos ;
/* node i is pivotal; push it onto the recursive stack
* and immediately break so we can recurse on node i. */
Stack [++head] = i ;
break ;
}
else
{
/* node i is not pivotal (no outgoing edges). */
/* Flag as visited and store directly into L,
* and continue with current node j. */
Flag [i] = k ;
Lik [l_length] = i ;
l_length++ ;
}
}
}
if (pos == -1)
{
/* if all adjacent nodes of j are already visited, pop j from
* recursive stack and push j onto output stack */
head-- ;
Stack[--top] = j ;
PRINTF ((" end dfs at %d ] head : %d\n", j, head)) ;
}
}
*plength = l_length ;
return (top) ;
}
/* ========================================================================== */
/* === lsolve_symbolic ====================================================== */
/* ========================================================================== */
/* Finds the pattern of x, for the solution of Lx=b */
/* ========================================================================== */
/* === construct_column ===================================================== */
/* ========================================================================== */
/* Construct the kth column of A, and the off-diagonal part, if requested.
* Scatter the numerical values into the workspace X, and construct the
* corresponding column of the off-diagonal matrix. */
static Int lsolve_symbolic_and_construct_column
(
/* input, not modified on output: */
Int n, /* L is n-by-n, where n >= 0 */
Int k, /* also used as the mark value, for the Flag array */
Int Ap [ ],
Int Ai [ ],
Int Q [ ],
Int Pinv [ ], /* Pinv [i] = k if i is kth pivot row, or EMPTY if row i
* is not yet pivotal. */
/* workspace, not defined on input or output */
Int Stack [ ], /* size n */
/* workspace, defined on input and output */
Int Flag [ ], /* size n. Initially, all of Flag [0..n-1] < k. After
* lsolve_symbolic is done, Flag [i] == k if i is in
* the pattern of the output, and Flag [0..n-1] <= k. */
/* other */
Int Lpend [ ], /* for symmetric pruning */
Int Ap_pos [ ], /* workspace used in dfs */
Unit LU [ ], /* LU factors (pattern and values) */
Int lup, /* pointer to free space in LU */
Int Llen [ ], /* size n, Llen [k] = # nonzeros in column k of L */
Int Lip [ ], /* size n, Lip [k] is position in LU of column k of L */
/* ---- the following are only used in the BTF case --- */
Int k1, /* the block of A is from k1 to k2-1 */
Int PSinv [ ], /* inverse of P from symbolic factorization */
/* inputs, not modified on output */
Entry Ax [ ],
/* zero on input, modified on output */
Entry X [ ],
/* ---- the following are only used in the BTF case --- */
/* inputs, not modified on output */
double Rs [ ], /* scale factors for A */
Int scale, /* 0: no scaling, nonzero: scale the rows with Rs */
/* inputs, modified on output */
Int Offp [ ], /* off-diagonal matrix (modified by this routine) */
Int Offi [ ],
Entry Offx [ ]
)
{
Entry aik ;
Int *Lik ;
Int i, p, pend, oldcol, kglobal, poff, oldrow, top, l_length ;
top = n ;
l_length = 0 ;
Lik = (Int *) (LU + lup);
/* ---------------------------------------------------------------------- */
/* BTF factorization of A (k1:k2-1, k1:k2-1) */
/* ---------------------------------------------------------------------- */
kglobal = k + k1 ; /* column k of the block is col kglobal of A */
poff = Offp [kglobal] ; /* start of off-diagonal column */
oldcol = Q [kglobal] ; /* Q must be present for BTF case */
pend = Ap [oldcol+1] ;
if (scale <= 0)
{
for (p = Ap [oldcol] ; p < pend ; p++)
{
oldrow = Ai [p] ;
i = PSinv [oldrow] - k1 ;
if (i < 0)
{
/* ---------------------------------------------------------------------- */
/* Scatter the column into X. */
/* ---------------------------------------------------------------------- */
aik = Ax [p] ;
/* this is an entry in the off-diagonal part */
Offi [poff] = oldrow ;
Offx [poff] = aik ;
poff++ ;
}
else
{
/* (i,k) is an entry in the block. start a DFS at node i */
PRINTF (("\n ===== DFS at node %d in b, inew: %d\n", i, Pinv [i])) ;
if (Flag [i] != k)
{
if (Pinv [i] >= 0)
{
top = dfs (i, k, Pinv, Llen, Lip, Stack, Flag,
Lpend, top, LU, Lik, &l_length, Ap_pos) ;
}
else
{
/* i is not pivotal, and not flagged. Flag and put in L */
Flag [i] = k ;
Lik [l_length] = i ;
l_length++;
}
}
/* ---------------------------------------------------------------------- */
/* Scatter the column into X. */
/* ---------------------------------------------------------------------- */
/* no scaling */
aik = Ax [p] ;
/* (i,k) is an entry in the block. scatter into X */
X [i] = aik ;
}
}
}
else
{
for (p = Ap [oldcol] ; p < pend ; p++)
{
oldrow = Ai [p] ;
i = PSinv [oldrow] - k1 ;
if (i < 0)
{
/* ---------------------------------------------------------------------- */
/* Scale and scatter the column into X. */
/* ---------------------------------------------------------------------- */
/* row scaling */
aik = Ax [p] ;
SCALE_DIV (aik, Rs [oldrow]) ;
/* this is an entry in the off-diagonal part */
Offi [poff] = oldrow ;
Offx [poff] = aik ;
poff++ ;
}
else
{
/* (i,k) is an entry in the block. start a DFS at node i */
PRINTF (("\n ===== DFS at node %d in b, inew: %d\n", i, Pinv [i])) ;
if (Flag [i] != k)
{
if (Pinv [i] >= 0)
{
top = dfs (i, k, Pinv, Llen, Lip, Stack, Flag,
Lpend, top, LU, Lik, &l_length, Ap_pos) ;
}
else
{
/* i is not pivotal, and not flagged. Flag and put in L */
Flag [i] = k ;
Lik [l_length] = i ;
l_length++;
}
}
/* ---------------------------------------------------------------------- */
/* Scale and scatter the column into X. */
/* ---------------------------------------------------------------------- */
/* row scaling */
aik = Ax [p] ;
SCALE_DIV (aik, Rs [oldrow]) ;
/* (i,k) is an entry in the block. scatter into X */
X [i] = aik ;
}
}
}
Offp [kglobal+1] = poff ; /* start of the next col of off-diag part */
/* If Llen [k] is zero, the matrix is structurally singular */
Llen [k] = l_length ;
return (top) ;
}
/* ========================================================================== */
/* === lsolve_numeric ======================================================= */
/* ========================================================================== */
/* Computes the numerical values of x, for the solution of Lx=b. Note that x
* may include explicit zeros if numerical cancelation occurs. L is assumed
* to be unit-diagonal, with possibly unsorted columns (but the first entry in
* the column must always be the diagonal entry). */
static void lsolve_numeric
(
/* input, not modified on output: */
Int Pinv [ ], /* Pinv [i] = k if i is kth pivot row, or EMPTY if row i
* is not yet pivotal. */
Unit *LU, /* LU factors (pattern and values) */
Int Stack [ ], /* stack for dfs */
Int Lip [ ], /* size n, Lip [k] is position in LU of column k of L */
Int top, /* top of stack on input */
Int n, /* A is n-by-n */
Int Llen [ ], /* size n, Llen [k] = # nonzeros in column k of L */
/* output, must be zero on input: */
Entry X [ ] /* size n, initially zero. On output,
* X [Ui [up1..up-1]] and X [Li [lp1..lp-1]]
* contains the solution. */
)
{
Entry xj ;
Entry *Lx ;
Int *Li ;
Int p, s, j, jnew, len ;
/* solve Lx=b */
for (s = top ; s < n ; s++)
{
/* forward solve with column j of L */
j = Stack [s] ;
jnew = Pinv [j] ;
ASSERT (jnew >= 0) ;
xj = X [j] ;
GET_POINTER (LU, Lip, Llen, Li, Lx, jnew, len) ;
ASSERT (Lip [jnew] <= Lip [jnew+1]) ;
for (p = 0 ; p < len ; p++)
{
/*X [Li [p]] -= Lx [p] * xj ; */
MULT_SUB (X [Li [p]], Lx [p], xj) ;
}
}
}
/* ========================================================================== */
/* === lpivot =============================================================== */
/* ========================================================================== */
/* Find a pivot via partial pivoting, and scale the column of L. */
static Int lpivot
(
Int diagrow,
Int *p_pivrow,
Entry *p_pivot,
double *p_abs_pivot,
double tol,
Entry X [ ],
Unit *LU, /* LU factors (pattern and values) */
Int Lip [ ],
Int Llen [ ],
Int k,
Int n,
Int Pinv [ ], /* Pinv [i] = k if row i is kth pivot row, or EMPTY if
* row i is not yet pivotal. */
Int *p_firstrow,
KLU_common *Common
)
{
Entry x, pivot, *Lx ;
double abs_pivot, xabs ;
Int p, i, ppivrow, pdiag, pivrow, *Li, last_row_index, firstrow, len ;
pivrow = EMPTY ;
if (Llen [k] == 0)
{
/* matrix is structurally singular */
if (Common->halt_if_singular)
{
return (FALSE) ;
}
for (firstrow = *p_firstrow ; firstrow < n ; firstrow++)
{
PRINTF (("check %d\n", firstrow)) ;
if (Pinv [firstrow] < 0)
{
/* found the lowest-numbered non-pivotal row. Pick it. */
pivrow = firstrow ;
PRINTF (("Got pivotal row: %d\n", pivrow)) ;
break ;
}
}
ASSERT (pivrow >= 0 && pivrow < n) ;
CLEAR (pivot) ;
*p_pivrow = pivrow ;
*p_pivot = pivot ;
*p_abs_pivot = 0 ;
*p_firstrow = firstrow ;
return (FALSE) ;
}
pdiag = EMPTY ;
ppivrow = EMPTY ;
abs_pivot = EMPTY ;
i = Llen [k] - 1 ;
GET_POINTER (LU, Lip, Llen, Li, Lx, k, len) ;
last_row_index = Li [i] ;
/* decrement the length by 1 */
Llen [k] = i ;
GET_POINTER (LU, Lip, Llen, Li, Lx, k, len) ;
/* look in Li [0 ..Llen [k] - 1 ] for a pivot row */
for (p = 0 ; p < len ; p++)
{
/* gather the entry from X and store in L */
i = Li [p] ;
x = X [i] ;
CLEAR (X [i]) ;
Lx [p] = x ;
/* xabs = ABS (x) ; */
ABS (xabs, x) ;
/* find the diagonal */
if (i == diagrow)
{
pdiag = p ;
}
/* find the partial-pivoting choice */
if (xabs > abs_pivot)
{
abs_pivot = xabs ;
ppivrow = p ;
}
}
/* xabs = ABS (X [last_row_index]) ;*/
ABS (xabs, X [last_row_index]) ;
if (xabs > abs_pivot)
{
abs_pivot = xabs ;
ppivrow = EMPTY ;
}
/* compare the diagonal with the largest entry */
if (last_row_index == diagrow)
{
if (xabs >= tol * abs_pivot)
{
abs_pivot = xabs ;
ppivrow = EMPTY ;
}
}
else if (pdiag != EMPTY)
{
/* xabs = ABS (Lx [pdiag]) ;*/
ABS (xabs, Lx [pdiag]) ;
if (xabs >= tol * abs_pivot)
{
/* the diagonal is large enough */
abs_pivot = xabs ;
ppivrow = pdiag ;
}
}
if (ppivrow != EMPTY)
{
pivrow = Li [ppivrow] ;
pivot = Lx [ppivrow] ;
/* overwrite the ppivrow values with last index values */
Li [ppivrow] = last_row_index ;
Lx [ppivrow] = X [last_row_index] ;
}
else
{
pivrow = last_row_index ;
pivot = X [last_row_index] ;
}
CLEAR (X [last_row_index]) ;
*p_pivrow = pivrow ;
*p_pivot = pivot ;
*p_abs_pivot = abs_pivot ;
ASSERT (pivrow >= 0 && pivrow < n) ;
if (IS_ZERO (pivot) && Common->halt_if_singular)
{
/* numerically singular case */
return (FALSE) ;
}
/* divide L by the pivot value */
for (p = 0 ; p < Llen [k] ; p++)
{
/* Lx [p] /= pivot ; */
DIV (Lx [p], Lx [p], pivot) ;
}
return (TRUE) ;
}
/* ========================================================================== */
/* === prune ================================================================ */
/* ========================================================================== */
/* Prune the columns of L to reduce work in subsequent depth-first searches */
static void prune
(
/* input/output: */
Int Lpend [ ], /* Lpend [j] marks symmetric pruning point for L(:,j) */
/* input: */
Int Pinv [ ], /* Pinv [i] = k if row i is kth pivot row, or EMPTY if
* row i is not yet pivotal. */
Int k, /* prune using column k of U */
Int pivrow, /* current pivot row */
/* input/output: */
Unit *LU, /* LU factors (pattern and values) */
/* input */
Int Uip [ ], /* size n, column pointers for U */
Int Lip [ ], /* size n, column pointers for L */
Int Ulen [ ], /* size n, column length of U */
Int Llen [ ] /* size n, column length of L */
)
{
Entry x ;
Entry *Lx, *Ux ;
Int *Li, *Ui ;
Int p, i, j, p2, phead, ptail, llen, ulen ;
/* check to see if any column of L can be pruned */
GET_POINTER (LU, Uip, Ulen, Ui, Ux, k, ulen) ;
for (p = 0 ; p < ulen ; p++)
{
j = Ui [p] ;
ASSERT (j < k) ;
PRINTF (("%d is pruned: %d. Lpend[j] %d Lip[j+1] %d\n",
j, Lpend [j] != EMPTY, Lpend [j], Lip [j+1])) ;
if (Lpend [j] == EMPTY)
{
/* scan column j of L for the pivot row */
GET_POINTER (LU, Lip, Llen, Li, Lx, j, llen) ;
for (p2 = 0 ; p2 < llen ; p2++)
{
if (pivrow == Li [p2])
{
/* found it! This column can be pruned */
#ifndef NDEBUG
PRINTF (("==== PRUNE: col j %d of L\n", j)) ;
{
Int p3 ;
for (p3 = 0 ; p3 < Llen [j] ; p3++)
{
PRINTF (("before: %i pivotal: %d\n", Li [p3],
Pinv [Li [p3]] >= 0)) ;
}
}
#endif
/* partition column j of L. The unit diagonal of L
* is not stored in the column of L. */
phead = 0 ;
ptail = Llen [j] ;
while (phead < ptail)
{
i = Li [phead] ;
if (Pinv [i] >= 0)
{
/* leave at the head */
phead++ ;
}
else
{
/* swap with the tail */
ptail-- ;
Li [phead] = Li [ptail] ;
Li [ptail] = i ;
x = Lx [phead] ;
Lx [phead] = Lx [ptail] ;
Lx [ptail] = x ;
}
}
/* set Lpend to one past the last entry in the
* first part of the column of L. Entries in
* Li [0 ... Lpend [j]-1] are the only part of
* column j of L that needs to be scanned in the DFS.
* Lpend [j] was EMPTY; setting it >= 0 also flags
* column j as pruned. */
Lpend [j] = ptail ;
#ifndef NDEBUG
{
Int p3 ;
for (p3 = 0 ; p3 < Llen [j] ; p3++)
{
if (p3 == Lpend [j]) PRINTF (("----\n")) ;
PRINTF (("after: %i pivotal: %d\n", Li [p3],
Pinv [Li [p3]] >= 0)) ;
}
}
#endif
break ;
}
}
}
}
}
/* ========================================================================== */
/* === KLU_kernel =========================================================== */
/* ========================================================================== */
size_t KLU_kernel /* final size of LU on output */
(
/* input, not modified */
Int n, /* A is n-by-n */
Int Ap [ ], /* size n+1, column pointers for A */
Int Ai [ ], /* size nz = Ap [n], row indices for A */
Entry Ax [ ], /* size nz, values of A */
Int Q [ ], /* size n, optional input permutation */
size_t lusize, /* initial size of LU on input */
/* output, not defined on input */
Int Pinv [ ], /* size n, inverse row permutation, where Pinv [i] = k if
* row i is the kth pivot row */
Int P [ ], /* size n, row permutation, where P [k] = i if row i is the
* kth pivot row. */
Unit **p_LU, /* LU array, size lusize on input */
Entry Udiag [ ], /* size n, diagonal of U */
Int Llen [ ], /* size n, column length of L */
Int Ulen [ ], /* size n, column length of U */
Int Lip [ ], /* size n, column pointers for L */
Int Uip [ ], /* size n, column pointers for U */
Int *lnz, /* size of L*/
Int *unz, /* size of U*/
/* workspace, not defined on input */
Entry X [ ], /* size n, undefined on input, zero on output */
/* workspace, not defined on input or output */
Int Stack [ ], /* size n */
Int Flag [ ], /* size n */
Int Ap_pos [ ], /* size n */
/* other workspace: */
Int Lpend [ ], /* size n workspace, for pruning only */
/* inputs, not modified on output */
Int k1, /* the block of A is from k1 to k2-1 */
Int PSinv [ ], /* inverse of P from symbolic factorization */
double Rs [ ], /* scale factors for A */
/* inputs, modified on output */
Int Offp [ ], /* off-diagonal matrix (modified by this routine) */
Int Offi [ ],
Entry Offx [ ],
/* --------------- */
KLU_common *Common
)
{
Entry pivot ;
double abs_pivot, xsize, nunits, tol, memgrow ;
Entry *Ux ;
Int *Li, *Ui ;
Unit *LU ; /* LU factors (pattern and values) */
Int k, p, i, j, pivrow = 0, kbar, diagrow, firstrow, lup, top, scale, len ;
size_t newlusize ;
#ifndef NDEBUG
Entry *Lx ;
#endif
ASSERT (Common != NULL) ;
scale = Common->scale ;
tol = Common->tol ;
memgrow = Common->memgrow ;
*lnz = 0 ;
*unz = 0 ;
CLEAR (pivot) ;
/* ---------------------------------------------------------------------- */
/* get initial Li, Lx, Ui, and Ux */
/* ---------------------------------------------------------------------- */
PRINTF (("input: lusize %d \n", lusize)) ;
ASSERT (lusize > 0) ;
LU = *p_LU ;
/* ---------------------------------------------------------------------- */
/* initializations */
/* ---------------------------------------------------------------------- */
firstrow = 0 ;
lup = 0 ;
for (k = 0 ; k < n ; k++)
{
/* X [k] = 0 ; */
CLEAR (X [k]) ;
Flag [k] = EMPTY ;
Lpend [k] = EMPTY ; /* flag k as not pruned */
}
/* ---------------------------------------------------------------------- */
/* mark all rows as non-pivotal and determine initial diagonal mapping */
/* ---------------------------------------------------------------------- */
/* PSinv does the symmetric permutation, so don't do it here */
for (k = 0 ; k < n ; k++)
{
P [k] = k ;
Pinv [k] = FLIP (k) ; /* mark all rows as non-pivotal */
}
/* initialize the construction of the off-diagonal matrix */
Offp [0] = 0 ;
/* P [k] = row means that UNFLIP (Pinv [row]) = k, and visa versa.
* If row is pivotal, then Pinv [row] >= 0. A row is initially "flipped"
* (Pinv [k] < EMPTY), and then marked "unflipped" when it becomes
* pivotal. */
#ifndef NDEBUG
for (k = 0 ; k < n ; k++)
{
PRINTF (("Initial P [%d] = %d\n", k, P [k])) ;
}
#endif
/* ---------------------------------------------------------------------- */
/* factorize */
/* ---------------------------------------------------------------------- */
for (k = 0 ; k < n ; k++)
{
PRINTF (("\n\n==================================== k: %d\n", k)) ;
/* ------------------------------------------------------------------ */
/* determine if LU factors have grown too big */
/* ------------------------------------------------------------------ */
/* (n - k) entries for L and k entries for U */
nunits = DUNITS (Int, n - k) + DUNITS (Int, k) +
DUNITS (Entry, n - k) + DUNITS (Entry, k) ;
/* LU can grow by at most 'nunits' entries if the column is dense */
PRINTF (("lup %d lusize %g lup+nunits: %g\n", lup, (double) lusize,
lup+nunits));
xsize = ((double) lup) + nunits ;
if (xsize > (double) lusize)
{
/* check here how much to grow */
xsize = (memgrow * ((double) lusize) + 4*n + 1) ;
if (INT_OVERFLOW (xsize))
{
PRINTF (("Matrix is too large (Int overflow)\n")) ;
Common->status = KLU_TOO_LARGE ;
return (lusize) ;
}
newlusize = memgrow * lusize + 2*n + 1 ;
/* Future work: retry mechanism in case of malloc failure */
LU = KLU_realloc (newlusize, lusize, sizeof (Unit), LU, Common) ;
Common->nrealloc++ ;
*p_LU = LU ;
if (Common->status == KLU_OUT_OF_MEMORY)
{
PRINTF (("Matrix is too large (LU)\n")) ;
return (lusize) ;
}
lusize = newlusize ;
PRINTF (("inc LU to %d done\n", lusize)) ;
}
/* ------------------------------------------------------------------ */
/* start the kth column of L and U */
/* ------------------------------------------------------------------ */
Lip [k] = lup ;
/* ------------------------------------------------------------------ */
/* compute the nonzero pattern of the kth column of L and U */
/* ------------------------------------------------------------------ */
#ifndef NDEBUG
for (i = 0 ; i < n ; i++)
{
ASSERT (Flag [i] < k) ;
/* ASSERT (X [i] == 0) ; */
ASSERT (IS_ZERO (X [i])) ;
}
#endif
/* Francesco - Compressed 'lsolve_symbolic' and 'cosntruct_column' in one routine */
top = lsolve_symbolic_and_construct_column (n, k, Ap, Ai, Q, Pinv, Stack, Flag, Lpend,
Ap_pos, LU, lup, Llen, Lip, k1, PSinv, Ax, X, Rs, scale, Offp, Offi, Offx) ;
/* ------------------------------------------------------------------ */
/* get the column of the matrix to factorize and scatter into X */
/* ------------------------------------------------------------------ */
/*
#ifndef NDEBUG
PRINTF (("--- in U:\n")) ;
for (p = top ; p < n ; p++)
{
PRINTF (("pattern of X for U: %d : %d pivot row: %d\n",
p, Stack [p], Pinv [Stack [p]])) ;
ASSERT (Flag [Stack [p]] == k) ;
}
PRINTF (("--- in L:\n")) ;
Li = (Int *) (LU + Lip [k]);
for (p = 0 ; p < Llen [k] ; p++)
{
PRINTF (("pattern of X in L: %d : %d pivot row: %d\n",
p, Li [p], Pinv [Li [p]])) ;
ASSERT (Flag [Li [p]] == k) ;
}
p = 0 ;
for (i = 0 ; i < n ; i++)
{
ASSERT (Flag [i] <= k) ;
if (Flag [i] == k) p++ ;
}
#endif
*/
/* ------------------------------------------------------------------ */
/* compute the numerical values of the kth column (s = L \ A (:,k)) */
/* ------------------------------------------------------------------ */
lsolve_numeric (Pinv, LU, Stack, Lip, top, n, Llen, X) ;
#ifndef NDEBUG
for (p = top ; p < n ; p++)
{
PRINTF (("X for U %d : ", Stack [p])) ;
PRINT_ENTRY (X [Stack [p]]) ;
}
Li = (Int *) (LU + Lip [k]) ;
for (p = 0 ; p < Llen [k] ; p++)
{
PRINTF (("X for L %d : ", Li [p])) ;
PRINT_ENTRY (X [Li [p]]) ;
}
#endif
/* ------------------------------------------------------------------ */
/* partial pivoting with diagonal preference */
/* ------------------------------------------------------------------ */
/* determine what the "diagonal" is */
diagrow = P [k] ; /* might already be pivotal */
PRINTF (("k %d, diagrow = %d, UNFLIP (diagrow) = %d\n",
k, diagrow, UNFLIP (diagrow))) ;
/* find a pivot and scale the pivot column */
if (!lpivot (diagrow, &pivrow, &pivot, &abs_pivot, tol, X, LU, Lip,
Llen, k, n, Pinv, &firstrow, Common))
{
/* matrix is structurally or numerically singular */
Common->status = KLU_SINGULAR ;
if (Common->numerical_rank == EMPTY)
{
Common->numerical_rank = k+k1 ;
Common->singular_col = Q [k+k1] ;
}
if (Common->halt_if_singular)
{
/* do not continue the factorization */
return (lusize) ;
}
}
/* we now have a valid pivot row, even if the column has NaN's or
* has no entries on or below the diagonal at all. */
PRINTF (("\nk %d : Pivot row %d : ", k, pivrow)) ;
PRINT_ENTRY (pivot) ;
ASSERT (pivrow >= 0 && pivrow < n) ;
ASSERT (Pinv [pivrow] < 0) ;
/* set the Uip pointer */
Uip [k] = Lip [k] + UNITS (Int, Llen [k]) + UNITS (Entry, Llen [k]) ;
/* move the lup pointer to the position where indices of U
* should be stored */
lup += UNITS (Int, Llen [k]) + UNITS (Entry, Llen [k]) ;
Ulen [k] = n - top ;
/* extract Stack [top..n-1] to Ui and the values to Ux and clear X */
GET_POINTER (LU, Uip, Ulen, Ui, Ux, k, len) ;
for (p = top, i = 0 ; p < n ; p++, i++)
{
j = Stack [p] ;
Ui [i] = Pinv [j] ;
Ux [i] = X [j] ;
CLEAR (X [j]) ;
}
/* position the lu index at the starting point for next column */
lup += UNITS (Int, Ulen [k]) + UNITS (Entry, Ulen [k]) ;
/* U(k,k) = pivot */
Udiag [k] = pivot ;
/* ------------------------------------------------------------------ */
/* log the pivot permutation */
/* ------------------------------------------------------------------ */
ASSERT (UNFLIP (Pinv [diagrow]) < n) ;
ASSERT (P [UNFLIP (Pinv [diagrow])] == diagrow) ;
if (pivrow != diagrow)
{
/* an off-diagonal pivot has been chosen */
Common->noffdiag++ ;
PRINTF ((">>>>>>>>>>>>>>>>> pivrow %d k %d off-diagonal\n",
pivrow, k)) ;
if (Pinv [diagrow] < 0)
{
/* the former diagonal row index, diagrow, has not yet been
* chosen as a pivot row. Log this diagrow as the "diagonal"
* entry in the column kbar for which the chosen pivot row,
* pivrow, was originally logged as the "diagonal" */
kbar = FLIP (Pinv [pivrow]) ;
P [kbar] = diagrow ;
Pinv [diagrow] = FLIP (kbar) ;
}
}
P [k] = pivrow ;
Pinv [pivrow] = k ;
#ifndef NDEBUG
for (i = 0 ; i < n ; i++) { ASSERT (IS_ZERO (X [i])) ;}
GET_POINTER (LU, Uip, Ulen, Ui, Ux, k, len) ;
for (p = 0 ; p < len ; p++)
{
PRINTF (("Column %d of U: %d : ", k, Ui [p])) ;
PRINT_ENTRY (Ux [p]) ;
}
GET_POINTER (LU, Lip, Llen, Li, Lx, k, len) ;
for (p = 0 ; p < len ; p++)
{
PRINTF (("Column %d of L: %d : ", k, Li [p])) ;
PRINT_ENTRY (Lx [p]) ;
}
#endif
/* ------------------------------------------------------------------ */
/* symmetric pruning */
/* ------------------------------------------------------------------ */
prune (Lpend, Pinv, k, pivrow, LU, Uip, Lip, Ulen, Llen) ;
*lnz += Llen [k] + 1 ; /* 1 added to lnz for diagonal */
*unz += Ulen [k] + 1 ; /* 1 added to unz for diagonal */
}
/* ---------------------------------------------------------------------- */
/* finalize column pointers for L and U, and put L in the pivotal order */
/* ---------------------------------------------------------------------- */
for (p = 0 ; p < n ; p++)
{
Li = (Int *) (LU + Lip [p]) ;
for (i = 0 ; i < Llen [p] ; i++)
{
Li [i] = Pinv [Li [i]] ;
}
}
#ifndef NDEBUG
for (i = 0 ; i < n ; i++)
{
PRINTF (("P [%d] = %d Pinv [%d] = %d\n", i, P [i], i, Pinv [i])) ;
}
for (i = 0 ; i < n ; i++)
{
ASSERT (Pinv [i] >= 0 && Pinv [i] < n) ;
ASSERT (P [i] >= 0 && P [i] < n) ;
ASSERT (P [Pinv [i]] == i) ;
ASSERT (IS_ZERO (X [i])) ;
}
#endif
/* ---------------------------------------------------------------------- */
/* shrink the LU factors to just the required size */
/* ---------------------------------------------------------------------- */
newlusize = lup ;
ASSERT ((size_t) newlusize <= lusize) ;
/* this cannot fail, since the block is descreasing in size */
LU = KLU_realloc (newlusize, lusize, sizeof (Unit), LU, Common) ;
*p_LU = LU ;
return (newlusize) ;
}