/usr/share/axiom-20170501/src/algebra/SEM.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 | )abbrev domain SEM SparseEchelonMatrix
++ Description:
++ \spad{SparseEchelonMatrix(C, D)} implements sparse matrices whose columns
++ are enumerated by the \spadtype{OrderedSet} \spad{C} and whose entries
++ belong to the \spadtype{GcdDomain} \spad{D}. The basic operation of
++ this domain is the computation of an row echelon form. The used algorithm
++ tries to maintain the sparsity and is especially adapted to matrices who
++ are already close to a row echelon form.
SparseEchelonMatrix(C,D) : SIG == CODE where
C : OrderedSet
D : Ring
Sy ==> Symbol
L ==> List
V ==> Vector
VD ==> Vector D
MD ==> Matrix D
FD ==> Fraction D
MFD ==> Matrix FD
I ==> Integer
NNI ==> NonNegativeInteger
B ==> Boolean
OUT ==> OutputForm
ROWREC ==> Record(Indices : L C, Entries : L D)
iter ==> "iterated"::Sy
rand ==> "random"::Sy
SIG ==> CoercibleTo OUT with
shallowlyMutable
++ Matrices may be altered destructively.
finiteAggregate
++ Matrices are finite.
coerce : % -> MD
++ \spad{coerce(A)} yields the matrix \spad{A} in the usual matrix type.
copy : % -> %
++ \spad{copy(A)} returns a copy of the matrix \spad{A}.
ncols : % -> NNI
++ \spad{ncols(A)} returns the number of columns of the matrix \spad{A}.
nrows : % -> NNI
++ \spad{nrows(A)} returns the number of rows of the matrix \spad{A}.
allIndices : % -> L C
++ \spad{allIndices(A)} returns all indices used for enumerating the
++ columns of the matrix \spad{A}.
elimZeroCols! : % -> Void
++ \spad{elimZeroCols!(A)} removes columns which contain only zeros.
++ This effects basically only the value of \spad{allIndices(A)}.
purge! : (%, C-> B) -> Void
++ \spad{purge!(A, crit)} eliminates all columns belonging to an index
++ \spad{c} such that \spad{crit(c)} yields \spad{true}.
sortedPurge! : (%, C-> B) -> Void
++ \spad{sortedPurge!(A, crit)} is like \spad{purge}, however, with the
++ additional assumption that \spad{crit} respects the ordering of the
++ indices.
new : (L C, I) -> %
++ \spad{new(inds, nrows)} generates a new matrix with \spad{nrows}
++ rows and columns enumerated by the indices \spad{inds}. The matrix
++ is empty, the zero matrix.
elt : (%, I, C) -> D
++ \spad{elt(A, i, c)} returns the entry of the matrix \spad{A} in row
++ \spad{i} and in the column with index \spad{c}.
setelt! : (%, I, C, D) -> Void
++ \spad{setelt!(A, i, c, d)} sets the entry of the matrix \spad{A} in
++ row \spad{i} and in the column with index \spad{c} to the value
++ \spad{d}.
row : (%, I) -> ROWREC
++ \spad{row(A, i)} returns the \spad{i}-th row of the matrix \spad{A}.
setRow! : (%, I, ROWREC) -> Void
++ \spad{setRow!(A, i, ind, ent)} sets the \spad{i}-th row of the matrix
++ \spad{A} to the value \spad{r}.
setRow! : (%, I, L C, L D) -> Void
++ \spad{setRow!(A, i, ind, ent)} sets the \spad{i}-th row of the matrix
++ \spad{A}. Its indices are \spad{ind}; the entries \spad{ent}.
deleteRow! : (%, I) -> Void
++ \spad{deleteRow(A, i)} deletes the \spad{i}-th row of the matrix
++ \spad{A}.
consRow! : (%, ROWREC) -> Void
++ \spad{consRow!(A, r)} inserts the row \spad{r} at the top of the
++ matrix \spad{A}.
appendRow! : (%, ROWREC) -> Void
++ \spad{appendRow!(A, r)} appends the row \spad{r} at the end of the
++ matrix \spad{A}.
extract : (%, I, I) -> %
++ \spad{extract(A, i1, i2)} extracts the rows \spad{i1} to \spad{i2}
++ and returns them as a new matrix.
rowEchelon : % -> Record(Ech : %, Lt : MD, Pivots : L D, Rank : NNI)
++ \spad{primitiveRowEchelon(A)} computes a row echelon form for the
++ matrix \spad{A}. The algorithm used is fraction-free elimination.
++ It is especially adapted to matrices already close to row echelon
++ form. The transformation matrix, the used pivots and the rank of the
++ matrix are also returned.
if D has GcdDomain then
setGcdMode : Sy -> Sy
++ \spad{setGcdMode(s)} sets a new value for the flag deciding on
++ the method used to compute gcd`s for lists. Possible values for
++ \spad{s} are \spad{iterated} and \spad{random}.
primitiveRowEchelon: % -> Record(Ech:%, Lt:MFD, Pivots:L D, Rank:NNI)
++ \spad{primitiveRowEchelon(A)} computes a row echelon form for the
++ matrix \spad{A}. The algorithm used is fraction-free elimination.
++ Every row is made primitive by division by the gcd. The algorithm
++ is especially adapted to matrices already close to row echelon
++ form. The transformation matrix, the used pivots and the rank of the
++ matrix are also returned.
pivot : (%, I) -> Record(Index : C, Entry : D)
++ \spad{pivot(A, i)} returns the leading entry of the \spad{i}-th row
++ of the matrix \spad{A} together with its index.
pivots : % -> ROWREC
++ \spad{pivots(A)} returns all leading entries of the matrix \spad{A}
++ together with their indices.
"*" : (MD, %) -> %
++ \spad{L*A} implements left multiplication with a usual matrix.
if D has IntegralDomain then
"*" : (MFD, %) -> %
++ \spad{L*A} implements left multiplication with a usual matrix over
++ the quotient field of \spad{D}.
join : (%, %) -> %
++ \spad{join(A, B)} vertically concats the matrices \spad{A} and
++ \spad{B}.
horizJoin : (%, %) -> %
++ \spad{horizJoin(A, B)} horizontally concats the matrices \spad{A} and
++ \spad{B}. It is assumed that all indices of \spad{B} are smaller than
++ those of \spad{A}.
horizSplit : (%, C) -> Record(Left : %, Right : %)
++ \spad{horizSplit(A, c)} splits the matrix \spad{A} into two at the
++ column given by \spad{c}. The first column of the right matrix is
++ enumerated by the first index less or equal to \spad{c}.
CODE ==> add
minInd : I := minIndex([i for i in 1..1])
offset : I := minInd-1
emptyRec : ROWREC := [empty, empty]
noChecks? : B := D has lazyRepresentation -- flag for lazy representation
seed : I := 113 -- seed for random number generation
GCDmode : Sy := iter -- flag for gcd algorithm
greater(r1 : ROWREC, r2 : ROWREC) : B ==
empty? r1.Indices => false
empty? r2.Indices => true
first(r2.Indices) < first(r1.Indices)
-- -------------- --
-- Representation --
-- -------------- --
-- For efficiency reasons most checks for correct index ranges are omitted.
Rep := Record(NCols : NNI, NRows : NNI, AllInds : L C, Rows : V ROWREC)
ncols(A : %) : NNI == A.NCols
nrows(A : %) : NNI == A.NRows
allIndices(A : %) : L C == copy A.AllInds
row(A : %, i : I) : ROWREC ==
-- i < 0 or i > A.NRows => error "index out of range"
qelt(A.Rows, i)
setRow!(A : %, i : I, r : ROWREC) : Void ==
-- i < 0 or i > A.NRows => error "index out of range"
qsetelt!(A.Rows, i, r)
void
setRow!(A : %, i : I, inds : L C, ents : L D) : Void ==
-- i < 0 or i > A.NRows => error "index out of range"
-- #inds ^= #ents => error "improper row"
qsetelt!(A.Rows, i, [inds, ents])
void
new(inds : L C, n : I) : % ==
[#inds, n::NNI, inds, [copy emptyRec for i in 1..n]]
elt(A : %, i : I, c : C) : D ==
r := row(A, i)
pos := position(c, r.Indices)
pos < minInd => 0$D
qelt(r.Entries, pos)
setelt!(A : %, i : I, c : C, d : D) : Void ==
r := row(A, i)
pos := position(c, r.Indices)
if pos >= minInd then
qsetelt!(r.Entries, pos, d)
else
j := minInd
for ind in r.Indices while c < ind repeat
j := j+1
r.Indices := insert!(c, r.Indices, j)
r.Entries := insert!(d, r.Entries, j)
qsetelt!(A.Rows, i, r)
void
coerce(A : %) : MD ==
zero? A.NCols => error "cannot coerce matrix with zero columns"
AA : MD := new(A.NRows, A.NCols, 0$D)
for r in entries(A.Rows) for i in minRowIndex(AA).. repeat
inds := r.Indices
ents := r.Entries
for ind in A.AllInds for j in minColIndex(AA).. _
while not empty? inds repeat
if ind = first inds then
qsetelt!(AA, i, j, first ents)
inds := rest inds
ents := rest ents
AA
coerce(A : %) : OUT ==
zero? A.NCols => 0$D ::OUT
A::MD::OUT
copy(A : %) : % ==
resRows : V ROWREC := new(A.NRows, emptyRec)
for l in 1..A.NRows repeat
r := qelt(A.Rows, l)
qsetelt!(resRows, l, [copy r.Indices, copy r.Entries])
[A.NCols, A.NRows, copy A.AllInds, resRows]
-- ----------------------- --
-- Basic Matrix Operations --
-- ----------------------- --
elimZeroCols!(A : %) : Void ==
newInds : L C := empty
for r in entries(A.Rows) repeat
newInds := removeDuplicates! merge!((x, y) +-> y < x,
newInds, r.Indices)
A.AllInds := newInds
void
purge!(A : %, crit : C-> B) : Void ==
newInds : L C := empty
for c in A.AllInds repeat
if not crit c then
newInds := cons(c, newInds)
newInds := reverse! newInds
if #newInds ^= #A.AllInds then
A.AllInds := newInds
for l in 1..A.NRows repeat
r := qelt(A.Rows, l)
newInds : L C := empty
newEnts : L D := empty
for c in r.Indices for e in r.Entries repeat
if not crit c then
newInds := cons(c, newInds)
newEnts := cons(e, newEnts)
qsetelt!(A.Rows, l, [reverse! newInds, reverse! newEnts])
void
sortedPurge!(A : %, crit : C-> B) : Void ==
if crit first A.AllInds then
while not(empty? A.AllInds) and crit first A.AllInds repeat
A.AllInds := rest A.AllInds
for l in 1..A.NRows repeat
r := qelt(A.Rows, l)
while not(empty? r.Indices) and crit first r.Indices repeat
r.Indices := rest r.Indices
r.Entries := rest r.Entries
qsetelt!(A.Rows, l, r)
void
deleteRow!(A : %, i : I) : Void ==
i > A.NRows => A
nr := (A.NRows-1)::NNI
resRows : V ROWREC := new(nr, emptyRec)
for l in 1..(i-1) repeat
qsetelt!(resRows, l, qelt(A.Rows, l))
for l in (i+1)..A.NRows repeat
qsetelt!(resRows, l-1, qelt(A.Rows, l))
A.NRows := nr
A.Rows := resRows
void
consRow!(A : %, r : ROWREC) : Void ==
A.NRows := A.NRows + 1
newRows : L ROWREC := cons(r, entries A.Rows)
A.Rows := construct newRows
newInds := setDifference(r.Indices, A.AllInds)
if not empty? newInds then
A.AllInds := merge((x, y) +-> y < x, A.AllInds,
sort!((x, y) +-> y < x, newInds))
void
appendRow!(A : %, r : ROWREC) : Void ==
A.NRows := A.NRows + 1
newRows : L ROWREC := concat(entries A.Rows, r)
A.Rows := construct newRows
newInds := setDifference(r.Indices, A.AllInds)
if not empty? newInds then
A.AllInds := merge((x, y) +-> y < x, A.AllInds,
sort!((x, y) +-> y < x, newInds))
void
extract(A : %, i1 : I, i2 : I) : % ==
nr := (i2-i1+1)::NNI
resRows : V ROWREC := new(nr, emptyRec)
newInds : L C := empty
for i in i1..i2 repeat
qsetelt!(resRows, i-i1+1, row(A, i))
newInds := removeDuplicates! merge((x, y) +-> y < x,
newInds, row(A, i).Indices)
[A.NCols, nr, newInds, resRows]
join(A1 : %, A2 : %) : % ==
newInds := removeDuplicates! merge((x : C, y : C) : Boolean +-> y < x,
A1.AllInds, A2.AllInds)
newNRows := A1.NRows + A2.NRows
newRows : V ROWREC := new(newNRows, emptyRec)
for l in 1..A1.NRows repeat
qsetelt!(newRows, l, qelt(A1.Rows, l))
for l in 1..A2.NRows repeat
qsetelt!(newRows, A1.NRows+l, qelt(A2.Rows, l))
[#newInds, newNRows, newInds, newRows]
horizJoin(A1 : %, A2 : %) : % ==
A1.NRows ^= A2.NRows => error "incompatible dimensions in horizJoin"
newInds := append(A1.AllInds, A2.AllInds)
res : % := new(newInds, A1.NRows)
for i in 1..A1.NRows repeat
r1 := row(A1, i)
r2 := row(A2, i)
setRow!(res, i, append(r1.Indices, r2.Indices), _
append(r1.Entries, r2.Entries))
res
horizSplit(A : %, c : C) : Record(Left : %, Right : %) ==
rinds : L C := allIndices A
linds : L C := empty
while not(empty? rinds) and (first(rinds) > c) repeat
linds := cons(first(rinds), linds)
rinds := rest rinds
empty? linds => [new(linds, A.NRows), A]
linds := reverse! linds
empty? rinds => [A, new(rinds, A.NRows)]
LA : % := new(linds, A.NRows)
RA : % := new(rinds, A.NRows)
for i in 1..A.NRows repeat
r := row(A, i)
ri : L C := r.Indices
re : L D := r.Entries
li : L C := empty
le : L D := empty
while not(empty? ri) and (first(ri) > c) repeat
li := cons(first(ri), li)
le := cons(first re, le)
ri := rest ri
re := rest re
if not empty? li then
li := reverse! li
le := reverse! le
setRow!(LA, i, li, le)
if not empty? ri then
setRow!(RA, i, ri, re)
[LA, RA]
-- ----------- --
-- Row Echelon --
-- ----------- --
addRows(d1 : D, r1 : ROWREC, d2 : D, r2 : ROWREC) : ROWREC ==
-- Computes linear combination of two rows.
-- Local function.
empty? r1.Indices =>
one? d2 => r2
[r2.Indices, [d2*e2 for e2 in r2.Entries]]
empty? r2.Indices =>
one? d1 => r1
[r1.Indices, [d1*e1 for e1 in r1.Entries]]
resI : L C := empty
resE : L D := empty
lent1 : L D
lent2 : L D
if not(noChecks?) and one? d1 then
lent1 := r1.Entries
else
lent1 := [d1*e1 for e1 in r1.Entries]
if not(noChecks?) and one? d2 then
lent2 := copy r2.Entries
else
lent2 := [d2*e2 for e2 in r2.Entries]
lind2 := copy r2.Indices
for c1 in r1.Indices for e1 in lent1 repeat
while not(empty? lind2) and c1 < first(lind2) repeat
resI := cons(first lind2, resI)
resE := cons(first(lent2), resE)
lind2 := rest lind2
lent2 := rest lent2
if not(empty? lind2) and first(lind2) = c1 then
sum := e1+first(lent2)
if noChecks? or not zero? sum then
resI := cons(c1, resI)
resE := cons(sum, resE)
lind2 := rest lind2
lent2 := rest lent2
else
resI := cons(c1, resI)
resE := cons(e1, resE)
resI := concat!(reverse! resI, lind2)
resE := concat!(reverse! resE, lent2)
while not(empty? resE) and zero? first resE repeat
resI := rest resI
resE := rest resE
[resI, resE]
pivot(A : %, i : I) : Record(Index : C, Entry : D) ==
r := row(A, i)
empty? r.Indices => error "empty row"
[first r.Indices, first r.Entries]
pivots(A : %) : ROWREC ==
resI : L C := empty
resE : L D := empty
for r in entries A.Rows | not empty? r.Indices repeat
resI := cons(first r.Indices, resI)
resE := cons(first r.Entries, resE)
[reverse! resI, reverse! resE]
rowEchelon(AA : %) : Record(Ech : %, Lt : MD, Pivots : L D, Rank : NNI) ==
A := copy AA
LTr : MD := diagonalMatrix [1$D for i in 1..A.NRows]
Pivs : L D := empty
-- check pivots
for i in 1..A.NRows repeat
r := qelt(A.Rows, i)
changed? : B := false
while not(empty? r.Entries) and zero? first r.Entries repeat
r.Entries := rest r.Entries
r.Indices := rest r.Indices
changed? := true
if changed? then
qsetelt!(A.Rows, i, r)
-- sort rows by pivots (bubble sort)
sorted? : B := false
until sorted? repeat
sorted? := true
oldr := qelt(A.Rows, 1)
for i in 2..A.NRows repeat
newr := qelt(A.Rows, i)
if greater(newr, oldr) then
qsetelt!(A.Rows, i, oldr)
qsetelt!(A.Rows, i-1, newr)
swapRows!(LTr, i-1, i)
sorted? := false
else
oldr := newr
-- fraction-free elimination
finished? : B := false
pivlen, pivrow, rk : NNI
for i in 1..A.NRows until finished? repeat
r := qelt(A.Rows, i)
finished? := empty? r.Indices
if finished? then
rk : NNI := (i-1)::NNI
else -- search good pivot
pivind := first r.Indices
pivlen := #r.Indices
pivrow := i
k : I := 0
for j in (i+1)..A.NRows _
while not(empty? qelt(A.Rows, j).Indices) and _
pivind = first(qelt(A.Rows, j).Indices) repeat
len := #qelt(A.Rows, j).Indices
k := k+1
if len < pivlen then
pivlen := len
pivrow := j
piv : D := first qelt(A.Rows, pivrow).Entries
Pivs := cons(piv, Pivs)
-- elimination necessary?
if k > 0 then
if pivrow ^= i then
pr := qelt(A.Rows, pivrow)
qsetelt!(A.Rows, pivrow, qelt(A.Rows, i))
qsetelt!(A.Rows, i, pr)
swapRows!(LTr, i, pivrow)
-- elimination (and resorting of rows)
pr := copy qelt(A.Rows, i)
pr.Indices := rest pr.Indices
pr.Entries := rest pr.Entries
for j in (i+1)..(i+k) repeat
r := copy qelt(A.Rows, i+1)
c := first r.Entries
r.Indices := rest r.Indices
r.Entries := rest r.Entries
r := addRows(piv, r, -c, pr)
for l in 1..A.NRows repeat
f := piv*qelt(LTr, i+1, l) - c*qelt(LTr, i, l)
qsetelt!(LTr, i+1, l, f)
for l in (i+2)..(2*i+k+1-j) repeat
qsetelt!(A.Rows, l-1, qelt(A.Rows, l))
swapRows!(LTr, l-1, l)
for l in (2*i+k+2-j)..A.NRows _
while greater(qelt(A.Rows, l), r) repeat
qsetelt!(A.Rows, l-1, qelt(A.Rows, l))
swapRows!(LTr, l-1, l)
qsetelt!(A.Rows, l-1, r)
if not finished? then
rk : NNI := A.NRows
[A, LTr, Pivs, rk]
if D has GcdDomain then
setGcdMode(s : Sy) : Sy ==
tmp := GCDmode
(s = iter) or (s = rand) =>
GCDmode := s
tmp
error "unknown gcd mode"
randomGCD(le : L D) : D ==
-- Probabilistic technique.
#le = 2 => gcd(first le, second le)
f := first le
g := second le
l := rest rest le
while not empty? l repeat
one? first l => return 1$D
f := f + (1+random(113)$I)*first(l)
l := rest l
if not empty? l then
one? first l => return 1$D
g := g + (1+random(113)$I)*first(l)
l := rest l
h := gcd(f, g)
l := [h]
for e in le repeat
tmp := e exquo h
if tmp case "failed" then
l := cons(e, l)
one?(#l) => h
randomGCD l
iteratedGCD(le : L D) : D ==
-- Computes gcd iteratively
res := gcd(first le, second le)
l := rest rest le
while not(empty?(l) or one?(res)) repeat
res := gcd(res, first l)
l := rest l
res
makePrimitive(r : ROWREC) : Record(GCD : D, Row : ROWREC) ==
-- remove common gcd of row
le := r.Entries
one?(#le) => [first le, [r.Indices, [1$D]]]
g : D
if GCDmode = 'iterated then
g := iteratedGCD le
else
g := randomGCD le
one? g => [1, r]
le := [(e exquo g)::D for e in le]
[g, [r.Indices, le]]
primitiveRowEchelon(AA : %) : _
Record(Ech : %, Lt : MFD, Pivots : L D, Rank : NNI) ==
A := copy AA
LTr : MFD := diagonalMatrix [1$FD for i in 1..A.NRows]
Pivs : L D := empty
-- check pivots
for i in 1..A.NRows repeat
r := qelt(A.Rows, i)
changed? : B := false
while not(empty? r.Entries) and zero? first r.Entries repeat
r.Entries := rest r.Entries
r.Indices := rest r.Indices
changed? := true
if changed? then
qsetelt!(A.Rows, i, r)
-- sort rows by pivots (bubble sort)
sorted? : B := false
until sorted? repeat
sorted? := true
oldr := qelt(A.Rows, 1)
for i in 2..A.NRows repeat
newr := qelt(A.Rows, i)
if greater(newr, oldr) then
qsetelt!(A.Rows, i, oldr)
qsetelt!(A.Rows, i-1, newr)
swapRows!(LTr, i-1, i)
sorted? := false
else
oldr := newr
-- primitive fraction-free elimination
finished? : B := false
pivlen, pivrow, rk : NNI
for i in 1..A.NRows until finished? repeat
r := qelt(A.Rows, i)
finished? := empty? r.Indices
if finished? then
rk : NNI := (i-1)::NNI
else -- search good pivot
pivind := first r.Indices
pivlen := #r.Indices
pivrow := i
k : I := 0
for j in (i+1)..A.NRows _
while not(empty? qelt(A.Rows, j).Indices) and _
pivind = first(qelt(A.Rows, j).Indices) repeat
len := #qelt(A.Rows, j).Indices
k := k+1
if len < pivlen then
pivlen := len
pivrow := j
-- make row primitive
tmp := makePrimitive qelt(A.Rows, pivrow)
if not one? tmp.GCD then
qsetelt!(A.Rows, pivrow, tmp.Row)
q : FD := 1/tmp.GCD
for l in 1..A.NRows | not zero? qelt(LTr, pivrow, l) _
repeat
qsetelt!(LTr, pivrow, l, q*qelt(LTr, pivrow, l))
piv : D := first qelt(A.Rows, pivrow).Entries
Pivs := cons(piv, Pivs)
-- elimination necessary?
if k > 0 then
if pivrow ^= i then
pr := qelt(A.Rows, pivrow)
qsetelt!(A.Rows, pivrow, qelt(A.Rows, i))
qsetelt!(A.Rows, i, pr)
swapRows!(LTr, i, pivrow)
-- elimination (and resorting of rows)
pr := copy tmp.Row
pr.Indices := rest pr.Indices
pr.Entries := rest pr.Entries
for j in (i+1)..(i+k) repeat
r := copy qelt(A.Rows, i+1)
c := first r.Entries
r.Indices := rest r.Indices
r.Entries := rest r.Entries
r := addRows(piv, r, -c, pr)
for l in 1..A.NRows repeat
fd : FD := piv *$FD qelt(LTr, i+1, l) - _
(c*qelt(LTr, i, l))::FD
qsetelt!(LTr, i+1, l, fd)
for l in (i+2)..(2*i+k+1-j) repeat
qsetelt!(A.Rows, l-1, qelt(A.Rows, l))
swapRows!(LTr, l-1, l)
for l in (2*i+k+2-j)..A.NRows _
while greater(qelt(A.Rows, l), r) repeat
qsetelt!(A.Rows, l-1, qelt(A.Rows, l))
swapRows!(LTr, l-1, l)
qsetelt!(A.Rows, l-1, r)
if not finished? then
rk : NNI := A.NRows
[A, LTr, Pivs, rk]
-- -------------- --
-- Multiplication --
-- -------------- --
L : MD * AA : % ==
ncols(L) ^= AA.NRows => error "improper matrix dimensions"
A := copy AA
rlen := nrows L
res : % := new(A.AllInds, rlen)
for c in A.AllInds repeat
tmp : V D := new(rlen, 0$D)
for i in 1..A.NRows repeat
r := qelt(A.Rows, i)
inds := r.Indices
if not(empty? inds) and first(inds) = c then
for k in 1..rlen | not zero? qelt(L, k, i) repeat
qsetelt!(tmp, k, qelt(tmp, k) + qelt(L, k, i)* _
first(r.Entries))
r.Entries := rest r.Entries
r.Indices := rest inds
qsetelt!(A.Rows, i, r)
for k in 1..rlen | not zero? qelt(tmp, k) repeat
r := qelt(res.Rows, k)
r.Indices := cons(c, r.Indices)
r.Entries := cons(qelt(tmp, k), r.Entries)
qsetelt!(res.Rows, k, r)
for k in 1..rlen repeat
r := qelt(res.Rows, k)
r.Indices := reverse! r.Indices
r.Entries := reverse! r.Entries
qsetelt!(res.Rows, k, r)
res
if D has IntegralDomain then
mult(f : FD, d : D) : D ==
res := numer(f)*d
tmp := res exquo denom(f)
tmp case "failed" => error "cannot divide in mult"
tmp::D
L : MFD * AA : % ==
ncols(L) ^= AA.NRows => error "improper matrix dimensions"
A := copy AA
rlen := nrows L
res : % := new(A.AllInds, rlen)
for c in A.AllInds repeat
tmp : V FD := new(rlen, 0$FD)
for i in 1..A.NRows repeat
r := qelt(A.Rows, i)
inds := r.Indices
if not(empty? inds) and first(inds) = c then
for k in 1..rlen | not zero? qelt(L, k, i) repeat
qsetelt!(tmp, k, qelt(tmp, k) + qelt(L, k, i)* _
first(r.Entries))
r.Entries := rest r.Entries
r.Indices := rest inds
qsetelt!(A.Rows, i, r)
for k in 1..rlen | not zero? qelt(tmp, k) repeat
d : Union(D, "failed") := retractIfCan qelt(tmp, k)
d case "failed" => error "cannot divide in *"
r := qelt(res.Rows, k)
r.Indices := cons(c, r.Indices)
r.Entries := cons(d::D, r.Entries)
qsetelt!(res.Rows, k, r)
for k in 1..rlen repeat
r := qelt(res.Rows, k)
r.Indices := reverse! r.Indices
r.Entries := reverse! r.Entries
qsetelt!(res.Rows, k, r)
res
|