1*> \brief \b SLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method. 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download SLASYF + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slasyf.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slasyf.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slasyf.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) 22* 23* .. Scalar Arguments .. 24* CHARACTER UPLO 25* INTEGER INFO, KB, LDA, LDW, N, NB 26* .. 27* .. Array Arguments .. 28* INTEGER IPIV( * ) 29* REAL A( LDA, * ), W( LDW, * ) 30* .. 31* 32* 33*> \par Purpose: 34* ============= 35*> 36*> \verbatim 37*> 38*> SLASYF computes a partial factorization of a real symmetric matrix A 39*> using the Bunch-Kaufman diagonal pivoting method. The partial 40*> factorization has the form: 41*> 42*> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or: 43*> ( 0 U22 ) ( 0 D ) ( U12**T U22**T ) 44*> 45*> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L' 46*> ( L21 I ) ( 0 A22 ) ( 0 I ) 47*> 48*> where the order of D is at most NB. The actual order is returned in 49*> the argument KB, and is either NB or NB-1, or N if N <= NB. 50*> 51*> SLASYF is an auxiliary routine called by SSYTRF. It uses blocked code 52*> (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or 53*> A22 (if UPLO = 'L'). 54*> \endverbatim 55* 56* Arguments: 57* ========== 58* 59*> \param[in] UPLO 60*> \verbatim 61*> UPLO is CHARACTER*1 62*> Specifies whether the upper or lower triangular part of the 63*> symmetric matrix A is stored: 64*> = 'U': Upper triangular 65*> = 'L': Lower triangular 66*> \endverbatim 67*> 68*> \param[in] N 69*> \verbatim 70*> N is INTEGER 71*> The order of the matrix A. N >= 0. 72*> \endverbatim 73*> 74*> \param[in] NB 75*> \verbatim 76*> NB is INTEGER 77*> The maximum number of columns of the matrix A that should be 78*> factored. NB should be at least 2 to allow for 2-by-2 pivot 79*> blocks. 80*> \endverbatim 81*> 82*> \param[out] KB 83*> \verbatim 84*> KB is INTEGER 85*> The number of columns of A that were actually factored. 86*> KB is either NB-1 or NB, or N if N <= NB. 87*> \endverbatim 88*> 89*> \param[in,out] A 90*> \verbatim 91*> A is REAL array, dimension (LDA,N) 92*> On entry, the symmetric matrix A. If UPLO = 'U', the leading 93*> n-by-n upper triangular part of A contains the upper 94*> triangular part of the matrix A, and the strictly lower 95*> triangular part of A is not referenced. If UPLO = 'L', the 96*> leading n-by-n lower triangular part of A contains the lower 97*> triangular part of the matrix A, and the strictly upper 98*> triangular part of A is not referenced. 99*> On exit, A contains details of the partial factorization. 100*> \endverbatim 101*> 102*> \param[in] LDA 103*> \verbatim 104*> LDA is INTEGER 105*> The leading dimension of the array A. LDA >= max(1,N). 106*> \endverbatim 107*> 108*> \param[out] IPIV 109*> \verbatim 110*> IPIV is INTEGER array, dimension (N) 111*> Details of the interchanges and the block structure of D. 112*> 113*> If UPLO = 'U': 114*> Only the last KB elements of IPIV are set. 115*> 116*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were 117*> interchanged and D(k,k) is a 1-by-1 diagonal block. 118*> 119*> If IPIV(k) = IPIV(k-1) < 0, then rows and columns 120*> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) 121*> is a 2-by-2 diagonal block. 122*> 123*> If UPLO = 'L': 124*> Only the first KB elements of IPIV are set. 125*> 126*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were 127*> interchanged and D(k,k) is a 1-by-1 diagonal block. 128*> 129*> If IPIV(k) = IPIV(k+1) < 0, then rows and columns 130*> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) 131*> is a 2-by-2 diagonal block. 132*> \endverbatim 133*> 134*> \param[out] W 135*> \verbatim 136*> W is REAL array, dimension (LDW,NB) 137*> \endverbatim 138*> 139*> \param[in] LDW 140*> \verbatim 141*> LDW is INTEGER 142*> The leading dimension of the array W. LDW >= max(1,N). 143*> \endverbatim 144*> 145*> \param[out] INFO 146*> \verbatim 147*> INFO is INTEGER 148*> = 0: successful exit 149*> > 0: if INFO = k, D(k,k) is exactly zero. The factorization 150*> has been completed, but the block diagonal matrix D is 151*> exactly singular. 152*> \endverbatim 153* 154* Authors: 155* ======== 156* 157*> \author Univ. of Tennessee 158*> \author Univ. of California Berkeley 159*> \author Univ. of Colorado Denver 160*> \author NAG Ltd. 161* 162*> \date November 2013 163* 164*> \ingroup realSYcomputational 165* 166*> \par Contributors: 167* ================== 168*> 169*> \verbatim 170*> 171*> November 2013, Igor Kozachenko, 172*> Computer Science Division, 173*> University of California, Berkeley 174*> \endverbatim 175* 176* ===================================================================== 177 SUBROUTINE SLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) 178* 179* -- LAPACK computational routine (version 3.5.0) -- 180* -- LAPACK is a software package provided by Univ. of Tennessee, -- 181* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 182* November 2013 183* 184* .. Scalar Arguments .. 185 CHARACTER UPLO 186 INTEGER INFO, KB, LDA, LDW, N, NB 187* .. 188* .. Array Arguments .. 189 INTEGER IPIV( * ) 190 REAL A( LDA, * ), W( LDW, * ) 191* .. 192* 193* ===================================================================== 194* 195* .. Parameters .. 196 REAL ZERO, ONE 197 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 198 REAL EIGHT, SEVTEN 199 PARAMETER ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 ) 200* .. 201* .. Local Scalars .. 202 INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP, 203 $ KSTEP, KW 204 REAL ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1, 205 $ ROWMAX, T 206* .. 207* .. External Functions .. 208 LOGICAL LSAME 209 INTEGER ISAMAX 210 EXTERNAL LSAME, ISAMAX 211* .. 212* .. External Subroutines .. 213 EXTERNAL SCOPY, SGEMM, SGEMV, SSCAL, SSWAP 214* .. 215* .. Intrinsic Functions .. 216 INTRINSIC ABS, MAX, MIN, SQRT 217* .. 218* .. Executable Statements .. 219* 220 INFO = 0 221* 222* Initialize ALPHA for use in choosing pivot block size. 223* 224 ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT 225* 226 IF( LSAME( UPLO, 'U' ) ) THEN 227* 228* Factorize the trailing columns of A using the upper triangle 229* of A and working backwards, and compute the matrix W = U12*D 230* for use in updating A11 231* 232* K is the main loop index, decreasing from N in steps of 1 or 2 233* 234* KW is the column of W which corresponds to column K of A 235* 236 K = N 237 10 CONTINUE 238 KW = NB + K - N 239* 240* Exit from loop 241* 242 IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 ) 243 $ GO TO 30 244* 245* Copy column K of A to column KW of W and update it 246* 247 CALL SCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 ) 248 IF( K.LT.N ) 249 $ CALL SGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), LDA, 250 $ W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 ) 251* 252 KSTEP = 1 253* 254* Determine rows and columns to be interchanged and whether 255* a 1-by-1 or 2-by-2 pivot block will be used 256* 257 ABSAKK = ABS( W( K, KW ) ) 258* 259* IMAX is the row-index of the largest off-diagonal element in 260* column K, and COLMAX is its absolute value. 261* Determine both COLMAX and IMAX. 262* 263 IF( K.GT.1 ) THEN 264 IMAX = ISAMAX( K-1, W( 1, KW ), 1 ) 265 COLMAX = ABS( W( IMAX, KW ) ) 266 ELSE 267 COLMAX = ZERO 268 END IF 269* 270 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 271* 272* Column K is zero or underflow: set INFO and continue 273* 274 IF( INFO.EQ.0 ) 275 $ INFO = K 276 KP = K 277 ELSE 278 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 279* 280* no interchange, use 1-by-1 pivot block 281* 282 KP = K 283 ELSE 284* 285* Copy column IMAX to column KW-1 of W and update it 286* 287 CALL SCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) 288 CALL SCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA, 289 $ W( IMAX+1, KW-1 ), 1 ) 290 IF( K.LT.N ) 291 $ CALL SGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), 292 $ LDA, W( IMAX, KW+1 ), LDW, ONE, 293 $ W( 1, KW-1 ), 1 ) 294* 295* JMAX is the column-index of the largest off-diagonal 296* element in row IMAX, and ROWMAX is its absolute value 297* 298 JMAX = IMAX + ISAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 ) 299 ROWMAX = ABS( W( JMAX, KW-1 ) ) 300 IF( IMAX.GT.1 ) THEN 301 JMAX = ISAMAX( IMAX-1, W( 1, KW-1 ), 1 ) 302 ROWMAX = MAX( ROWMAX, ABS( W( JMAX, KW-1 ) ) ) 303 END IF 304* 305 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 306* 307* no interchange, use 1-by-1 pivot block 308* 309 KP = K 310 ELSE IF( ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN 311* 312* interchange rows and columns K and IMAX, use 1-by-1 313* pivot block 314* 315 KP = IMAX 316* 317* copy column KW-1 of W to column KW of W 318* 319 CALL SCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) 320 ELSE 321* 322* interchange rows and columns K-1 and IMAX, use 2-by-2 323* pivot block 324* 325 KP = IMAX 326 KSTEP = 2 327 END IF 328 END IF 329* 330* ============================================================ 331* 332* KK is the column of A where pivoting step stopped 333* 334 KK = K - KSTEP + 1 335* 336* KKW is the column of W which corresponds to column KK of A 337* 338 KKW = NB + KK - N 339* 340* Interchange rows and columns KP and KK. 341* Updated column KP is already stored in column KKW of W. 342* 343 IF( KP.NE.KK ) THEN 344* 345* Copy non-updated column KK to column KP of submatrix A 346* at step K. No need to copy element into column K 347* (or K and K-1 for 2-by-2 pivot) of A, since these columns 348* will be later overwritten. 349* 350 A( KP, KP ) = A( KK, KK ) 351 CALL SCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), 352 $ LDA ) 353 IF( KP.GT.1 ) 354 $ CALL SCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) 355* 356* Interchange rows KK and KP in last K+1 to N columns of A 357* (columns K (or K and K-1 for 2-by-2 pivot) of A will be 358* later overwritten). Interchange rows KK and KP 359* in last KKW to NB columns of W. 360* 361 IF( K.LT.N ) 362 $ CALL SSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ), 363 $ LDA ) 364 CALL SSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ), 365 $ LDW ) 366 END IF 367* 368 IF( KSTEP.EQ.1 ) THEN 369* 370* 1-by-1 pivot block D(k): column kw of W now holds 371* 372* W(kw) = U(k)*D(k), 373* 374* where U(k) is the k-th column of U 375* 376* Store subdiag. elements of column U(k) 377* and 1-by-1 block D(k) in column k of A. 378* NOTE: Diagonal element U(k,k) is a UNIT element 379* and not stored. 380* A(k,k) := D(k,k) = W(k,kw) 381* A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k) 382* 383 CALL SCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) 384 R1 = ONE / A( K, K ) 385 CALL SSCAL( K-1, R1, A( 1, K ), 1 ) 386* 387 ELSE 388* 389* 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold 390* 391* ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k) 392* 393* where U(k) and U(k-1) are the k-th and (k-1)-th columns 394* of U 395* 396* Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2 397* block D(k-1:k,k-1:k) in columns k-1 and k of A. 398* NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT 399* block and not stored. 400* A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw) 401* A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) = 402* = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) ) 403* 404 IF( K.GT.2 ) THEN 405* 406* Compose the columns of the inverse of 2-by-2 pivot 407* block D in the following way to reduce the number 408* of FLOPS when we myltiply panel ( W(kw-1) W(kw) ) by 409* this inverse 410* 411* D**(-1) = ( d11 d21 )**(-1) = 412* ( d21 d22 ) 413* 414* = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) = 415* ( (-d21 ) ( d11 ) ) 416* 417* = 1/d21 * 1/((d11/d21)*(d22/d21)-1) * 418* 419* * ( ( d22/d21 ) ( -1 ) ) = 420* ( ( -1 ) ( d11/d21 ) ) 421* 422* = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) = 423* ( ( -1 ) ( D22 ) ) 424* 425* = 1/d21 * T * ( ( D11 ) ( -1 ) ) 426* ( ( -1 ) ( D22 ) ) 427* 428* = D21 * ( ( D11 ) ( -1 ) ) 429* ( ( -1 ) ( D22 ) ) 430* 431 D21 = W( K-1, KW ) 432 D11 = W( K, KW ) / D21 433 D22 = W( K-1, KW-1 ) / D21 434 T = ONE / ( D11*D22-ONE ) 435 D21 = T / D21 436* 437* Update elements in columns A(k-1) and A(k) as 438* dot products of rows of ( W(kw-1) W(kw) ) and columns 439* of D**(-1) 440* 441 DO 20 J = 1, K - 2 442 A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) 443 A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) ) 444 20 CONTINUE 445 END IF 446* 447* Copy D(k) to A 448* 449 A( K-1, K-1 ) = W( K-1, KW-1 ) 450 A( K-1, K ) = W( K-1, KW ) 451 A( K, K ) = W( K, KW ) 452* 453 END IF 454* 455 END IF 456* 457* Store details of the interchanges in IPIV 458* 459 IF( KSTEP.EQ.1 ) THEN 460 IPIV( K ) = KP 461 ELSE 462 IPIV( K ) = -KP 463 IPIV( K-1 ) = -KP 464 END IF 465* 466* Decrease K and return to the start of the main loop 467* 468 K = K - KSTEP 469 GO TO 10 470* 471 30 CONTINUE 472* 473* Update the upper triangle of A11 (= A(1:k,1:k)) as 474* 475* A11 := A11 - U12*D*U12**T = A11 - U12*W**T 476* 477* computing blocks of NB columns at a time 478* 479 DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB 480 JB = MIN( NB, K-J+1 ) 481* 482* Update the upper triangle of the diagonal block 483* 484 DO 40 JJ = J, J + JB - 1 485 CALL SGEMV( 'No transpose', JJ-J+1, N-K, -ONE, 486 $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE, 487 $ A( J, JJ ), 1 ) 488 40 CONTINUE 489* 490* Update the rectangular superdiagonal block 491* 492 CALL SGEMM( 'No transpose', 'Transpose', J-1, JB, N-K, -ONE, 493 $ A( 1, K+1 ), LDA, W( J, KW+1 ), LDW, ONE, 494 $ A( 1, J ), LDA ) 495 50 CONTINUE 496* 497* Put U12 in standard form by partially undoing the interchanges 498* in columns k+1:n looping backwards from k+1 to n 499* 500 J = K + 1 501 60 CONTINUE 502* 503* Undo the interchanges (if any) of rows JJ and JP at each 504* step J 505* 506* (Here, J is a diagonal index) 507 JJ = J 508 JP = IPIV( J ) 509 IF( JP.LT.0 ) THEN 510 JP = -JP 511* (Here, J is a diagonal index) 512 J = J + 1 513 END IF 514* (NOTE: Here, J is used to determine row length. Length N-J+1 515* of the rows to swap back doesn't include diagonal element) 516 J = J + 1 517 IF( JP.NE.JJ .AND. J.LE.N ) 518 $ CALL SSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) 519 IF( J.LT.N ) 520 $ GO TO 60 521* 522* Set KB to the number of columns factorized 523* 524 KB = N - K 525* 526 ELSE 527* 528* Factorize the leading columns of A using the lower triangle 529* of A and working forwards, and compute the matrix W = L21*D 530* for use in updating A22 531* 532* K is the main loop index, increasing from 1 in steps of 1 or 2 533* 534 K = 1 535 70 CONTINUE 536* 537* Exit from loop 538* 539 IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N ) 540 $ GO TO 90 541* 542* Copy column K of A to column K of W and update it 543* 544 CALL SCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 ) 545 CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), LDA, 546 $ W( K, 1 ), LDW, ONE, W( K, K ), 1 ) 547* 548 KSTEP = 1 549* 550* Determine rows and columns to be interchanged and whether 551* a 1-by-1 or 2-by-2 pivot block will be used 552* 553 ABSAKK = ABS( W( K, K ) ) 554* 555* IMAX is the row-index of the largest off-diagonal element in 556* column K, and COLMAX is its absolute value. 557* Determine both COLMAX and IMAX. 558* 559 IF( K.LT.N ) THEN 560 IMAX = K + ISAMAX( N-K, W( K+1, K ), 1 ) 561 COLMAX = ABS( W( IMAX, K ) ) 562 ELSE 563 COLMAX = ZERO 564 END IF 565* 566 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 567* 568* Column K is zero or underflow: set INFO and continue 569* 570 IF( INFO.EQ.0 ) 571 $ INFO = K 572 KP = K 573 ELSE 574 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 575* 576* no interchange, use 1-by-1 pivot block 577* 578 KP = K 579 ELSE 580* 581* Copy column IMAX to column K+1 of W and update it 582* 583 CALL SCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 ) 584 CALL SCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ), 585 $ 1 ) 586 CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), 587 $ LDA, W( IMAX, 1 ), LDW, ONE, W( K, K+1 ), 1 ) 588* 589* JMAX is the column-index of the largest off-diagonal 590* element in row IMAX, and ROWMAX is its absolute value 591* 592 JMAX = K - 1 + ISAMAX( IMAX-K, W( K, K+1 ), 1 ) 593 ROWMAX = ABS( W( JMAX, K+1 ) ) 594 IF( IMAX.LT.N ) THEN 595 JMAX = IMAX + ISAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 ) 596 ROWMAX = MAX( ROWMAX, ABS( W( JMAX, K+1 ) ) ) 597 END IF 598* 599 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 600* 601* no interchange, use 1-by-1 pivot block 602* 603 KP = K 604 ELSE IF( ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN 605* 606* interchange rows and columns K and IMAX, use 1-by-1 607* pivot block 608* 609 KP = IMAX 610* 611* copy column K+1 of W to column K of W 612* 613 CALL SCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) 614 ELSE 615* 616* interchange rows and columns K+1 and IMAX, use 2-by-2 617* pivot block 618* 619 KP = IMAX 620 KSTEP = 2 621 END IF 622 END IF 623* 624* ============================================================ 625* 626* KK is the column of A where pivoting step stopped 627* 628 KK = K + KSTEP - 1 629* 630* Interchange rows and columns KP and KK. 631* Updated column KP is already stored in column KK of W. 632* 633 IF( KP.NE.KK ) THEN 634* 635* Copy non-updated column KK to column KP of submatrix A 636* at step K. No need to copy element into column K 637* (or K and K+1 for 2-by-2 pivot) of A, since these columns 638* will be later overwritten. 639* 640 A( KP, KP ) = A( KK, KK ) 641 CALL SCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ), 642 $ LDA ) 643 IF( KP.LT.N ) 644 $ CALL SCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) 645* 646* Interchange rows KK and KP in first K-1 columns of A 647* (columns K (or K and K+1 for 2-by-2 pivot) of A will be 648* later overwritten). Interchange rows KK and KP 649* in first KK columns of W. 650* 651 IF( K.GT.1 ) 652 $ CALL SSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) 653 CALL SSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW ) 654 END IF 655* 656 IF( KSTEP.EQ.1 ) THEN 657* 658* 1-by-1 pivot block D(k): column k of W now holds 659* 660* W(k) = L(k)*D(k), 661* 662* where L(k) is the k-th column of L 663* 664* Store subdiag. elements of column L(k) 665* and 1-by-1 block D(k) in column k of A. 666* (NOTE: Diagonal element L(k,k) is a UNIT element 667* and not stored) 668* A(k,k) := D(k,k) = W(k,k) 669* A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k) 670* 671 CALL SCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) 672 IF( K.LT.N ) THEN 673 R1 = ONE / A( K, K ) 674 CALL SSCAL( N-K, R1, A( K+1, K ), 1 ) 675 END IF 676* 677 ELSE 678* 679* 2-by-2 pivot block D(k): columns k and k+1 of W now hold 680* 681* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) 682* 683* where L(k) and L(k+1) are the k-th and (k+1)-th columns 684* of L 685* 686* Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2 687* block D(k:k+1,k:k+1) in columns k and k+1 of A. 688* (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT 689* block and not stored) 690* A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1) 691* A(k+2:N,k:k+1) := L(k+2:N,k:k+1) = 692* = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) ) 693* 694 IF( K.LT.N-1 ) THEN 695* 696* Compose the columns of the inverse of 2-by-2 pivot 697* block D in the following way to reduce the number 698* of FLOPS when we myltiply panel ( W(k) W(k+1) ) by 699* this inverse 700* 701* D**(-1) = ( d11 d21 )**(-1) = 702* ( d21 d22 ) 703* 704* = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) = 705* ( (-d21 ) ( d11 ) ) 706* 707* = 1/d21 * 1/((d11/d21)*(d22/d21)-1) * 708* 709* * ( ( d22/d21 ) ( -1 ) ) = 710* ( ( -1 ) ( d11/d21 ) ) 711* 712* = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) = 713* ( ( -1 ) ( D22 ) ) 714* 715* = 1/d21 * T * ( ( D11 ) ( -1 ) ) 716* ( ( -1 ) ( D22 ) ) 717* 718* = D21 * ( ( D11 ) ( -1 ) ) 719* ( ( -1 ) ( D22 ) ) 720* 721 D21 = W( K+1, K ) 722 D11 = W( K+1, K+1 ) / D21 723 D22 = W( K, K ) / D21 724 T = ONE / ( D11*D22-ONE ) 725 D21 = T / D21 726* 727* Update elements in columns A(k) and A(k+1) as 728* dot products of rows of ( W(k) W(k+1) ) and columns 729* of D**(-1) 730* 731 DO 80 J = K + 2, N 732 A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) ) 733 A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) ) 734 80 CONTINUE 735 END IF 736* 737* Copy D(k) to A 738* 739 A( K, K ) = W( K, K ) 740 A( K+1, K ) = W( K+1, K ) 741 A( K+1, K+1 ) = W( K+1, K+1 ) 742* 743 END IF 744* 745 END IF 746* 747* Store details of the interchanges in IPIV 748* 749 IF( KSTEP.EQ.1 ) THEN 750 IPIV( K ) = KP 751 ELSE 752 IPIV( K ) = -KP 753 IPIV( K+1 ) = -KP 754 END IF 755* 756* Increase K and return to the start of the main loop 757* 758 K = K + KSTEP 759 GO TO 70 760* 761 90 CONTINUE 762* 763* Update the lower triangle of A22 (= A(k:n,k:n)) as 764* 765* A22 := A22 - L21*D*L21**T = A22 - L21*W**T 766* 767* computing blocks of NB columns at a time 768* 769 DO 110 J = K, N, NB 770 JB = MIN( NB, N-J+1 ) 771* 772* Update the lower triangle of the diagonal block 773* 774 DO 100 JJ = J, J + JB - 1 775 CALL SGEMV( 'No transpose', J+JB-JJ, K-1, -ONE, 776 $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE, 777 $ A( JJ, JJ ), 1 ) 778 100 CONTINUE 779* 780* Update the rectangular subdiagonal block 781* 782 IF( J+JB.LE.N ) 783 $ CALL SGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB, 784 $ K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW, 785 $ ONE, A( J+JB, J ), LDA ) 786 110 CONTINUE 787* 788* Put L21 in standard form by partially undoing the interchanges 789* of rows in columns 1:k-1 looping backwards from k-1 to 1 790* 791 J = K - 1 792 120 CONTINUE 793* 794* Undo the interchanges (if any) of rows JJ and JP at each 795* step J 796* 797* (Here, J is a diagonal index) 798 JJ = J 799 JP = IPIV( J ) 800 IF( JP.LT.0 ) THEN 801 JP = -JP 802* (Here, J is a diagonal index) 803 J = J - 1 804 END IF 805* (NOTE: Here, J is used to determine row length. Length J 806* of the rows to swap back doesn't include diagonal element) 807 J = J - 1 808 IF( JP.NE.JJ .AND. J.GE.1 ) 809 $ CALL SSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) 810 IF( J.GT.1 ) 811 $ GO TO 120 812* 813* Set KB to the number of columns factorized 814* 815 KB = K - 1 816* 817 END IF 818 RETURN 819* 820* End of SLASYF 821* 822 END 823