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*> \ingroup realSYcomputational 163* 164*> \par Contributors: 165* ================== 166*> 167*> \verbatim 168*> 169*> November 2013, Igor Kozachenko, 170*> Computer Science Division, 171*> University of California, Berkeley 172*> \endverbatim 173* 174* ===================================================================== 175 SUBROUTINE SLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO ) 176* 177* -- LAPACK computational routine -- 178* -- LAPACK is a software package provided by Univ. of Tennessee, -- 179* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 180* 181* .. Scalar Arguments .. 182 CHARACTER UPLO 183 INTEGER INFO, KB, LDA, LDW, N, NB 184* .. 185* .. Array Arguments .. 186 INTEGER IPIV( * ) 187 REAL A( LDA, * ), W( LDW, * ) 188* .. 189* 190* ===================================================================== 191* 192* .. Parameters .. 193 REAL ZERO, ONE 194 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 195 REAL EIGHT, SEVTEN 196 PARAMETER ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 ) 197* .. 198* .. Local Scalars .. 199 INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP, 200 $ KSTEP, KW 201 REAL ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1, 202 $ ROWMAX, T 203* .. 204* .. External Functions .. 205 LOGICAL LSAME 206 INTEGER ISAMAX 207 EXTERNAL LSAME, ISAMAX 208* .. 209* .. External Subroutines .. 210 EXTERNAL SCOPY, SGEMM, SGEMV, SSCAL, SSWAP 211* .. 212* .. Intrinsic Functions .. 213 INTRINSIC ABS, MAX, MIN, SQRT 214* .. 215* .. Executable Statements .. 216* 217 INFO = 0 218* 219* Initialize ALPHA for use in choosing pivot block size. 220* 221 ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT 222* 223 IF( LSAME( UPLO, 'U' ) ) THEN 224* 225* Factorize the trailing columns of A using the upper triangle 226* of A and working backwards, and compute the matrix W = U12*D 227* for use in updating A11 228* 229* K is the main loop index, decreasing from N in steps of 1 or 2 230* 231* KW is the column of W which corresponds to column K of A 232* 233 K = N 234 10 CONTINUE 235 KW = NB + K - N 236* 237* Exit from loop 238* 239 IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 ) 240 $ GO TO 30 241* 242* Copy column K of A to column KW of W and update it 243* 244 CALL SCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 ) 245 IF( K.LT.N ) 246 $ CALL SGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), LDA, 247 $ W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 ) 248* 249 KSTEP = 1 250* 251* Determine rows and columns to be interchanged and whether 252* a 1-by-1 or 2-by-2 pivot block will be used 253* 254 ABSAKK = ABS( W( K, KW ) ) 255* 256* IMAX is the row-index of the largest off-diagonal element in 257* column K, and COLMAX is its absolute value. 258* Determine both COLMAX and IMAX. 259* 260 IF( K.GT.1 ) THEN 261 IMAX = ISAMAX( K-1, W( 1, KW ), 1 ) 262 COLMAX = ABS( W( IMAX, KW ) ) 263 ELSE 264 COLMAX = ZERO 265 END IF 266* 267 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 268* 269* Column K is zero or underflow: set INFO and continue 270* 271 IF( INFO.EQ.0 ) 272 $ INFO = K 273 KP = K 274 ELSE 275 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 276* 277* no interchange, use 1-by-1 pivot block 278* 279 KP = K 280 ELSE 281* 282* Copy column IMAX to column KW-1 of W and update it 283* 284 CALL SCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 ) 285 CALL SCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA, 286 $ W( IMAX+1, KW-1 ), 1 ) 287 IF( K.LT.N ) 288 $ CALL SGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), 289 $ LDA, W( IMAX, KW+1 ), LDW, ONE, 290 $ W( 1, KW-1 ), 1 ) 291* 292* JMAX is the column-index of the largest off-diagonal 293* element in row IMAX, and ROWMAX is its absolute value 294* 295 JMAX = IMAX + ISAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 ) 296 ROWMAX = ABS( W( JMAX, KW-1 ) ) 297 IF( IMAX.GT.1 ) THEN 298 JMAX = ISAMAX( IMAX-1, W( 1, KW-1 ), 1 ) 299 ROWMAX = MAX( ROWMAX, ABS( W( JMAX, KW-1 ) ) ) 300 END IF 301* 302 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 303* 304* no interchange, use 1-by-1 pivot block 305* 306 KP = K 307 ELSE IF( ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN 308* 309* interchange rows and columns K and IMAX, use 1-by-1 310* pivot block 311* 312 KP = IMAX 313* 314* copy column KW-1 of W to column KW of W 315* 316 CALL SCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 ) 317 ELSE 318* 319* interchange rows and columns K-1 and IMAX, use 2-by-2 320* pivot block 321* 322 KP = IMAX 323 KSTEP = 2 324 END IF 325 END IF 326* 327* ============================================================ 328* 329* KK is the column of A where pivoting step stopped 330* 331 KK = K - KSTEP + 1 332* 333* KKW is the column of W which corresponds to column KK of A 334* 335 KKW = NB + KK - N 336* 337* Interchange rows and columns KP and KK. 338* Updated column KP is already stored in column KKW of W. 339* 340 IF( KP.NE.KK ) THEN 341* 342* Copy non-updated column KK to column KP of submatrix A 343* at step K. No need to copy element into column K 344* (or K and K-1 for 2-by-2 pivot) of A, since these columns 345* will be later overwritten. 346* 347 A( KP, KP ) = A( KK, KK ) 348 CALL SCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ), 349 $ LDA ) 350 IF( KP.GT.1 ) 351 $ CALL SCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) 352* 353* Interchange rows KK and KP in last K+1 to N columns of A 354* (columns K (or K and K-1 for 2-by-2 pivot) of A will be 355* later overwritten). Interchange rows KK and KP 356* in last KKW to NB columns of W. 357* 358 IF( K.LT.N ) 359 $ CALL SSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ), 360 $ LDA ) 361 CALL SSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ), 362 $ LDW ) 363 END IF 364* 365 IF( KSTEP.EQ.1 ) THEN 366* 367* 1-by-1 pivot block D(k): column kw of W now holds 368* 369* W(kw) = U(k)*D(k), 370* 371* where U(k) is the k-th column of U 372* 373* Store subdiag. elements of column U(k) 374* and 1-by-1 block D(k) in column k of A. 375* NOTE: Diagonal element U(k,k) is a UNIT element 376* and not stored. 377* A(k,k) := D(k,k) = W(k,kw) 378* A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k) 379* 380 CALL SCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 ) 381 R1 = ONE / A( K, K ) 382 CALL SSCAL( K-1, R1, A( 1, K ), 1 ) 383* 384 ELSE 385* 386* 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold 387* 388* ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k) 389* 390* where U(k) and U(k-1) are the k-th and (k-1)-th columns 391* of U 392* 393* Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2 394* block D(k-1:k,k-1:k) in columns k-1 and k of A. 395* NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT 396* block and not stored. 397* A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw) 398* A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) = 399* = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) ) 400* 401 IF( K.GT.2 ) THEN 402* 403* Compose the columns of the inverse of 2-by-2 pivot 404* block D in the following way to reduce the number 405* of FLOPS when we myltiply panel ( W(kw-1) W(kw) ) by 406* this inverse 407* 408* D**(-1) = ( d11 d21 )**(-1) = 409* ( d21 d22 ) 410* 411* = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) = 412* ( (-d21 ) ( d11 ) ) 413* 414* = 1/d21 * 1/((d11/d21)*(d22/d21)-1) * 415* 416* * ( ( d22/d21 ) ( -1 ) ) = 417* ( ( -1 ) ( d11/d21 ) ) 418* 419* = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) = 420* ( ( -1 ) ( D22 ) ) 421* 422* = 1/d21 * T * ( ( D11 ) ( -1 ) ) 423* ( ( -1 ) ( D22 ) ) 424* 425* = D21 * ( ( D11 ) ( -1 ) ) 426* ( ( -1 ) ( D22 ) ) 427* 428 D21 = W( K-1, KW ) 429 D11 = W( K, KW ) / D21 430 D22 = W( K-1, KW-1 ) / D21 431 T = ONE / ( D11*D22-ONE ) 432 D21 = T / D21 433* 434* Update elements in columns A(k-1) and A(k) as 435* dot products of rows of ( W(kw-1) W(kw) ) and columns 436* of D**(-1) 437* 438 DO 20 J = 1, K - 2 439 A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) ) 440 A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) ) 441 20 CONTINUE 442 END IF 443* 444* Copy D(k) to A 445* 446 A( K-1, K-1 ) = W( K-1, KW-1 ) 447 A( K-1, K ) = W( K-1, KW ) 448 A( K, K ) = W( K, KW ) 449* 450 END IF 451* 452 END IF 453* 454* Store details of the interchanges in IPIV 455* 456 IF( KSTEP.EQ.1 ) THEN 457 IPIV( K ) = KP 458 ELSE 459 IPIV( K ) = -KP 460 IPIV( K-1 ) = -KP 461 END IF 462* 463* Decrease K and return to the start of the main loop 464* 465 K = K - KSTEP 466 GO TO 10 467* 468 30 CONTINUE 469* 470* Update the upper triangle of A11 (= A(1:k,1:k)) as 471* 472* A11 := A11 - U12*D*U12**T = A11 - U12*W**T 473* 474* computing blocks of NB columns at a time 475* 476 DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB 477 JB = MIN( NB, K-J+1 ) 478* 479* Update the upper triangle of the diagonal block 480* 481 DO 40 JJ = J, J + JB - 1 482 CALL SGEMV( 'No transpose', JJ-J+1, N-K, -ONE, 483 $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE, 484 $ A( J, JJ ), 1 ) 485 40 CONTINUE 486* 487* Update the rectangular superdiagonal block 488* 489 CALL SGEMM( 'No transpose', 'Transpose', J-1, JB, N-K, -ONE, 490 $ A( 1, K+1 ), LDA, W( J, KW+1 ), LDW, ONE, 491 $ A( 1, J ), LDA ) 492 50 CONTINUE 493* 494* Put U12 in standard form by partially undoing the interchanges 495* in columns k+1:n looping backwards from k+1 to n 496* 497 J = K + 1 498 60 CONTINUE 499* 500* Undo the interchanges (if any) of rows JJ and JP at each 501* step J 502* 503* (Here, J is a diagonal index) 504 JJ = J 505 JP = IPIV( J ) 506 IF( JP.LT.0 ) THEN 507 JP = -JP 508* (Here, J is a diagonal index) 509 J = J + 1 510 END IF 511* (NOTE: Here, J is used to determine row length. Length N-J+1 512* of the rows to swap back doesn't include diagonal element) 513 J = J + 1 514 IF( JP.NE.JJ .AND. J.LE.N ) 515 $ CALL SSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA ) 516 IF( J.LT.N ) 517 $ GO TO 60 518* 519* Set KB to the number of columns factorized 520* 521 KB = N - K 522* 523 ELSE 524* 525* Factorize the leading columns of A using the lower triangle 526* of A and working forwards, and compute the matrix W = L21*D 527* for use in updating A22 528* 529* K is the main loop index, increasing from 1 in steps of 1 or 2 530* 531 K = 1 532 70 CONTINUE 533* 534* Exit from loop 535* 536 IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N ) 537 $ GO TO 90 538* 539* Copy column K of A to column K of W and update it 540* 541 CALL SCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 ) 542 CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), LDA, 543 $ W( K, 1 ), LDW, ONE, W( K, K ), 1 ) 544* 545 KSTEP = 1 546* 547* Determine rows and columns to be interchanged and whether 548* a 1-by-1 or 2-by-2 pivot block will be used 549* 550 ABSAKK = ABS( W( K, K ) ) 551* 552* IMAX is the row-index of the largest off-diagonal element in 553* column K, and COLMAX is its absolute value. 554* Determine both COLMAX and IMAX. 555* 556 IF( K.LT.N ) THEN 557 IMAX = K + ISAMAX( N-K, W( K+1, K ), 1 ) 558 COLMAX = ABS( W( IMAX, K ) ) 559 ELSE 560 COLMAX = ZERO 561 END IF 562* 563 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 564* 565* Column K is zero or underflow: set INFO and continue 566* 567 IF( INFO.EQ.0 ) 568 $ INFO = K 569 KP = K 570 ELSE 571 IF( ABSAKK.GE.ALPHA*COLMAX ) THEN 572* 573* no interchange, use 1-by-1 pivot block 574* 575 KP = K 576 ELSE 577* 578* Copy column IMAX to column K+1 of W and update it 579* 580 CALL SCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 ) 581 CALL SCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ), 582 $ 1 ) 583 CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), 584 $ LDA, W( IMAX, 1 ), LDW, ONE, W( K, K+1 ), 1 ) 585* 586* JMAX is the column-index of the largest off-diagonal 587* element in row IMAX, and ROWMAX is its absolute value 588* 589 JMAX = K - 1 + ISAMAX( IMAX-K, W( K, K+1 ), 1 ) 590 ROWMAX = ABS( W( JMAX, K+1 ) ) 591 IF( IMAX.LT.N ) THEN 592 JMAX = IMAX + ISAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 ) 593 ROWMAX = MAX( ROWMAX, ABS( W( JMAX, K+1 ) ) ) 594 END IF 595* 596 IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN 597* 598* no interchange, use 1-by-1 pivot block 599* 600 KP = K 601 ELSE IF( ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN 602* 603* interchange rows and columns K and IMAX, use 1-by-1 604* pivot block 605* 606 KP = IMAX 607* 608* copy column K+1 of W to column K of W 609* 610 CALL SCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 ) 611 ELSE 612* 613* interchange rows and columns K+1 and IMAX, use 2-by-2 614* pivot block 615* 616 KP = IMAX 617 KSTEP = 2 618 END IF 619 END IF 620* 621* ============================================================ 622* 623* KK is the column of A where pivoting step stopped 624* 625 KK = K + KSTEP - 1 626* 627* Interchange rows and columns KP and KK. 628* Updated column KP is already stored in column KK of W. 629* 630 IF( KP.NE.KK ) THEN 631* 632* Copy non-updated column KK to column KP of submatrix A 633* at step K. No need to copy element into column K 634* (or K and K+1 for 2-by-2 pivot) of A, since these columns 635* will be later overwritten. 636* 637 A( KP, KP ) = A( KK, KK ) 638 CALL SCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ), 639 $ LDA ) 640 IF( KP.LT.N ) 641 $ CALL SCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) 642* 643* Interchange rows KK and KP in first K-1 columns of A 644* (columns K (or K and K+1 for 2-by-2 pivot) of A will be 645* later overwritten). Interchange rows KK and KP 646* in first KK columns of W. 647* 648 IF( K.GT.1 ) 649 $ CALL SSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) 650 CALL SSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW ) 651 END IF 652* 653 IF( KSTEP.EQ.1 ) THEN 654* 655* 1-by-1 pivot block D(k): column k of W now holds 656* 657* W(k) = L(k)*D(k), 658* 659* where L(k) is the k-th column of L 660* 661* Store subdiag. elements of column L(k) 662* and 1-by-1 block D(k) in column k of A. 663* (NOTE: Diagonal element L(k,k) is a UNIT element 664* and not stored) 665* A(k,k) := D(k,k) = W(k,k) 666* A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k) 667* 668 CALL SCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 ) 669 IF( K.LT.N ) THEN 670 R1 = ONE / A( K, K ) 671 CALL SSCAL( N-K, R1, A( K+1, K ), 1 ) 672 END IF 673* 674 ELSE 675* 676* 2-by-2 pivot block D(k): columns k and k+1 of W now hold 677* 678* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) 679* 680* where L(k) and L(k+1) are the k-th and (k+1)-th columns 681* of L 682* 683* Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2 684* block D(k:k+1,k:k+1) in columns k and k+1 of A. 685* (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT 686* block and not stored) 687* A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1) 688* A(k+2:N,k:k+1) := L(k+2:N,k:k+1) = 689* = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) ) 690* 691 IF( K.LT.N-1 ) THEN 692* 693* Compose the columns of the inverse of 2-by-2 pivot 694* block D in the following way to reduce the number 695* of FLOPS when we myltiply panel ( W(k) W(k+1) ) by 696* this inverse 697* 698* D**(-1) = ( d11 d21 )**(-1) = 699* ( d21 d22 ) 700* 701* = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) = 702* ( (-d21 ) ( d11 ) ) 703* 704* = 1/d21 * 1/((d11/d21)*(d22/d21)-1) * 705* 706* * ( ( d22/d21 ) ( -1 ) ) = 707* ( ( -1 ) ( d11/d21 ) ) 708* 709* = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) = 710* ( ( -1 ) ( D22 ) ) 711* 712* = 1/d21 * T * ( ( D11 ) ( -1 ) ) 713* ( ( -1 ) ( D22 ) ) 714* 715* = D21 * ( ( D11 ) ( -1 ) ) 716* ( ( -1 ) ( D22 ) ) 717* 718 D21 = W( K+1, K ) 719 D11 = W( K+1, K+1 ) / D21 720 D22 = W( K, K ) / D21 721 T = ONE / ( D11*D22-ONE ) 722 D21 = T / D21 723* 724* Update elements in columns A(k) and A(k+1) as 725* dot products of rows of ( W(k) W(k+1) ) and columns 726* of D**(-1) 727* 728 DO 80 J = K + 2, N 729 A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) ) 730 A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) ) 731 80 CONTINUE 732 END IF 733* 734* Copy D(k) to A 735* 736 A( K, K ) = W( K, K ) 737 A( K+1, K ) = W( K+1, K ) 738 A( K+1, K+1 ) = W( K+1, K+1 ) 739* 740 END IF 741* 742 END IF 743* 744* Store details of the interchanges in IPIV 745* 746 IF( KSTEP.EQ.1 ) THEN 747 IPIV( K ) = KP 748 ELSE 749 IPIV( K ) = -KP 750 IPIV( K+1 ) = -KP 751 END IF 752* 753* Increase K and return to the start of the main loop 754* 755 K = K + KSTEP 756 GO TO 70 757* 758 90 CONTINUE 759* 760* Update the lower triangle of A22 (= A(k:n,k:n)) as 761* 762* A22 := A22 - L21*D*L21**T = A22 - L21*W**T 763* 764* computing blocks of NB columns at a time 765* 766 DO 110 J = K, N, NB 767 JB = MIN( NB, N-J+1 ) 768* 769* Update the lower triangle of the diagonal block 770* 771 DO 100 JJ = J, J + JB - 1 772 CALL SGEMV( 'No transpose', J+JB-JJ, K-1, -ONE, 773 $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE, 774 $ A( JJ, JJ ), 1 ) 775 100 CONTINUE 776* 777* Update the rectangular subdiagonal block 778* 779 IF( J+JB.LE.N ) 780 $ CALL SGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB, 781 $ K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW, 782 $ ONE, A( J+JB, J ), LDA ) 783 110 CONTINUE 784* 785* Put L21 in standard form by partially undoing the interchanges 786* of rows in columns 1:k-1 looping backwards from k-1 to 1 787* 788 J = K - 1 789 120 CONTINUE 790* 791* Undo the interchanges (if any) of rows JJ and JP at each 792* step J 793* 794* (Here, J is a diagonal index) 795 JJ = J 796 JP = IPIV( J ) 797 IF( JP.LT.0 ) THEN 798 JP = -JP 799* (Here, J is a diagonal index) 800 J = J - 1 801 END IF 802* (NOTE: Here, J is used to determine row length. Length J 803* of the rows to swap back doesn't include diagonal element) 804 J = J - 1 805 IF( JP.NE.JJ .AND. J.GE.1 ) 806 $ CALL SSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA ) 807 IF( J.GT.1 ) 808 $ GO TO 120 809* 810* Set KB to the number of columns factorized 811* 812 KB = K - 1 813* 814 END IF 815 RETURN 816* 817* End of SLASYF 818* 819 END 820