1*> \brief \b CHETF2_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm). 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download CHETF2_RK + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chetf2_rk.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chetf2_rk.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chetf2_rk.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE CHETF2_RK( UPLO, N, A, LDA, E, IPIV, INFO ) 22* 23* .. Scalar Arguments .. 24* CHARACTER UPLO 25* INTEGER INFO, LDA, N 26* .. 27* .. Array Arguments .. 28* INTEGER IPIV( * ) 29* COMPLEX A( LDA, * ), E ( * ) 30* .. 31* 32* 33*> \par Purpose: 34* ============= 35*> 36*> \verbatim 37*> CHETF2_RK computes the factorization of a complex Hermitian matrix A 38*> using the bounded Bunch-Kaufman (rook) diagonal pivoting method: 39*> 40*> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T), 41*> 42*> where U (or L) is unit upper (or lower) triangular matrix, 43*> U**H (or L**H) is the conjugate of U (or L), P is a permutation 44*> matrix, P**T is the transpose of P, and D is Hermitian and block 45*> diagonal with 1-by-1 and 2-by-2 diagonal blocks. 46*> 47*> This is the unblocked version of the algorithm, calling Level 2 BLAS. 48*> For more information see Further Details section. 49*> \endverbatim 50* 51* Arguments: 52* ========== 53* 54*> \param[in] UPLO 55*> \verbatim 56*> UPLO is CHARACTER*1 57*> Specifies whether the upper or lower triangular part of the 58*> Hermitian matrix A is stored: 59*> = 'U': Upper triangular 60*> = 'L': Lower triangular 61*> \endverbatim 62*> 63*> \param[in] N 64*> \verbatim 65*> N is INTEGER 66*> The order of the matrix A. N >= 0. 67*> \endverbatim 68*> 69*> \param[in,out] A 70*> \verbatim 71*> A is COMPLEX array, dimension (LDA,N) 72*> On entry, the Hermitian matrix A. 73*> If UPLO = 'U': the leading N-by-N upper triangular part 74*> of A contains the upper triangular part of the matrix A, 75*> and the strictly lower triangular part of A is not 76*> referenced. 77*> 78*> If UPLO = 'L': the leading N-by-N lower triangular part 79*> of A contains the lower triangular part of the matrix A, 80*> and the strictly upper triangular part of A is not 81*> referenced. 82*> 83*> On exit, contains: 84*> a) ONLY diagonal elements of the Hermitian block diagonal 85*> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); 86*> (superdiagonal (or subdiagonal) elements of D 87*> are stored on exit in array E), and 88*> b) If UPLO = 'U': factor U in the superdiagonal part of A. 89*> If UPLO = 'L': factor L in the subdiagonal part of A. 90*> \endverbatim 91*> 92*> \param[in] LDA 93*> \verbatim 94*> LDA is INTEGER 95*> The leading dimension of the array A. LDA >= max(1,N). 96*> \endverbatim 97*> 98*> \param[out] E 99*> \verbatim 100*> E is COMPLEX array, dimension (N) 101*> On exit, contains the superdiagonal (or subdiagonal) 102*> elements of the Hermitian block diagonal matrix D 103*> with 1-by-1 or 2-by-2 diagonal blocks, where 104*> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; 105*> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. 106*> 107*> NOTE: For 1-by-1 diagonal block D(k), where 108*> 1 <= k <= N, the element E(k) is set to 0 in both 109*> UPLO = 'U' or UPLO = 'L' cases. 110*> \endverbatim 111*> 112*> \param[out] IPIV 113*> \verbatim 114*> IPIV is INTEGER array, dimension (N) 115*> IPIV describes the permutation matrix P in the factorization 116*> of matrix A as follows. The absolute value of IPIV(k) 117*> represents the index of row and column that were 118*> interchanged with the k-th row and column. The value of UPLO 119*> describes the order in which the interchanges were applied. 120*> Also, the sign of IPIV represents the block structure of 121*> the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2 122*> diagonal blocks which correspond to 1 or 2 interchanges 123*> at each factorization step. For more info see Further 124*> Details section. 125*> 126*> If UPLO = 'U', 127*> ( in factorization order, k decreases from N to 1 ): 128*> a) A single positive entry IPIV(k) > 0 means: 129*> D(k,k) is a 1-by-1 diagonal block. 130*> If IPIV(k) != k, rows and columns k and IPIV(k) were 131*> interchanged in the matrix A(1:N,1:N); 132*> If IPIV(k) = k, no interchange occurred. 133*> 134*> b) A pair of consecutive negative entries 135*> IPIV(k) < 0 and IPIV(k-1) < 0 means: 136*> D(k-1:k,k-1:k) is a 2-by-2 diagonal block. 137*> (NOTE: negative entries in IPIV appear ONLY in pairs). 138*> 1) If -IPIV(k) != k, rows and columns 139*> k and -IPIV(k) were interchanged 140*> in the matrix A(1:N,1:N). 141*> If -IPIV(k) = k, no interchange occurred. 142*> 2) If -IPIV(k-1) != k-1, rows and columns 143*> k-1 and -IPIV(k-1) were interchanged 144*> in the matrix A(1:N,1:N). 145*> If -IPIV(k-1) = k-1, no interchange occurred. 146*> 147*> c) In both cases a) and b), always ABS( IPIV(k) ) <= k. 148*> 149*> d) NOTE: Any entry IPIV(k) is always NONZERO on output. 150*> 151*> If UPLO = 'L', 152*> ( in factorization order, k increases from 1 to N ): 153*> a) A single positive entry IPIV(k) > 0 means: 154*> D(k,k) is a 1-by-1 diagonal block. 155*> If IPIV(k) != k, rows and columns k and IPIV(k) were 156*> interchanged in the matrix A(1:N,1:N). 157*> If IPIV(k) = k, no interchange occurred. 158*> 159*> b) A pair of consecutive negative entries 160*> IPIV(k) < 0 and IPIV(k+1) < 0 means: 161*> D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 162*> (NOTE: negative entries in IPIV appear ONLY in pairs). 163*> 1) If -IPIV(k) != k, rows and columns 164*> k and -IPIV(k) were interchanged 165*> in the matrix A(1:N,1:N). 166*> If -IPIV(k) = k, no interchange occurred. 167*> 2) If -IPIV(k+1) != k+1, rows and columns 168*> k-1 and -IPIV(k-1) were interchanged 169*> in the matrix A(1:N,1:N). 170*> If -IPIV(k+1) = k+1, no interchange occurred. 171*> 172*> c) In both cases a) and b), always ABS( IPIV(k) ) >= k. 173*> 174*> d) NOTE: Any entry IPIV(k) is always NONZERO on output. 175*> \endverbatim 176*> 177*> \param[out] INFO 178*> \verbatim 179*> INFO is INTEGER 180*> = 0: successful exit 181*> 182*> < 0: If INFO = -k, the k-th argument had an illegal value 183*> 184*> > 0: If INFO = k, the matrix A is singular, because: 185*> If UPLO = 'U': column k in the upper 186*> triangular part of A contains all zeros. 187*> If UPLO = 'L': column k in the lower 188*> triangular part of A contains all zeros. 189*> 190*> Therefore D(k,k) is exactly zero, and superdiagonal 191*> elements of column k of U (or subdiagonal elements of 192*> column k of L ) are all zeros. The factorization has 193*> been completed, but the block diagonal matrix D is 194*> exactly singular, and division by zero will occur if 195*> it is used to solve a system of equations. 196*> 197*> NOTE: INFO only stores the first occurrence of 198*> a singularity, any subsequent occurrence of singularity 199*> is not stored in INFO even though the factorization 200*> always completes. 201*> \endverbatim 202* 203* Authors: 204* ======== 205* 206*> \author Univ. of Tennessee 207*> \author Univ. of California Berkeley 208*> \author Univ. of Colorado Denver 209*> \author NAG Ltd. 210* 211*> \date December 2016 212* 213*> \ingroup complexHEcomputational 214* 215*> \par Further Details: 216* ===================== 217*> 218*> \verbatim 219*> TODO: put further details 220*> \endverbatim 221* 222*> \par Contributors: 223* ================== 224*> 225*> \verbatim 226*> 227*> December 2016, Igor Kozachenko, 228*> Computer Science Division, 229*> University of California, Berkeley 230*> 231*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas, 232*> School of Mathematics, 233*> University of Manchester 234*> 235*> 01-01-96 - Based on modifications by 236*> J. Lewis, Boeing Computer Services Company 237*> A. Petitet, Computer Science Dept., 238*> Univ. of Tenn., Knoxville abd , USA 239*> \endverbatim 240* 241* ===================================================================== 242 SUBROUTINE CHETF2_RK( UPLO, N, A, LDA, E, IPIV, INFO ) 243* 244* -- LAPACK computational routine (version 3.7.0) -- 245* -- LAPACK is a software package provided by Univ. of Tennessee, -- 246* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 247* December 2016 248* 249* .. Scalar Arguments .. 250 CHARACTER UPLO 251 INTEGER INFO, LDA, N 252* .. 253* .. Array Arguments .. 254 INTEGER IPIV( * ) 255 COMPLEX A( LDA, * ), E( * ) 256* .. 257* 258* ====================================================================== 259* 260* .. Parameters .. 261 REAL ZERO, ONE 262 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 263 REAL EIGHT, SEVTEN 264 PARAMETER ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 ) 265 COMPLEX CZERO 266 PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ) ) 267* .. 268* .. Local Scalars .. 269 LOGICAL DONE, UPPER 270 INTEGER I, II, IMAX, ITEMP, J, JMAX, K, KK, KP, KSTEP, 271 $ P 272 REAL ABSAKK, ALPHA, COLMAX, D, D11, D22, R1, STEMP, 273 $ ROWMAX, TT, SFMIN 274 COMPLEX D12, D21, T, WK, WKM1, WKP1, Z 275* .. 276* .. External Functions .. 277* 278 LOGICAL LSAME 279 INTEGER ICAMAX 280 REAL SLAMCH, SLAPY2 281 EXTERNAL LSAME, ICAMAX, SLAMCH, SLAPY2 282* .. 283* .. External Subroutines .. 284 EXTERNAL XERBLA, CSSCAL, CHER, CSWAP 285* .. 286* .. Intrinsic Functions .. 287 INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, REAL, SQRT 288* .. 289* .. Statement Functions .. 290 REAL CABS1 291* .. 292* .. Statement Function definitions .. 293 CABS1( Z ) = ABS( REAL( Z ) ) + ABS( AIMAG( Z ) ) 294* .. 295* .. Executable Statements .. 296* 297* Test the input parameters. 298* 299 INFO = 0 300 UPPER = LSAME( UPLO, 'U' ) 301 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 302 INFO = -1 303 ELSE IF( N.LT.0 ) THEN 304 INFO = -2 305 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 306 INFO = -4 307 END IF 308 IF( INFO.NE.0 ) THEN 309 CALL XERBLA( 'CHETF2_RK', -INFO ) 310 RETURN 311 END IF 312* 313* Initialize ALPHA for use in choosing pivot block size. 314* 315 ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT 316* 317* Compute machine safe minimum 318* 319 SFMIN = SLAMCH( 'S' ) 320* 321 IF( UPPER ) THEN 322* 323* Factorize A as U*D*U**H using the upper triangle of A 324* 325* Initialize the first entry of array E, where superdiagonal 326* elements of D are stored 327* 328 E( 1 ) = CZERO 329* 330* K is the main loop index, decreasing from N to 1 in steps of 331* 1 or 2 332* 333 K = N 334 10 CONTINUE 335* 336* If K < 1, exit from loop 337* 338 IF( K.LT.1 ) 339 $ GO TO 34 340 KSTEP = 1 341 P = K 342* 343* Determine rows and columns to be interchanged and whether 344* a 1-by-1 or 2-by-2 pivot block will be used 345* 346 ABSAKK = ABS( REAL( A( K, K ) ) ) 347* 348* IMAX is the row-index of the largest off-diagonal element in 349* column K, and COLMAX is its absolute value. 350* Determine both COLMAX and IMAX. 351* 352 IF( K.GT.1 ) THEN 353 IMAX = ICAMAX( K-1, A( 1, K ), 1 ) 354 COLMAX = CABS1( A( IMAX, K ) ) 355 ELSE 356 COLMAX = ZERO 357 END IF 358* 359 IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN 360* 361* Column K is zero or underflow: set INFO and continue 362* 363 IF( INFO.EQ.0 ) 364 $ INFO = K 365 KP = K 366 A( K, K ) = REAL( A( K, K ) ) 367* 368* Set E( K ) to zero 369* 370 IF( K.GT.1 ) 371 $ E( K ) = CZERO 372* 373 ELSE 374* 375* ============================================================ 376* 377* BEGIN pivot search 378* 379* Case(1) 380* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX 381* (used to handle NaN and Inf) 382* 383 IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN 384* 385* no interchange, use 1-by-1 pivot block 386* 387 KP = K 388* 389 ELSE 390* 391 DONE = .FALSE. 392* 393* Loop until pivot found 394* 395 12 CONTINUE 396* 397* BEGIN pivot search loop body 398* 399* 400* JMAX is the column-index of the largest off-diagonal 401* element in row IMAX, and ROWMAX is its absolute value. 402* Determine both ROWMAX and JMAX. 403* 404 IF( IMAX.NE.K ) THEN 405 JMAX = IMAX + ICAMAX( K-IMAX, A( IMAX, IMAX+1 ), 406 $ LDA ) 407 ROWMAX = CABS1( A( IMAX, JMAX ) ) 408 ELSE 409 ROWMAX = ZERO 410 END IF 411* 412 IF( IMAX.GT.1 ) THEN 413 ITEMP = ICAMAX( IMAX-1, A( 1, IMAX ), 1 ) 414 STEMP = CABS1( A( ITEMP, IMAX ) ) 415 IF( STEMP.GT.ROWMAX ) THEN 416 ROWMAX = STEMP 417 JMAX = ITEMP 418 END IF 419 END IF 420* 421* Case(2) 422* Equivalent to testing for 423* ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX 424* (used to handle NaN and Inf) 425* 426 IF( .NOT.( ABS( REAL( A( IMAX, IMAX ) ) ) 427 $ .LT.ALPHA*ROWMAX ) ) THEN 428* 429* interchange rows and columns K and IMAX, 430* use 1-by-1 pivot block 431* 432 KP = IMAX 433 DONE = .TRUE. 434* 435* Case(3) 436* Equivalent to testing for ROWMAX.EQ.COLMAX, 437* (used to handle NaN and Inf) 438* 439 ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) ) 440 $ THEN 441* 442* interchange rows and columns K-1 and IMAX, 443* use 2-by-2 pivot block 444* 445 KP = IMAX 446 KSTEP = 2 447 DONE = .TRUE. 448* 449* Case(4) 450 ELSE 451* 452* Pivot not found: set params and repeat 453* 454 P = IMAX 455 COLMAX = ROWMAX 456 IMAX = JMAX 457 END IF 458* 459* END pivot search loop body 460* 461 IF( .NOT.DONE ) GOTO 12 462* 463 END IF 464* 465* END pivot search 466* 467* ============================================================ 468* 469* KK is the column of A where pivoting step stopped 470* 471 KK = K - KSTEP + 1 472* 473* For only a 2x2 pivot, interchange rows and columns K and P 474* in the leading submatrix A(1:k,1:k) 475* 476 IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN 477* (1) Swap columnar parts 478 IF( P.GT.1 ) 479 $ CALL CSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 ) 480* (2) Swap and conjugate middle parts 481 DO 14 J = P + 1, K - 1 482 T = CONJG( A( J, K ) ) 483 A( J, K ) = CONJG( A( P, J ) ) 484 A( P, J ) = T 485 14 CONTINUE 486* (3) Swap and conjugate corner elements at row-col interserction 487 A( P, K ) = CONJG( A( P, K ) ) 488* (4) Swap diagonal elements at row-col intersection 489 R1 = REAL( A( K, K ) ) 490 A( K, K ) = REAL( A( P, P ) ) 491 A( P, P ) = R1 492* 493* Convert upper triangle of A into U form by applying 494* the interchanges in columns k+1:N. 495* 496 IF( K.LT.N ) 497 $ CALL CSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ), LDA ) 498* 499 END IF 500* 501* For both 1x1 and 2x2 pivots, interchange rows and 502* columns KK and KP in the leading submatrix A(1:k,1:k) 503* 504 IF( KP.NE.KK ) THEN 505* (1) Swap columnar parts 506 IF( KP.GT.1 ) 507 $ CALL CSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 ) 508* (2) Swap and conjugate middle parts 509 DO 15 J = KP + 1, KK - 1 510 T = CONJG( A( J, KK ) ) 511 A( J, KK ) = CONJG( A( KP, J ) ) 512 A( KP, J ) = T 513 15 CONTINUE 514* (3) Swap and conjugate corner elements at row-col interserction 515 A( KP, KK ) = CONJG( A( KP, KK ) ) 516* (4) Swap diagonal elements at row-col intersection 517 R1 = REAL( A( KK, KK ) ) 518 A( KK, KK ) = REAL( A( KP, KP ) ) 519 A( KP, KP ) = R1 520* 521 IF( KSTEP.EQ.2 ) THEN 522* (*) Make sure that diagonal element of pivot is real 523 A( K, K ) = REAL( A( K, K ) ) 524* (5) Swap row elements 525 T = A( K-1, K ) 526 A( K-1, K ) = A( KP, K ) 527 A( KP, K ) = T 528 END IF 529* 530* Convert upper triangle of A into U form by applying 531* the interchanges in columns k+1:N. 532* 533 IF( K.LT.N ) 534 $ CALL CSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ), 535 $ LDA ) 536* 537 ELSE 538* (*) Make sure that diagonal element of pivot is real 539 A( K, K ) = REAL( A( K, K ) ) 540 IF( KSTEP.EQ.2 ) 541 $ A( K-1, K-1 ) = REAL( A( K-1, K-1 ) ) 542 END IF 543* 544* Update the leading submatrix 545* 546 IF( KSTEP.EQ.1 ) THEN 547* 548* 1-by-1 pivot block D(k): column k now holds 549* 550* W(k) = U(k)*D(k) 551* 552* where U(k) is the k-th column of U 553* 554 IF( K.GT.1 ) THEN 555* 556* Perform a rank-1 update of A(1:k-1,1:k-1) and 557* store U(k) in column k 558* 559 IF( ABS( REAL( A( K, K ) ) ).GE.SFMIN ) THEN 560* 561* Perform a rank-1 update of A(1:k-1,1:k-1) as 562* A := A - U(k)*D(k)*U(k)**T 563* = A - W(k)*1/D(k)*W(k)**T 564* 565 D11 = ONE / REAL( A( K, K ) ) 566 CALL CHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA ) 567* 568* Store U(k) in column k 569* 570 CALL CSSCAL( K-1, D11, A( 1, K ), 1 ) 571 ELSE 572* 573* Store L(k) in column K 574* 575 D11 = REAL( A( K, K ) ) 576 DO 16 II = 1, K - 1 577 A( II, K ) = A( II, K ) / D11 578 16 CONTINUE 579* 580* Perform a rank-1 update of A(k+1:n,k+1:n) as 581* A := A - U(k)*D(k)*U(k)**T 582* = A - W(k)*(1/D(k))*W(k)**T 583* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T 584* 585 CALL CHER( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA ) 586 END IF 587* 588* Store the superdiagonal element of D in array E 589* 590 E( K ) = CZERO 591* 592 END IF 593* 594 ELSE 595* 596* 2-by-2 pivot block D(k): columns k and k-1 now hold 597* 598* ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k) 599* 600* where U(k) and U(k-1) are the k-th and (k-1)-th columns 601* of U 602* 603* Perform a rank-2 update of A(1:k-2,1:k-2) as 604* 605* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T 606* = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T 607* 608* and store L(k) and L(k+1) in columns k and k+1 609* 610 IF( K.GT.2 ) THEN 611* D = |A12| 612 D = SLAPY2( REAL( A( K-1, K ) ), 613 $ AIMAG( A( K-1, K ) ) ) 614 D11 = A( K, K ) / D 615 D22 = A( K-1, K-1 ) / D 616 D12 = A( K-1, K ) / D 617 TT = ONE / ( D11*D22-ONE ) 618* 619 DO 30 J = K - 2, 1, -1 620* 621* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J 622* 623 WKM1 = TT*( D11*A( J, K-1 )-CONJG( D12 )* 624 $ A( J, K ) ) 625 WK = TT*( D22*A( J, K )-D12*A( J, K-1 ) ) 626* 627* Perform a rank-2 update of A(1:k-2,1:k-2) 628* 629 DO 20 I = J, 1, -1 630 A( I, J ) = A( I, J ) - 631 $ ( A( I, K ) / D )*CONJG( WK ) - 632 $ ( A( I, K-1 ) / D )*CONJG( WKM1 ) 633 20 CONTINUE 634* 635* Store U(k) and U(k-1) in cols k and k-1 for row J 636* 637 A( J, K ) = WK / D 638 A( J, K-1 ) = WKM1 / D 639* (*) Make sure that diagonal element of pivot is real 640 A( J, J ) = CMPLX( REAL( A( J, J ) ), ZERO ) 641* 642 30 CONTINUE 643* 644 END IF 645* 646* Copy superdiagonal elements of D(K) to E(K) and 647* ZERO out superdiagonal entry of A 648* 649 E( K ) = A( K-1, K ) 650 E( K-1 ) = CZERO 651 A( K-1, K ) = CZERO 652* 653 END IF 654* 655* End column K is nonsingular 656* 657 END IF 658* 659* Store details of the interchanges in IPIV 660* 661 IF( KSTEP.EQ.1 ) THEN 662 IPIV( K ) = KP 663 ELSE 664 IPIV( K ) = -P 665 IPIV( K-1 ) = -KP 666 END IF 667* 668* Decrease K and return to the start of the main loop 669* 670 K = K - KSTEP 671 GO TO 10 672* 673 34 CONTINUE 674* 675 ELSE 676* 677* Factorize A as L*D*L**H using the lower triangle of A 678* 679* Initialize the unused last entry of the subdiagonal array E. 680* 681 E( N ) = CZERO 682* 683* K is the main loop index, increasing from 1 to N in steps of 684* 1 or 2 685* 686 K = 1 687 40 CONTINUE 688* 689* If K > N, exit from loop 690* 691 IF( K.GT.N ) 692 $ GO TO 64 693 KSTEP = 1 694 P = K 695* 696* Determine rows and columns to be interchanged and whether 697* a 1-by-1 or 2-by-2 pivot block will be used 698* 699 ABSAKK = ABS( REAL( A( K, K ) ) ) 700* 701* IMAX is the row-index of the largest off-diagonal element in 702* column K, and COLMAX is its absolute value. 703* Determine both COLMAX and IMAX. 704* 705 IF( K.LT.N ) THEN 706 IMAX = K + ICAMAX( N-K, A( K+1, K ), 1 ) 707 COLMAX = CABS1( A( IMAX, K ) ) 708 ELSE 709 COLMAX = ZERO 710 END IF 711* 712 IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN 713* 714* Column K is zero or underflow: set INFO and continue 715* 716 IF( INFO.EQ.0 ) 717 $ INFO = K 718 KP = K 719 A( K, K ) = REAL( A( K, K ) ) 720* 721* Set E( K ) to zero 722* 723 IF( K.LT.N ) 724 $ E( K ) = CZERO 725* 726 ELSE 727* 728* ============================================================ 729* 730* BEGIN pivot search 731* 732* Case(1) 733* Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX 734* (used to handle NaN and Inf) 735* 736 IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN 737* 738* no interchange, use 1-by-1 pivot block 739* 740 KP = K 741* 742 ELSE 743* 744 DONE = .FALSE. 745* 746* Loop until pivot found 747* 748 42 CONTINUE 749* 750* BEGIN pivot search loop body 751* 752* 753* JMAX is the column-index of the largest off-diagonal 754* element in row IMAX, and ROWMAX is its absolute value. 755* Determine both ROWMAX and JMAX. 756* 757 IF( IMAX.NE.K ) THEN 758 JMAX = K - 1 + ICAMAX( IMAX-K, A( IMAX, K ), LDA ) 759 ROWMAX = CABS1( A( IMAX, JMAX ) ) 760 ELSE 761 ROWMAX = ZERO 762 END IF 763* 764 IF( IMAX.LT.N ) THEN 765 ITEMP = IMAX + ICAMAX( N-IMAX, A( IMAX+1, IMAX ), 766 $ 1 ) 767 STEMP = CABS1( A( ITEMP, IMAX ) ) 768 IF( STEMP.GT.ROWMAX ) THEN 769 ROWMAX = STEMP 770 JMAX = ITEMP 771 END IF 772 END IF 773* 774* Case(2) 775* Equivalent to testing for 776* ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX 777* (used to handle NaN and Inf) 778* 779 IF( .NOT.( ABS( REAL( A( IMAX, IMAX ) ) ) 780 $ .LT.ALPHA*ROWMAX ) ) THEN 781* 782* interchange rows and columns K and IMAX, 783* use 1-by-1 pivot block 784* 785 KP = IMAX 786 DONE = .TRUE. 787* 788* Case(3) 789* Equivalent to testing for ROWMAX.EQ.COLMAX, 790* (used to handle NaN and Inf) 791* 792 ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) ) 793 $ THEN 794* 795* interchange rows and columns K+1 and IMAX, 796* use 2-by-2 pivot block 797* 798 KP = IMAX 799 KSTEP = 2 800 DONE = .TRUE. 801* 802* Case(4) 803 ELSE 804* 805* Pivot not found: set params and repeat 806* 807 P = IMAX 808 COLMAX = ROWMAX 809 IMAX = JMAX 810 END IF 811* 812* 813* END pivot search loop body 814* 815 IF( .NOT.DONE ) GOTO 42 816* 817 END IF 818* 819* END pivot search 820* 821* ============================================================ 822* 823* KK is the column of A where pivoting step stopped 824* 825 KK = K + KSTEP - 1 826* 827* For only a 2x2 pivot, interchange rows and columns K and P 828* in the trailing submatrix A(k:n,k:n) 829* 830 IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN 831* (1) Swap columnar parts 832 IF( P.LT.N ) 833 $ CALL CSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 ) 834* (2) Swap and conjugate middle parts 835 DO 44 J = K + 1, P - 1 836 T = CONJG( A( J, K ) ) 837 A( J, K ) = CONJG( A( P, J ) ) 838 A( P, J ) = T 839 44 CONTINUE 840* (3) Swap and conjugate corner elements at row-col interserction 841 A( P, K ) = CONJG( A( P, K ) ) 842* (4) Swap diagonal elements at row-col intersection 843 R1 = REAL( A( K, K ) ) 844 A( K, K ) = REAL( A( P, P ) ) 845 A( P, P ) = R1 846* 847* Convert lower triangle of A into L form by applying 848* the interchanges in columns 1:k-1. 849* 850 IF ( K.GT.1 ) 851 $ CALL CSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA ) 852* 853 END IF 854* 855* For both 1x1 and 2x2 pivots, interchange rows and 856* columns KK and KP in the trailing submatrix A(k:n,k:n) 857* 858 IF( KP.NE.KK ) THEN 859* (1) Swap columnar parts 860 IF( KP.LT.N ) 861 $ CALL CSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 ) 862* (2) Swap and conjugate middle parts 863 DO 45 J = KK + 1, KP - 1 864 T = CONJG( A( J, KK ) ) 865 A( J, KK ) = CONJG( A( KP, J ) ) 866 A( KP, J ) = T 867 45 CONTINUE 868* (3) Swap and conjugate corner elements at row-col interserction 869 A( KP, KK ) = CONJG( A( KP, KK ) ) 870* (4) Swap diagonal elements at row-col intersection 871 R1 = REAL( A( KK, KK ) ) 872 A( KK, KK ) = REAL( A( KP, KP ) ) 873 A( KP, KP ) = R1 874* 875 IF( KSTEP.EQ.2 ) THEN 876* (*) Make sure that diagonal element of pivot is real 877 A( K, K ) = REAL( A( K, K ) ) 878* (5) Swap row elements 879 T = A( K+1, K ) 880 A( K+1, K ) = A( KP, K ) 881 A( KP, K ) = T 882 END IF 883* 884* Convert lower triangle of A into L form by applying 885* the interchanges in columns 1:k-1. 886* 887 IF ( K.GT.1 ) 888 $ CALL CSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA ) 889* 890 ELSE 891* (*) Make sure that diagonal element of pivot is real 892 A( K, K ) = REAL( A( K, K ) ) 893 IF( KSTEP.EQ.2 ) 894 $ A( K+1, K+1 ) = REAL( A( K+1, K+1 ) ) 895 END IF 896* 897* Update the trailing submatrix 898* 899 IF( KSTEP.EQ.1 ) THEN 900* 901* 1-by-1 pivot block D(k): column k of A now holds 902* 903* W(k) = L(k)*D(k), 904* 905* where L(k) is the k-th column of L 906* 907 IF( K.LT.N ) THEN 908* 909* Perform a rank-1 update of A(k+1:n,k+1:n) and 910* store L(k) in column k 911* 912* Handle division by a small number 913* 914 IF( ABS( REAL( A( K, K ) ) ).GE.SFMIN ) THEN 915* 916* Perform a rank-1 update of A(k+1:n,k+1:n) as 917* A := A - L(k)*D(k)*L(k)**T 918* = A - W(k)*(1/D(k))*W(k)**T 919* 920 D11 = ONE / REAL( A( K, K ) ) 921 CALL CHER( UPLO, N-K, -D11, A( K+1, K ), 1, 922 $ A( K+1, K+1 ), LDA ) 923* 924* Store L(k) in column k 925* 926 CALL CSSCAL( N-K, D11, A( K+1, K ), 1 ) 927 ELSE 928* 929* Store L(k) in column k 930* 931 D11 = REAL( A( K, K ) ) 932 DO 46 II = K + 1, N 933 A( II, K ) = A( II, K ) / D11 934 46 CONTINUE 935* 936* Perform a rank-1 update of A(k+1:n,k+1:n) as 937* A := A - L(k)*D(k)*L(k)**T 938* = A - W(k)*(1/D(k))*W(k)**T 939* = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T 940* 941 CALL CHER( UPLO, N-K, -D11, A( K+1, K ), 1, 942 $ A( K+1, K+1 ), LDA ) 943 END IF 944* 945* Store the subdiagonal element of D in array E 946* 947 E( K ) = CZERO 948* 949 END IF 950* 951 ELSE 952* 953* 2-by-2 pivot block D(k): columns k and k+1 now hold 954* 955* ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k) 956* 957* where L(k) and L(k+1) are the k-th and (k+1)-th columns 958* of L 959* 960* 961* Perform a rank-2 update of A(k+2:n,k+2:n) as 962* 963* A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T 964* = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T 965* 966* and store L(k) and L(k+1) in columns k and k+1 967* 968 IF( K.LT.N-1 ) THEN 969* D = |A21| 970 D = SLAPY2( REAL( A( K+1, K ) ), 971 $ AIMAG( A( K+1, K ) ) ) 972 D11 = REAL( A( K+1, K+1 ) ) / D 973 D22 = REAL( A( K, K ) ) / D 974 D21 = A( K+1, K ) / D 975 TT = ONE / ( D11*D22-ONE ) 976* 977 DO 60 J = K + 2, N 978* 979* Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J 980* 981 WK = TT*( D11*A( J, K )-D21*A( J, K+1 ) ) 982 WKP1 = TT*( D22*A( J, K+1 )-CONJG( D21 )* 983 $ A( J, K ) ) 984* 985* Perform a rank-2 update of A(k+2:n,k+2:n) 986* 987 DO 50 I = J, N 988 A( I, J ) = A( I, J ) - 989 $ ( A( I, K ) / D )*CONJG( WK ) - 990 $ ( A( I, K+1 ) / D )*CONJG( WKP1 ) 991 50 CONTINUE 992* 993* Store L(k) and L(k+1) in cols k and k+1 for row J 994* 995 A( J, K ) = WK / D 996 A( J, K+1 ) = WKP1 / D 997* (*) Make sure that diagonal element of pivot is real 998 A( J, J ) = CMPLX( REAL( A( J, J ) ), ZERO ) 999* 1000 60 CONTINUE 1001* 1002 END IF 1003* 1004* Copy subdiagonal elements of D(K) to E(K) and 1005* ZERO out subdiagonal entry of A 1006* 1007 E( K ) = A( K+1, K ) 1008 E( K+1 ) = CZERO 1009 A( K+1, K ) = CZERO 1010* 1011 END IF 1012* 1013* End column K is nonsingular 1014* 1015 END IF 1016* 1017* Store details of the interchanges in IPIV 1018* 1019 IF( KSTEP.EQ.1 ) THEN 1020 IPIV( K ) = KP 1021 ELSE 1022 IPIV( K ) = -P 1023 IPIV( K+1 ) = -KP 1024 END IF 1025* 1026* Increase K and return to the start of the main loop 1027* 1028 K = K + KSTEP 1029 GO TO 40 1030* 1031 64 CONTINUE 1032* 1033 END IF 1034* 1035 RETURN 1036* 1037* End of CHETF2_RK 1038* 1039 END 1040