Annotation of rpl/lapack/lapack/zlahef_rook.f, revision 1.1
1.1 ! bertrand 1: * \brief \b ZLAHEF_ROOK computes a partial factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (blocked algorithm, calling Level 3 BLAS).
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZLAHEF_ROOK + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlahef_rook.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahef_rook.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahef_rook.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZLAHEF_ROOK( 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: * COMPLEX*16 A( LDA, * ), W( LDW, * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> ZLAHEF_ROOK computes a partial factorization of a complex Hermitian
! 39: *> matrix A using the bounded Bunch-Kaufman ("rook") diagonal pivoting
! 40: *> method. The partial factorization has the form:
! 41: *>
! 42: *> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
! 43: *> ( 0 U22 ) ( 0 D ) ( U12**H U22**H )
! 44: *>
! 45: *> A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) 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: *> Note that U**H denotes the conjugate transpose of U.
! 51: *>
! 52: *> ZLAHEF_ROOK is an auxiliary routine called by ZHETRF_ROOK. It uses
! 53: *> blocked code (calling Level 3 BLAS) to update the submatrix
! 54: *> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
! 55: *> \endverbatim
! 56: *
! 57: * Arguments:
! 58: * ==========
! 59: *
! 60: *> \param[in] UPLO
! 61: *> \verbatim
! 62: *> UPLO is CHARACTER*1
! 63: *> Specifies whether the upper or lower triangular part of the
! 64: *> Hermitian matrix A is stored:
! 65: *> = 'U': Upper triangular
! 66: *> = 'L': Lower triangular
! 67: *> \endverbatim
! 68: *>
! 69: *> \param[in] N
! 70: *> \verbatim
! 71: *> N is INTEGER
! 72: *> The order of the matrix A. N >= 0.
! 73: *> \endverbatim
! 74: *>
! 75: *> \param[in] NB
! 76: *> \verbatim
! 77: *> NB is INTEGER
! 78: *> The maximum number of columns of the matrix A that should be
! 79: *> factored. NB should be at least 2 to allow for 2-by-2 pivot
! 80: *> blocks.
! 81: *> \endverbatim
! 82: *>
! 83: *> \param[out] KB
! 84: *> \verbatim
! 85: *> KB is INTEGER
! 86: *> The number of columns of A that were actually factored.
! 87: *> KB is either NB-1 or NB, or N if N <= NB.
! 88: *> \endverbatim
! 89: *>
! 90: *> \param[in,out] A
! 91: *> \verbatim
! 92: *> A is COMPLEX*16 array, dimension (LDA,N)
! 93: *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
! 94: *> n-by-n upper triangular part of A contains the upper
! 95: *> triangular part of the matrix A, and the strictly lower
! 96: *> triangular part of A is not referenced. If UPLO = 'L', the
! 97: *> leading n-by-n lower triangular part of A contains the lower
! 98: *> triangular part of the matrix A, and the strictly upper
! 99: *> triangular part of A is not referenced.
! 100: *> On exit, A contains details of the partial factorization.
! 101: *> \endverbatim
! 102: *>
! 103: *> \param[in] LDA
! 104: *> \verbatim
! 105: *> LDA is INTEGER
! 106: *> The leading dimension of the array A. LDA >= max(1,N).
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[out] IPIV
! 110: *> \verbatim
! 111: *> IPIV is INTEGER array, dimension (N)
! 112: *> Details of the interchanges and the block structure of D.
! 113: *>
! 114: *> If UPLO = 'U':
! 115: *> Only the last KB elements of IPIV are set.
! 116: *>
! 117: *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
! 118: *> interchanged and D(k,k) is a 1-by-1 diagonal block.
! 119: *>
! 120: *> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
! 121: *> columns k and -IPIV(k) were interchanged and rows and
! 122: *> columns k-1 and -IPIV(k-1) were inerchaged,
! 123: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
! 124: *>
! 125: *> If UPLO = 'L':
! 126: *> Only the first KB elements of IPIV are set.
! 127: *>
! 128: *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
! 129: *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
! 130: *>
! 131: *> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
! 132: *> columns k and -IPIV(k) were interchanged and rows and
! 133: *> columns k+1 and -IPIV(k+1) were inerchaged,
! 134: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
! 135: *> \endverbatim
! 136: *>
! 137: *> \param[out] W
! 138: *> \verbatim
! 139: *> W is COMPLEX*16 array, dimension (LDW,NB)
! 140: *> \endverbatim
! 141: *>
! 142: *> \param[in] LDW
! 143: *> \verbatim
! 144: *> LDW is INTEGER
! 145: *> The leading dimension of the array W. LDW >= max(1,N).
! 146: *> \endverbatim
! 147: *>
! 148: *> \param[out] INFO
! 149: *> \verbatim
! 150: *> INFO is INTEGER
! 151: *> = 0: successful exit
! 152: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
! 153: *> has been completed, but the block diagonal matrix D is
! 154: *> exactly singular.
! 155: *> \endverbatim
! 156: *
! 157: * Authors:
! 158: * ========
! 159: *
! 160: *> \author Univ. of Tennessee
! 161: *> \author Univ. of California Berkeley
! 162: *> \author Univ. of Colorado Denver
! 163: *> \author NAG Ltd.
! 164: *
! 165: *> \date November 2013
! 166: *
! 167: *> \ingroup complex16HEcomputational
! 168: *
! 169: *> \par Contributors:
! 170: * ==================
! 171: *>
! 172: *> \verbatim
! 173: *>
! 174: *> November 2013, Igor Kozachenko,
! 175: *> Computer Science Division,
! 176: *> University of California, Berkeley
! 177: *>
! 178: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
! 179: *> School of Mathematics,
! 180: *> University of Manchester
! 181: *> \endverbatim
! 182: *
! 183: * =====================================================================
! 184: SUBROUTINE ZLAHEF_ROOK( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
! 185: $ INFO )
! 186: *
! 187: * -- LAPACK computational routine (version 3.5.0) --
! 188: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 189: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 190: * November 2013
! 191: *
! 192: * .. Scalar Arguments ..
! 193: CHARACTER UPLO
! 194: INTEGER INFO, KB, LDA, LDW, N, NB
! 195: * ..
! 196: * .. Array Arguments ..
! 197: INTEGER IPIV( * )
! 198: COMPLEX*16 A( LDA, * ), W( LDW, * )
! 199: * ..
! 200: *
! 201: * =====================================================================
! 202: *
! 203: * .. Parameters ..
! 204: DOUBLE PRECISION ZERO, ONE
! 205: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 206: COMPLEX*16 CONE
! 207: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
! 208: DOUBLE PRECISION EIGHT, SEVTEN
! 209: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
! 210: * ..
! 211: * .. Local Scalars ..
! 212: LOGICAL DONE
! 213: INTEGER IMAX, ITEMP, II, J, JB, JJ, JMAX, JP1, JP2, K,
! 214: $ KK, KKW, KP, KSTEP, KW, P
! 215: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, DTEMP, R1, ROWMAX, T,
! 216: $ SFMIN
! 217: COMPLEX*16 D11, D21, D22, Z
! 218: * ..
! 219: * .. External Functions ..
! 220: LOGICAL LSAME
! 221: INTEGER IZAMAX
! 222: DOUBLE PRECISION DLAMCH
! 223: EXTERNAL LSAME, IZAMAX, DLAMCH
! 224: * ..
! 225: * .. External Subroutines ..
! 226: EXTERNAL ZCOPY, ZDSCAL, ZGEMM, ZGEMV, ZLACGV, ZSWAP
! 227: * ..
! 228: * .. Intrinsic Functions ..
! 229: INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT
! 230: * ..
! 231: * .. Statement Functions ..
! 232: DOUBLE PRECISION CABS1
! 233: * ..
! 234: * .. Statement Function definitions ..
! 235: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
! 236: * ..
! 237: * .. Executable Statements ..
! 238: *
! 239: INFO = 0
! 240: *
! 241: * Initialize ALPHA for use in choosing pivot block size.
! 242: *
! 243: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
! 244: *
! 245: * Compute machine safe minimum
! 246: *
! 247: SFMIN = DLAMCH( 'S' )
! 248: *
! 249: IF( LSAME( UPLO, 'U' ) ) THEN
! 250: *
! 251: * Factorize the trailing columns of A using the upper triangle
! 252: * of A and working backwards, and compute the matrix W = U12*D
! 253: * for use in updating A11 (note that conjg(W) is actually stored)
! 254: *
! 255: * K is the main loop index, decreasing from N in steps of 1 or 2
! 256: *
! 257: K = N
! 258: 10 CONTINUE
! 259: *
! 260: * KW is the column of W which corresponds to column K of A
! 261: *
! 262: KW = NB + K - N
! 263: *
! 264: * Exit from loop
! 265: *
! 266: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
! 267: $ GO TO 30
! 268: *
! 269: KSTEP = 1
! 270: P = K
! 271: *
! 272: * Copy column K of A to column KW of W and update it
! 273: *
! 274: IF( K.GT.1 )
! 275: $ CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 )
! 276: W( K, KW ) = DBLE( A( K, K ) )
! 277: IF( K.LT.N ) THEN
! 278: CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
! 279: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
! 280: W( K, KW ) = DBLE( W( K, KW ) )
! 281: END IF
! 282: *
! 283: * Determine rows and columns to be interchanged and whether
! 284: * a 1-by-1 or 2-by-2 pivot block will be used
! 285: *
! 286: ABSAKK = ABS( DBLE( W( K, KW ) ) )
! 287: *
! 288: * IMAX is the row-index of the largest off-diagonal element in
! 289: * column K, and COLMAX is its absolute value.
! 290: * Determine both COLMAX and IMAX.
! 291: *
! 292: IF( K.GT.1 ) THEN
! 293: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
! 294: COLMAX = CABS1( W( IMAX, KW ) )
! 295: ELSE
! 296: COLMAX = ZERO
! 297: END IF
! 298: *
! 299: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 300: *
! 301: * Column K is zero or underflow: set INFO and continue
! 302: *
! 303: IF( INFO.EQ.0 )
! 304: $ INFO = K
! 305: KP = K
! 306: A( K, K ) = DBLE( W( K, KW ) )
! 307: IF( K.GT.1 )
! 308: $ CALL ZCOPY( K-1, W( 1, KW ), 1, A( 1, K ), 1 )
! 309: ELSE
! 310: *
! 311: * ============================================================
! 312: *
! 313: * BEGIN pivot search
! 314: *
! 315: * Case(1)
! 316: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 317: * (used to handle NaN and Inf)
! 318: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 319: *
! 320: * no interchange, use 1-by-1 pivot block
! 321: *
! 322: KP = K
! 323: *
! 324: ELSE
! 325: *
! 326: * Lop until pivot found
! 327: *
! 328: DONE = .FALSE.
! 329: *
! 330: 12 CONTINUE
! 331: *
! 332: * BEGIN pivot search loop body
! 333: *
! 334: *
! 335: * Copy column IMAX to column KW-1 of W and update it
! 336: *
! 337: IF( IMAX.GT.1 )
! 338: $ CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ),
! 339: $ 1 )
! 340: W( IMAX, KW-1 ) = DBLE( A( IMAX, IMAX ) )
! 341: *
! 342: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
! 343: $ W( IMAX+1, KW-1 ), 1 )
! 344: CALL ZLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 )
! 345: *
! 346: IF( K.LT.N ) THEN
! 347: CALL ZGEMV( 'No transpose', K, N-K, -CONE,
! 348: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
! 349: $ CONE, W( 1, KW-1 ), 1 )
! 350: W( IMAX, KW-1 ) = DBLE( W( IMAX, KW-1 ) )
! 351: END IF
! 352: *
! 353: * JMAX is the column-index of the largest off-diagonal
! 354: * element in row IMAX, and ROWMAX is its absolute value.
! 355: * Determine both ROWMAX and JMAX.
! 356: *
! 357: IF( IMAX.NE.K ) THEN
! 358: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
! 359: $ 1 )
! 360: ROWMAX = CABS1( W( JMAX, KW-1 ) )
! 361: ELSE
! 362: ROWMAX = ZERO
! 363: END IF
! 364: *
! 365: IF( IMAX.GT.1 ) THEN
! 366: ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
! 367: DTEMP = CABS1( W( ITEMP, KW-1 ) )
! 368: IF( DTEMP.GT.ROWMAX ) THEN
! 369: ROWMAX = DTEMP
! 370: JMAX = ITEMP
! 371: END IF
! 372: END IF
! 373: *
! 374: * Case(2)
! 375: * Equivalent to testing for
! 376: * ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
! 377: * (used to handle NaN and Inf)
! 378: *
! 379: IF( .NOT.( ABS( DBLE( W( IMAX,KW-1 ) ) )
! 380: $ .LT.ALPHA*ROWMAX ) ) THEN
! 381: *
! 382: * interchange rows and columns K and IMAX,
! 383: * use 1-by-1 pivot block
! 384: *
! 385: KP = IMAX
! 386: *
! 387: * copy column KW-1 of W to column KW of W
! 388: *
! 389: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
! 390: *
! 391: DONE = .TRUE.
! 392: *
! 393: * Case(3)
! 394: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 395: * (used to handle NaN and Inf)
! 396: *
! 397: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 398: $ THEN
! 399: *
! 400: * interchange rows and columns K-1 and IMAX,
! 401: * use 2-by-2 pivot block
! 402: *
! 403: KP = IMAX
! 404: KSTEP = 2
! 405: DONE = .TRUE.
! 406: *
! 407: * Case(4)
! 408: ELSE
! 409: *
! 410: * Pivot not found: set params and repeat
! 411: *
! 412: P = IMAX
! 413: COLMAX = ROWMAX
! 414: IMAX = JMAX
! 415: *
! 416: * Copy updated JMAXth (next IMAXth) column to Kth of W
! 417: *
! 418: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
! 419: *
! 420: END IF
! 421: *
! 422: *
! 423: * END pivot search loop body
! 424: *
! 425: IF( .NOT.DONE ) GOTO 12
! 426: *
! 427: END IF
! 428: *
! 429: * END pivot search
! 430: *
! 431: * ============================================================
! 432: *
! 433: * KK is the column of A where pivoting step stopped
! 434: *
! 435: KK = K - KSTEP + 1
! 436: *
! 437: * KKW is the column of W which corresponds to column KK of A
! 438: *
! 439: KKW = NB + KK - N
! 440: *
! 441: * Interchange rows and columns P and K.
! 442: * Updated column P is already stored in column KW of W.
! 443: *
! 444: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 445: *
! 446: * Copy non-updated column K to column P of submatrix A
! 447: * at step K. No need to copy element into columns
! 448: * K and K-1 of A for 2-by-2 pivot, since these columns
! 449: * will be later overwritten.
! 450: *
! 451: A( P, P ) = DBLE( A( K, K ) )
! 452: CALL ZCOPY( K-1-P, A( P+1, K ), 1, A( P, P+1 ),
! 453: $ LDA )
! 454: CALL ZLACGV( K-1-P, A( P, P+1 ), LDA )
! 455: IF( P.GT.1 )
! 456: $ CALL ZCOPY( P-1, A( 1, K ), 1, A( 1, P ), 1 )
! 457: *
! 458: * Interchange rows K and P in the last K+1 to N columns of A
! 459: * (columns K and K-1 of A for 2-by-2 pivot will be
! 460: * later overwritten). Interchange rows K and P
! 461: * in last KKW to NB columns of W.
! 462: *
! 463: IF( K.LT.N )
! 464: $ CALL ZSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ),
! 465: $ LDA )
! 466: CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ),
! 467: $ LDW )
! 468: END IF
! 469: *
! 470: * Interchange rows and columns KP and KK.
! 471: * Updated column KP is already stored in column KKW of W.
! 472: *
! 473: IF( KP.NE.KK ) THEN
! 474: *
! 475: * Copy non-updated column KK to column KP of submatrix A
! 476: * at step K. No need to copy element into column K
! 477: * (or K and K-1 for 2-by-2 pivot) of A, since these columns
! 478: * will be later overwritten.
! 479: *
! 480: A( KP, KP ) = DBLE( A( KK, KK ) )
! 481: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
! 482: $ LDA )
! 483: CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA )
! 484: IF( KP.GT.1 )
! 485: $ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
! 486: *
! 487: * Interchange rows KK and KP in last K+1 to N columns of A
! 488: * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
! 489: * later overwritten). Interchange rows KK and KP
! 490: * in last KKW to NB columns of W.
! 491: *
! 492: IF( K.LT.N )
! 493: $ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
! 494: $ LDA )
! 495: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
! 496: $ LDW )
! 497: END IF
! 498: *
! 499: IF( KSTEP.EQ.1 ) THEN
! 500: *
! 501: * 1-by-1 pivot block D(k): column kw of W now holds
! 502: *
! 503: * W(kw) = U(k)*D(k),
! 504: *
! 505: * where U(k) is the k-th column of U
! 506: *
! 507: * (1) Store subdiag. elements of column U(k)
! 508: * and 1-by-1 block D(k) in column k of A.
! 509: * (NOTE: Diagonal element U(k,k) is a UNIT element
! 510: * and not stored)
! 511: * A(k,k) := D(k,k) = W(k,kw)
! 512: * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
! 513: *
! 514: * (NOTE: No need to use for Hermitian matrix
! 515: * A( K, K ) = REAL( W( K, K) ) to separately copy diagonal
! 516: * element D(k,k) from W (potentially saves only one load))
! 517: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
! 518: IF( K.GT.1 ) THEN
! 519: *
! 520: * (NOTE: No need to check if A(k,k) is NOT ZERO,
! 521: * since that was ensured earlier in pivot search:
! 522: * case A(k,k) = 0 falls into 2x2 pivot case(3))
! 523: *
! 524: * Handle division by a small number
! 525: *
! 526: T = DBLE( A( K, K ) )
! 527: IF( ABS( T ).GE.SFMIN ) THEN
! 528: R1 = ONE / T
! 529: CALL ZDSCAL( K-1, R1, A( 1, K ), 1 )
! 530: ELSE
! 531: DO 14 II = 1, K-1
! 532: A( II, K ) = A( II, K ) / T
! 533: 14 CONTINUE
! 534: END IF
! 535: *
! 536: * (2) Conjugate column W(kw)
! 537: *
! 538: CALL ZLACGV( K-1, W( 1, KW ), 1 )
! 539: END IF
! 540: *
! 541: ELSE
! 542: *
! 543: * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
! 544: *
! 545: * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
! 546: *
! 547: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 548: * of U
! 549: *
! 550: * (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
! 551: * block D(k-1:k,k-1:k) in columns k-1 and k of A.
! 552: * (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
! 553: * block and not stored)
! 554: * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
! 555: * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
! 556: * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
! 557: *
! 558: IF( K.GT.2 ) THEN
! 559: *
! 560: * Factor out the columns of the inverse of 2-by-2 pivot
! 561: * block D, so that each column contains 1, to reduce the
! 562: * number of FLOPS when we multiply panel
! 563: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
! 564: *
! 565: * D**(-1) = ( d11 cj(d21) )**(-1) =
! 566: * ( d21 d22 )
! 567: *
! 568: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
! 569: * ( (-d21) ( d11 ) )
! 570: *
! 571: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
! 572: *
! 573: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
! 574: * ( ( -1 ) ( d11/conj(d21) ) )
! 575: *
! 576: * = 1/(|d21|**2) * 1/(D22*D11-1) *
! 577: *
! 578: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
! 579: * ( ( -1 ) ( D22 ) )
! 580: *
! 581: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
! 582: * ( ( -1 ) ( D22 ) )
! 583: *
! 584: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
! 585: * ( ( -1 ) ( D22 ) )
! 586: *
! 587: * Handle division by a small number. (NOTE: order of
! 588: * operations is important)
! 589: *
! 590: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
! 591: * ( (( -1 ) ) (( D22 ) ) ),
! 592: *
! 593: * where D11 = d22/d21,
! 594: * D22 = d11/conj(d21),
! 595: * D21 = d21,
! 596: * T = 1/(D22*D11-1).
! 597: *
! 598: * (NOTE: No need to check for division by ZERO,
! 599: * since that was ensured earlier in pivot search:
! 600: * (a) d21 != 0 in 2x2 pivot case(4),
! 601: * since |d21| should be larger than |d11| and |d22|;
! 602: * (b) (D22*D11 - 1) != 0, since from (a),
! 603: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
! 604: *
! 605: D21 = W( K-1, KW )
! 606: D11 = W( K, KW ) / DCONJG( D21 )
! 607: D22 = W( K-1, KW-1 ) / D21
! 608: T = ONE / ( DBLE( D11*D22 )-ONE )
! 609: *
! 610: * Update elements in columns A(k-1) and A(k) as
! 611: * dot products of rows of ( W(kw-1) W(kw) ) and columns
! 612: * of D**(-1)
! 613: *
! 614: DO 20 J = 1, K - 2
! 615: A( J, K-1 ) = T*( ( D11*W( J, KW-1 )-W( J, KW ) ) /
! 616: $ D21 )
! 617: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
! 618: $ DCONJG( D21 ) )
! 619: 20 CONTINUE
! 620: END IF
! 621: *
! 622: * Copy D(k) to A
! 623: *
! 624: A( K-1, K-1 ) = W( K-1, KW-1 )
! 625: A( K-1, K ) = W( K-1, KW )
! 626: A( K, K ) = W( K, KW )
! 627: *
! 628: * (2) Conjugate columns W(kw) and W(kw-1)
! 629: *
! 630: CALL ZLACGV( K-1, W( 1, KW ), 1 )
! 631: CALL ZLACGV( K-2, W( 1, KW-1 ), 1 )
! 632: *
! 633: END IF
! 634: *
! 635: END IF
! 636: *
! 637: * Store details of the interchanges in IPIV
! 638: *
! 639: IF( KSTEP.EQ.1 ) THEN
! 640: IPIV( K ) = KP
! 641: ELSE
! 642: IPIV( K ) = -P
! 643: IPIV( K-1 ) = -KP
! 644: END IF
! 645: *
! 646: * Decrease K and return to the start of the main loop
! 647: *
! 648: K = K - KSTEP
! 649: GO TO 10
! 650: *
! 651: 30 CONTINUE
! 652: *
! 653: * Update the upper triangle of A11 (= A(1:k,1:k)) as
! 654: *
! 655: * A11 := A11 - U12*D*U12**H = A11 - U12*W**H
! 656: *
! 657: * computing blocks of NB columns at a time (note that conjg(W) is
! 658: * actually stored)
! 659: *
! 660: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
! 661: JB = MIN( NB, K-J+1 )
! 662: *
! 663: * Update the upper triangle of the diagonal block
! 664: *
! 665: DO 40 JJ = J, J + JB - 1
! 666: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 667: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
! 668: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
! 669: $ A( J, JJ ), 1 )
! 670: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 671: 40 CONTINUE
! 672: *
! 673: * Update the rectangular superdiagonal block
! 674: *
! 675: IF( J.GE.2 )
! 676: $ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
! 677: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
! 678: $ CONE, A( 1, J ), LDA )
! 679: 50 CONTINUE
! 680: *
! 681: * Put U12 in standard form by partially undoing the interchanges
! 682: * in of rows in columns k+1:n looping backwards from k+1 to n
! 683: *
! 684: J = K + 1
! 685: 60 CONTINUE
! 686: *
! 687: * Undo the interchanges (if any) of rows J and JP2
! 688: * (or J and JP2, and J+1 and JP1) at each step J
! 689: *
! 690: KSTEP = 1
! 691: JP1 = 1
! 692: * (Here, J is a diagonal index)
! 693: JJ = J
! 694: JP2 = IPIV( J )
! 695: IF( JP2.LT.0 ) THEN
! 696: JP2 = -JP2
! 697: * (Here, J is a diagonal index)
! 698: J = J + 1
! 699: JP1 = -IPIV( J )
! 700: KSTEP = 2
! 701: END IF
! 702: * (NOTE: Here, J is used to determine row length. Length N-J+1
! 703: * of the rows to swap back doesn't include diagonal element)
! 704: J = J + 1
! 705: IF( JP2.NE.JJ .AND. J.LE.N )
! 706: $ CALL ZSWAP( N-J+1, A( JP2, J ), LDA, A( JJ, J ), LDA )
! 707: JJ = JJ + 1
! 708: IF( KSTEP.EQ.2 .AND. JP1.NE.JJ .AND. J.LE.N )
! 709: $ CALL ZSWAP( N-J+1, A( JP1, J ), LDA, A( JJ, J ), LDA )
! 710: IF( J.LT.N )
! 711: $ GO TO 60
! 712: *
! 713: * Set KB to the number of columns factorized
! 714: *
! 715: KB = N - K
! 716: *
! 717: ELSE
! 718: *
! 719: * Factorize the leading columns of A using the lower triangle
! 720: * of A and working forwards, and compute the matrix W = L21*D
! 721: * for use in updating A22 (note that conjg(W) is actually stored)
! 722: *
! 723: * K is the main loop index, increasing from 1 in steps of 1 or 2
! 724: *
! 725: K = 1
! 726: 70 CONTINUE
! 727: *
! 728: * Exit from loop
! 729: *
! 730: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
! 731: $ GO TO 90
! 732: *
! 733: KSTEP = 1
! 734: P = K
! 735: *
! 736: * Copy column K of A to column K of W and update column K of W
! 737: *
! 738: W( K, K ) = DBLE( A( K, K ) )
! 739: IF( K.LT.N )
! 740: $ CALL ZCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 )
! 741: IF( K.GT.1 ) THEN
! 742: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
! 743: $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
! 744: W( K, K ) = DBLE( W( K, K ) )
! 745: END IF
! 746: *
! 747: * Determine rows and columns to be interchanged and whether
! 748: * a 1-by-1 or 2-by-2 pivot block will be used
! 749: *
! 750: ABSAKK = ABS( DBLE( W( K, K ) ) )
! 751: *
! 752: * IMAX is the row-index of the largest off-diagonal element in
! 753: * column K, and COLMAX is its absolute value.
! 754: * Determine both COLMAX and IMAX.
! 755: *
! 756: IF( K.LT.N ) THEN
! 757: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
! 758: COLMAX = CABS1( W( IMAX, K ) )
! 759: ELSE
! 760: COLMAX = ZERO
! 761: END IF
! 762: *
! 763: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 764: *
! 765: * Column K is zero or underflow: set INFO and continue
! 766: *
! 767: IF( INFO.EQ.0 )
! 768: $ INFO = K
! 769: KP = K
! 770: A( K, K ) = DBLE( W( K, K ) )
! 771: IF( K.LT.N )
! 772: $ CALL ZCOPY( N-K, W( K+1, K ), 1, A( K+1, K ), 1 )
! 773: ELSE
! 774: *
! 775: * ============================================================
! 776: *
! 777: * BEGIN pivot search
! 778: *
! 779: * Case(1)
! 780: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 781: * (used to handle NaN and Inf)
! 782: *
! 783: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 784: *
! 785: * no interchange, use 1-by-1 pivot block
! 786: *
! 787: KP = K
! 788: *
! 789: ELSE
! 790: *
! 791: DONE = .FALSE.
! 792: *
! 793: * Loop until pivot found
! 794: *
! 795: 72 CONTINUE
! 796: *
! 797: * BEGIN pivot search loop body
! 798: *
! 799: *
! 800: * Copy column IMAX to column k+1 of W and update it
! 801: *
! 802: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
! 803: CALL ZLACGV( IMAX-K, W( K, K+1 ), 1 )
! 804: W( IMAX, K+1 ) = DBLE( A( IMAX, IMAX ) )
! 805: *
! 806: IF( IMAX.LT.N )
! 807: $ CALL ZCOPY( N-IMAX, A( IMAX+1, IMAX ), 1,
! 808: $ W( IMAX+1, K+1 ), 1 )
! 809: *
! 810: IF( K.GT.1 ) THEN
! 811: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
! 812: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
! 813: $ CONE, W( K, K+1 ), 1 )
! 814: W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) )
! 815: END IF
! 816: *
! 817: * JMAX is the column-index of the largest off-diagonal
! 818: * element in row IMAX, and ROWMAX is its absolute value.
! 819: * Determine both ROWMAX and JMAX.
! 820: *
! 821: IF( IMAX.NE.K ) THEN
! 822: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
! 823: ROWMAX = CABS1( W( JMAX, K+1 ) )
! 824: ELSE
! 825: ROWMAX = ZERO
! 826: END IF
! 827: *
! 828: IF( IMAX.LT.N ) THEN
! 829: ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
! 830: DTEMP = CABS1( W( ITEMP, K+1 ) )
! 831: IF( DTEMP.GT.ROWMAX ) THEN
! 832: ROWMAX = DTEMP
! 833: JMAX = ITEMP
! 834: END IF
! 835: END IF
! 836: *
! 837: * Case(2)
! 838: * Equivalent to testing for
! 839: * ABS( REAL( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX
! 840: * (used to handle NaN and Inf)
! 841: *
! 842: IF( .NOT.( ABS( DBLE( W( IMAX,K+1 ) ) )
! 843: $ .LT.ALPHA*ROWMAX ) ) THEN
! 844: *
! 845: * interchange rows and columns K and IMAX,
! 846: * use 1-by-1 pivot block
! 847: *
! 848: KP = IMAX
! 849: *
! 850: * copy column K+1 of W to column K of W
! 851: *
! 852: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
! 853: *
! 854: DONE = .TRUE.
! 855: *
! 856: * Case(3)
! 857: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 858: * (used to handle NaN and Inf)
! 859: *
! 860: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 861: $ THEN
! 862: *
! 863: * interchange rows and columns K+1 and IMAX,
! 864: * use 2-by-2 pivot block
! 865: *
! 866: KP = IMAX
! 867: KSTEP = 2
! 868: DONE = .TRUE.
! 869: *
! 870: * Case(4)
! 871: ELSE
! 872: *
! 873: * Pivot not found: set params and repeat
! 874: *
! 875: P = IMAX
! 876: COLMAX = ROWMAX
! 877: IMAX = JMAX
! 878: *
! 879: * Copy updated JMAXth (next IMAXth) column to Kth of W
! 880: *
! 881: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
! 882: *
! 883: END IF
! 884: *
! 885: *
! 886: * End pivot search loop body
! 887: *
! 888: IF( .NOT.DONE ) GOTO 72
! 889: *
! 890: END IF
! 891: *
! 892: * END pivot search
! 893: *
! 894: * ============================================================
! 895: *
! 896: * KK is the column of A where pivoting step stopped
! 897: *
! 898: KK = K + KSTEP - 1
! 899: *
! 900: * Interchange rows and columns P and K (only for 2-by-2 pivot).
! 901: * Updated column P is already stored in column K of W.
! 902: *
! 903: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 904: *
! 905: * Copy non-updated column KK-1 to column P of submatrix A
! 906: * at step K. No need to copy element into columns
! 907: * K and K+1 of A for 2-by-2 pivot, since these columns
! 908: * will be later overwritten.
! 909: *
! 910: A( P, P ) = DBLE( A( K, K ) )
! 911: CALL ZCOPY( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
! 912: CALL ZLACGV( P-K-1, A( P, K+1 ), LDA )
! 913: IF( P.LT.N )
! 914: $ CALL ZCOPY( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
! 915: *
! 916: * Interchange rows K and P in first K-1 columns of A
! 917: * (columns K and K+1 of A for 2-by-2 pivot will be
! 918: * later overwritten). Interchange rows K and P
! 919: * in first KK columns of W.
! 920: *
! 921: IF( K.GT.1 )
! 922: $ CALL ZSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
! 923: CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
! 924: END IF
! 925: *
! 926: * Interchange rows and columns KP and KK.
! 927: * Updated column KP is already stored in column KK of W.
! 928: *
! 929: IF( KP.NE.KK ) THEN
! 930: *
! 931: * Copy non-updated column KK to column KP of submatrix A
! 932: * at step K. No need to copy element into column K
! 933: * (or K and K+1 for 2-by-2 pivot) of A, since these columns
! 934: * will be later overwritten.
! 935: *
! 936: A( KP, KP ) = DBLE( A( KK, KK ) )
! 937: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
! 938: $ LDA )
! 939: CALL ZLACGV( KP-KK-1, A( KP, KK+1 ), LDA )
! 940: IF( KP.LT.N )
! 941: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
! 942: *
! 943: * Interchange rows KK and KP in first K-1 columns of A
! 944: * (column K (or K and K+1 for 2-by-2 pivot) of A will be
! 945: * later overwritten). Interchange rows KK and KP
! 946: * in first KK columns of W.
! 947: *
! 948: IF( K.GT.1 )
! 949: $ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
! 950: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
! 951: END IF
! 952: *
! 953: IF( KSTEP.EQ.1 ) THEN
! 954: *
! 955: * 1-by-1 pivot block D(k): column k of W now holds
! 956: *
! 957: * W(k) = L(k)*D(k),
! 958: *
! 959: * where L(k) is the k-th column of L
! 960: *
! 961: * (1) Store subdiag. elements of column L(k)
! 962: * and 1-by-1 block D(k) in column k of A.
! 963: * (NOTE: Diagonal element L(k,k) is a UNIT element
! 964: * and not stored)
! 965: * A(k,k) := D(k,k) = W(k,k)
! 966: * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
! 967: *
! 968: * (NOTE: No need to use for Hermitian matrix
! 969: * A( K, K ) = REAL( W( K, K) ) to separately copy diagonal
! 970: * element D(k,k) from W (potentially saves only one load))
! 971: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
! 972: IF( K.LT.N ) THEN
! 973: *
! 974: * (NOTE: No need to check if A(k,k) is NOT ZERO,
! 975: * since that was ensured earlier in pivot search:
! 976: * case A(k,k) = 0 falls into 2x2 pivot case(3))
! 977: *
! 978: * Handle division by a small number
! 979: *
! 980: T = DBLE( A( K, K ) )
! 981: IF( ABS( T ).GE.SFMIN ) THEN
! 982: R1 = ONE / T
! 983: CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 )
! 984: ELSE
! 985: DO 74 II = K + 1, N
! 986: A( II, K ) = A( II, K ) / T
! 987: 74 CONTINUE
! 988: END IF
! 989: *
! 990: * (2) Conjugate column W(k)
! 991: *
! 992: CALL ZLACGV( N-K, W( K+1, K ), 1 )
! 993: END IF
! 994: *
! 995: ELSE
! 996: *
! 997: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
! 998: *
! 999: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
! 1000: *
! 1001: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
! 1002: * of L
! 1003: *
! 1004: * (1) Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
! 1005: * block D(k:k+1,k:k+1) in columns k and k+1 of A.
! 1006: * NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
! 1007: * block and not stored.
! 1008: * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
! 1009: * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
! 1010: * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
! 1011: *
! 1012: IF( K.LT.N-1 ) THEN
! 1013: *
! 1014: * Factor out the columns of the inverse of 2-by-2 pivot
! 1015: * block D, so that each column contains 1, to reduce the
! 1016: * number of FLOPS when we multiply panel
! 1017: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
! 1018: *
! 1019: * D**(-1) = ( d11 cj(d21) )**(-1) =
! 1020: * ( d21 d22 )
! 1021: *
! 1022: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
! 1023: * ( (-d21) ( d11 ) )
! 1024: *
! 1025: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
! 1026: *
! 1027: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
! 1028: * ( ( -1 ) ( d11/conj(d21) ) )
! 1029: *
! 1030: * = 1/(|d21|**2) * 1/(D22*D11-1) *
! 1031: *
! 1032: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
! 1033: * ( ( -1 ) ( D22 ) )
! 1034: *
! 1035: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
! 1036: * ( ( -1 ) ( D22 ) )
! 1037: *
! 1038: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
! 1039: * ( ( -1 ) ( D22 ) )
! 1040: *
! 1041: * Handle division by a small number. (NOTE: order of
! 1042: * operations is important)
! 1043: *
! 1044: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
! 1045: * ( (( -1 ) ) (( D22 ) ) ),
! 1046: *
! 1047: * where D11 = d22/d21,
! 1048: * D22 = d11/conj(d21),
! 1049: * D21 = d21,
! 1050: * T = 1/(D22*D11-1).
! 1051: *
! 1052: * (NOTE: No need to check for division by ZERO,
! 1053: * since that was ensured earlier in pivot search:
! 1054: * (a) d21 != 0 in 2x2 pivot case(4),
! 1055: * since |d21| should be larger than |d11| and |d22|;
! 1056: * (b) (D22*D11 - 1) != 0, since from (a),
! 1057: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
! 1058: *
! 1059: D21 = W( K+1, K )
! 1060: D11 = W( K+1, K+1 ) / D21
! 1061: D22 = W( K, K ) / DCONJG( D21 )
! 1062: T = ONE / ( DBLE( D11*D22 )-ONE )
! 1063: *
! 1064: * Update elements in columns A(k) and A(k+1) as
! 1065: * dot products of rows of ( W(k) W(k+1) ) and columns
! 1066: * of D**(-1)
! 1067: *
! 1068: DO 80 J = K + 2, N
! 1069: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
! 1070: $ DCONJG( D21 ) )
! 1071: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
! 1072: $ D21 )
! 1073: 80 CONTINUE
! 1074: END IF
! 1075: *
! 1076: * Copy D(k) to A
! 1077: *
! 1078: A( K, K ) = W( K, K )
! 1079: A( K+1, K ) = W( K+1, K )
! 1080: A( K+1, K+1 ) = W( K+1, K+1 )
! 1081: *
! 1082: * (2) Conjugate columns W(k) and W(k+1)
! 1083: *
! 1084: CALL ZLACGV( N-K, W( K+1, K ), 1 )
! 1085: CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 )
! 1086: *
! 1087: END IF
! 1088: *
! 1089: END IF
! 1090: *
! 1091: * Store details of the interchanges in IPIV
! 1092: *
! 1093: IF( KSTEP.EQ.1 ) THEN
! 1094: IPIV( K ) = KP
! 1095: ELSE
! 1096: IPIV( K ) = -P
! 1097: IPIV( K+1 ) = -KP
! 1098: END IF
! 1099: *
! 1100: * Increase K and return to the start of the main loop
! 1101: *
! 1102: K = K + KSTEP
! 1103: GO TO 70
! 1104: *
! 1105: 90 CONTINUE
! 1106: *
! 1107: * Update the lower triangle of A22 (= A(k:n,k:n)) as
! 1108: *
! 1109: * A22 := A22 - L21*D*L21**H = A22 - L21*W**H
! 1110: *
! 1111: * computing blocks of NB columns at a time (note that conjg(W) is
! 1112: * actually stored)
! 1113: *
! 1114: DO 110 J = K, N, NB
! 1115: JB = MIN( NB, N-J+1 )
! 1116: *
! 1117: * Update the lower triangle of the diagonal block
! 1118: *
! 1119: DO 100 JJ = J, J + JB - 1
! 1120: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 1121: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
! 1122: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
! 1123: $ A( JJ, JJ ), 1 )
! 1124: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 1125: 100 CONTINUE
! 1126: *
! 1127: * Update the rectangular subdiagonal block
! 1128: *
! 1129: IF( J+JB.LE.N )
! 1130: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
! 1131: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
! 1132: $ LDW, CONE, A( J+JB, J ), LDA )
! 1133: 110 CONTINUE
! 1134: *
! 1135: * Put L21 in standard form by partially undoing the interchanges
! 1136: * of rows in columns 1:k-1 looping backwards from k-1 to 1
! 1137: *
! 1138: J = K - 1
! 1139: 120 CONTINUE
! 1140: *
! 1141: * Undo the interchanges (if any) of rows J and JP2
! 1142: * (or J and JP2, and J-1 and JP1) at each step J
! 1143: *
! 1144: KSTEP = 1
! 1145: JP1 = 1
! 1146: * (Here, J is a diagonal index)
! 1147: JJ = J
! 1148: JP2 = IPIV( J )
! 1149: IF( JP2.LT.0 ) THEN
! 1150: JP2 = -JP2
! 1151: * (Here, J is a diagonal index)
! 1152: J = J - 1
! 1153: JP1 = -IPIV( J )
! 1154: KSTEP = 2
! 1155: END IF
! 1156: * (NOTE: Here, J is used to determine row length. Length J
! 1157: * of the rows to swap back doesn't include diagonal element)
! 1158: J = J - 1
! 1159: IF( JP2.NE.JJ .AND. J.GE.1 )
! 1160: $ CALL ZSWAP( J, A( JP2, 1 ), LDA, A( JJ, 1 ), LDA )
! 1161: JJ = JJ -1
! 1162: IF( KSTEP.EQ.2 .AND. JP1.NE.JJ .AND. J.GE.1 )
! 1163: $ CALL ZSWAP( J, A( JP1, 1 ), LDA, A( JJ, 1 ), LDA )
! 1164: IF( J.GT.1 )
! 1165: $ GO TO 120
! 1166: *
! 1167: * Set KB to the number of columns factorized
! 1168: *
! 1169: KB = K - 1
! 1170: *
! 1171: END IF
! 1172: RETURN
! 1173: *
! 1174: * End of ZLAHEF_ROOK
! 1175: *
! 1176: END
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