Annotation of rpl/lapack/lapack/zlahef_rk.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) 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 ZLAHEF_RK + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlahef_rk.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlahef_rk.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlahef_rk.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZLAHEF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
! 22: * INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER UPLO
! 26: * INTEGER INFO, KB, LDA, LDW, N, NB
! 27: * ..
! 28: * .. Array Arguments ..
! 29: * INTEGER IPIV( * )
! 30: * COMPLEX*16 A( LDA, * ), E( * ), W( LDW, * )
! 31: * ..
! 32: *
! 33: *
! 34: *> \par Purpose:
! 35: * =============
! 36: *>
! 37: *> \verbatim
! 38: *> ZLAHEF_RK computes a partial factorization of a complex Hermitian
! 39: *> matrix A using the bounded Bunch-Kaufman (rook) diagonal
! 40: *> pivoting 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: *>
! 51: *> ZLAHEF_RK is an auxiliary routine called by ZHETRF_RK. It uses
! 52: *> blocked code (calling Level 3 BLAS) to update the submatrix
! 53: *> A11 (if UPLO = 'U') or 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: *> Hermitian 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 COMPLEX*16 array, dimension (LDA,N)
! 92: *> On entry, the Hermitian matrix A.
! 93: *> If UPLO = 'U': the leading N-by-N upper triangular part
! 94: *> of A contains the upper triangular part of the matrix A,
! 95: *> and the strictly lower triangular part of A is not
! 96: *> referenced.
! 97: *>
! 98: *> If UPLO = 'L': the leading N-by-N lower triangular part
! 99: *> of A contains the lower triangular part of the matrix A,
! 100: *> and the strictly upper triangular part of A is not
! 101: *> referenced.
! 102: *>
! 103: *> On exit, contains:
! 104: *> a) ONLY diagonal elements of the Hermitian block diagonal
! 105: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
! 106: *> (superdiagonal (or subdiagonal) elements of D
! 107: *> are stored on exit in array E), and
! 108: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
! 109: *> If UPLO = 'L': factor L in the subdiagonal part of A.
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[in] LDA
! 113: *> \verbatim
! 114: *> LDA is INTEGER
! 115: *> The leading dimension of the array A. LDA >= max(1,N).
! 116: *> \endverbatim
! 117: *>
! 118: *> \param[out] E
! 119: *> \verbatim
! 120: *> E is COMPLEX*16 array, dimension (N)
! 121: *> On exit, contains the superdiagonal (or subdiagonal)
! 122: *> elements of the Hermitian block diagonal matrix D
! 123: *> with 1-by-1 or 2-by-2 diagonal blocks, where
! 124: *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
! 125: *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
! 126: *>
! 127: *> NOTE: For 1-by-1 diagonal block D(k), where
! 128: *> 1 <= k <= N, the element E(k) is set to 0 in both
! 129: *> UPLO = 'U' or UPLO = 'L' cases.
! 130: *> \endverbatim
! 131: *>
! 132: *> \param[out] IPIV
! 133: *> \verbatim
! 134: *> IPIV is INTEGER array, dimension (N)
! 135: *> IPIV describes the permutation matrix P in the factorization
! 136: *> of matrix A as follows. The absolute value of IPIV(k)
! 137: *> represents the index of row and column that were
! 138: *> interchanged with the k-th row and column. The value of UPLO
! 139: *> describes the order in which the interchanges were applied.
! 140: *> Also, the sign of IPIV represents the block structure of
! 141: *> the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
! 142: *> diagonal blocks which correspond to 1 or 2 interchanges
! 143: *> at each factorization step.
! 144: *>
! 145: *> If UPLO = 'U',
! 146: *> ( in factorization order, k decreases from N to 1 ):
! 147: *> a) A single positive entry IPIV(k) > 0 means:
! 148: *> D(k,k) is a 1-by-1 diagonal block.
! 149: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
! 150: *> interchanged in the submatrix A(1:N,N-KB+1:N);
! 151: *> If IPIV(k) = k, no interchange occurred.
! 152: *>
! 153: *>
! 154: *> b) A pair of consecutive negative entries
! 155: *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
! 156: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
! 157: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
! 158: *> 1) If -IPIV(k) != k, rows and columns
! 159: *> k and -IPIV(k) were interchanged
! 160: *> in the matrix A(1:N,N-KB+1:N).
! 161: *> If -IPIV(k) = k, no interchange occurred.
! 162: *> 2) If -IPIV(k-1) != k-1, rows and columns
! 163: *> k-1 and -IPIV(k-1) were interchanged
! 164: *> in the submatrix A(1:N,N-KB+1:N).
! 165: *> If -IPIV(k-1) = k-1, no interchange occurred.
! 166: *>
! 167: *> c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
! 168: *>
! 169: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
! 170: *>
! 171: *> If UPLO = 'L',
! 172: *> ( in factorization order, k increases from 1 to N ):
! 173: *> a) A single positive entry IPIV(k) > 0 means:
! 174: *> D(k,k) is a 1-by-1 diagonal block.
! 175: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
! 176: *> interchanged in the submatrix A(1:N,1:KB).
! 177: *> If IPIV(k) = k, no interchange occurred.
! 178: *>
! 179: *> b) A pair of consecutive negative entries
! 180: *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
! 181: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
! 182: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
! 183: *> 1) If -IPIV(k) != k, rows and columns
! 184: *> k and -IPIV(k) were interchanged
! 185: *> in the submatrix A(1:N,1:KB).
! 186: *> If -IPIV(k) = k, no interchange occurred.
! 187: *> 2) If -IPIV(k+1) != k+1, rows and columns
! 188: *> k-1 and -IPIV(k-1) were interchanged
! 189: *> in the submatrix A(1:N,1:KB).
! 190: *> If -IPIV(k+1) = k+1, no interchange occurred.
! 191: *>
! 192: *> c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
! 193: *>
! 194: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
! 195: *> \endverbatim
! 196: *>
! 197: *> \param[out] W
! 198: *> \verbatim
! 199: *> W is COMPLEX*16 array, dimension (LDW,NB)
! 200: *> \endverbatim
! 201: *>
! 202: *> \param[in] LDW
! 203: *> \verbatim
! 204: *> LDW is INTEGER
! 205: *> The leading dimension of the array W. LDW >= max(1,N).
! 206: *> \endverbatim
! 207: *>
! 208: *> \param[out] INFO
! 209: *> \verbatim
! 210: *> INFO is INTEGER
! 211: *> = 0: successful exit
! 212: *>
! 213: *> < 0: If INFO = -k, the k-th argument had an illegal value
! 214: *>
! 215: *> > 0: If INFO = k, the matrix A is singular, because:
! 216: *> If UPLO = 'U': column k in the upper
! 217: *> triangular part of A contains all zeros.
! 218: *> If UPLO = 'L': column k in the lower
! 219: *> triangular part of A contains all zeros.
! 220: *>
! 221: *> Therefore D(k,k) is exactly zero, and superdiagonal
! 222: *> elements of column k of U (or subdiagonal elements of
! 223: *> column k of L ) are all zeros. The factorization has
! 224: *> been completed, but the block diagonal matrix D is
! 225: *> exactly singular, and division by zero will occur if
! 226: *> it is used to solve a system of equations.
! 227: *>
! 228: *> NOTE: INFO only stores the first occurrence of
! 229: *> a singularity, any subsequent occurrence of singularity
! 230: *> is not stored in INFO even though the factorization
! 231: *> always completes.
! 232: *> \endverbatim
! 233: *
! 234: * Authors:
! 235: * ========
! 236: *
! 237: *> \author Univ. of Tennessee
! 238: *> \author Univ. of California Berkeley
! 239: *> \author Univ. of Colorado Denver
! 240: *> \author NAG Ltd.
! 241: *
! 242: *> \date December 2016
! 243: *
! 244: *> \ingroup complex16HEcomputational
! 245: *
! 246: *> \par Contributors:
! 247: * ==================
! 248: *>
! 249: *> \verbatim
! 250: *>
! 251: *> December 2016, Igor Kozachenko,
! 252: *> Computer Science Division,
! 253: *> University of California, Berkeley
! 254: *>
! 255: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
! 256: *> School of Mathematics,
! 257: *> University of Manchester
! 258: *>
! 259: *> \endverbatim
! 260: *
! 261: * =====================================================================
! 262: SUBROUTINE ZLAHEF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
! 263: $ INFO )
! 264: *
! 265: * -- LAPACK computational routine (version 3.7.0) --
! 266: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 267: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 268: * December 2016
! 269: *
! 270: * .. Scalar Arguments ..
! 271: CHARACTER UPLO
! 272: INTEGER INFO, KB, LDA, LDW, N, NB
! 273: * ..
! 274: * .. Array Arguments ..
! 275: INTEGER IPIV( * )
! 276: COMPLEX*16 A( LDA, * ), W( LDW, * ), E( * )
! 277: * ..
! 278: *
! 279: * =====================================================================
! 280: *
! 281: * .. Parameters ..
! 282: DOUBLE PRECISION ZERO, ONE
! 283: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 284: COMPLEX*16 CONE
! 285: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
! 286: DOUBLE PRECISION EIGHT, SEVTEN
! 287: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
! 288: COMPLEX*16 CZERO
! 289: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) )
! 290: * ..
! 291: * .. Local Scalars ..
! 292: LOGICAL DONE
! 293: INTEGER IMAX, ITEMP, II, J, JB, JJ, JMAX, K, KK, KKW,
! 294: $ KP, KSTEP, KW, P
! 295: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, DTEMP, R1, ROWMAX, T,
! 296: $ SFMIN
! 297: COMPLEX*16 D11, D21, D22, Z
! 298: * ..
! 299: * .. External Functions ..
! 300: LOGICAL LSAME
! 301: INTEGER IZAMAX
! 302: DOUBLE PRECISION DLAMCH
! 303: EXTERNAL LSAME, IZAMAX, DLAMCH
! 304: * ..
! 305: * .. External Subroutines ..
! 306: EXTERNAL ZCOPY, ZDSCAL, ZGEMM, ZGEMV, ZLACGV, ZSWAP
! 307: * ..
! 308: * .. Intrinsic Functions ..
! 309: INTRINSIC ABS, DBLE, DCONJG, DIMAG, MAX, MIN, SQRT
! 310: * ..
! 311: * .. Statement Functions ..
! 312: DOUBLE PRECISION CABS1
! 313: * ..
! 314: * .. Statement Function definitions ..
! 315: CABS1( Z ) = ABS( DBLE( Z ) ) + ABS( DIMAG( Z ) )
! 316: * ..
! 317: * .. Executable Statements ..
! 318: *
! 319: INFO = 0
! 320: *
! 321: * Initialize ALPHA for use in choosing pivot block size.
! 322: *
! 323: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
! 324: *
! 325: * Compute machine safe minimum
! 326: *
! 327: SFMIN = DLAMCH( 'S' )
! 328: *
! 329: IF( LSAME( UPLO, 'U' ) ) THEN
! 330: *
! 331: * Factorize the trailing columns of A using the upper triangle
! 332: * of A and working backwards, and compute the matrix W = U12*D
! 333: * for use in updating A11 (note that conjg(W) is actually stored)
! 334: * Initilize the first entry of array E, where superdiagonal
! 335: * elements of D are stored
! 336: *
! 337: E( 1 ) = CZERO
! 338: *
! 339: * K is the main loop index, decreasing from N in steps of 1 or 2
! 340: *
! 341: K = N
! 342: 10 CONTINUE
! 343: *
! 344: * KW is the column of W which corresponds to column K of A
! 345: *
! 346: KW = NB + K - N
! 347: *
! 348: * Exit from loop
! 349: *
! 350: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
! 351: $ GO TO 30
! 352: *
! 353: KSTEP = 1
! 354: P = K
! 355: *
! 356: * Copy column K of A to column KW of W and update it
! 357: *
! 358: IF( K.GT.1 )
! 359: $ CALL ZCOPY( K-1, A( 1, K ), 1, W( 1, KW ), 1 )
! 360: W( K, KW ) = DBLE( A( K, K ) )
! 361: IF( K.LT.N ) THEN
! 362: CALL ZGEMV( 'No transpose', K, N-K, -CONE, A( 1, K+1 ), LDA,
! 363: $ W( K, KW+1 ), LDW, CONE, W( 1, KW ), 1 )
! 364: W( K, KW ) = DBLE( W( K, KW ) )
! 365: END IF
! 366: *
! 367: * Determine rows and columns to be interchanged and whether
! 368: * a 1-by-1 or 2-by-2 pivot block will be used
! 369: *
! 370: ABSAKK = ABS( DBLE( W( K, KW ) ) )
! 371: *
! 372: * IMAX is the row-index of the largest off-diagonal element in
! 373: * column K, and COLMAX is its absolute value.
! 374: * Determine both COLMAX and IMAX.
! 375: *
! 376: IF( K.GT.1 ) THEN
! 377: IMAX = IZAMAX( K-1, W( 1, KW ), 1 )
! 378: COLMAX = CABS1( W( IMAX, KW ) )
! 379: ELSE
! 380: COLMAX = ZERO
! 381: END IF
! 382: *
! 383: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 384: *
! 385: * Column K is zero or underflow: set INFO and continue
! 386: *
! 387: IF( INFO.EQ.0 )
! 388: $ INFO = K
! 389: KP = K
! 390: A( K, K ) = DBLE( W( K, KW ) )
! 391: IF( K.GT.1 )
! 392: $ CALL ZCOPY( K-1, W( 1, KW ), 1, A( 1, K ), 1 )
! 393: *
! 394: * Set E( K ) to zero
! 395: *
! 396: IF( K.GT.1 )
! 397: $ E( K ) = CZERO
! 398: *
! 399: ELSE
! 400: *
! 401: * ============================================================
! 402: *
! 403: * BEGIN pivot search
! 404: *
! 405: * Case(1)
! 406: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 407: * (used to handle NaN and Inf)
! 408: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 409: *
! 410: * no interchange, use 1-by-1 pivot block
! 411: *
! 412: KP = K
! 413: *
! 414: ELSE
! 415: *
! 416: * Lop until pivot found
! 417: *
! 418: DONE = .FALSE.
! 419: *
! 420: 12 CONTINUE
! 421: *
! 422: * BEGIN pivot search loop body
! 423: *
! 424: *
! 425: * Copy column IMAX to column KW-1 of W and update it
! 426: *
! 427: IF( IMAX.GT.1 )
! 428: $ CALL ZCOPY( IMAX-1, A( 1, IMAX ), 1, W( 1, KW-1 ),
! 429: $ 1 )
! 430: W( IMAX, KW-1 ) = DBLE( A( IMAX, IMAX ) )
! 431: *
! 432: CALL ZCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
! 433: $ W( IMAX+1, KW-1 ), 1 )
! 434: CALL ZLACGV( K-IMAX, W( IMAX+1, KW-1 ), 1 )
! 435: *
! 436: IF( K.LT.N ) THEN
! 437: CALL ZGEMV( 'No transpose', K, N-K, -CONE,
! 438: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
! 439: $ CONE, W( 1, KW-1 ), 1 )
! 440: W( IMAX, KW-1 ) = DBLE( W( IMAX, KW-1 ) )
! 441: END IF
! 442: *
! 443: * JMAX is the column-index of the largest off-diagonal
! 444: * element in row IMAX, and ROWMAX is its absolute value.
! 445: * Determine both ROWMAX and JMAX.
! 446: *
! 447: IF( IMAX.NE.K ) THEN
! 448: JMAX = IMAX + IZAMAX( K-IMAX, W( IMAX+1, KW-1 ),
! 449: $ 1 )
! 450: ROWMAX = CABS1( W( JMAX, KW-1 ) )
! 451: ELSE
! 452: ROWMAX = ZERO
! 453: END IF
! 454: *
! 455: IF( IMAX.GT.1 ) THEN
! 456: ITEMP = IZAMAX( IMAX-1, W( 1, KW-1 ), 1 )
! 457: DTEMP = CABS1( W( ITEMP, KW-1 ) )
! 458: IF( DTEMP.GT.ROWMAX ) THEN
! 459: ROWMAX = DTEMP
! 460: JMAX = ITEMP
! 461: END IF
! 462: END IF
! 463: *
! 464: * Case(2)
! 465: * Equivalent to testing for
! 466: * ABS( REAL( W( IMAX,KW-1 ) ) ).GE.ALPHA*ROWMAX
! 467: * (used to handle NaN and Inf)
! 468: *
! 469: IF( .NOT.( ABS( DBLE( W( IMAX,KW-1 ) ) )
! 470: $ .LT.ALPHA*ROWMAX ) ) THEN
! 471: *
! 472: * interchange rows and columns K and IMAX,
! 473: * use 1-by-1 pivot block
! 474: *
! 475: KP = IMAX
! 476: *
! 477: * copy column KW-1 of W to column KW of W
! 478: *
! 479: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
! 480: *
! 481: DONE = .TRUE.
! 482: *
! 483: * Case(3)
! 484: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 485: * (used to handle NaN and Inf)
! 486: *
! 487: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 488: $ THEN
! 489: *
! 490: * interchange rows and columns K-1 and IMAX,
! 491: * use 2-by-2 pivot block
! 492: *
! 493: KP = IMAX
! 494: KSTEP = 2
! 495: DONE = .TRUE.
! 496: *
! 497: * Case(4)
! 498: ELSE
! 499: *
! 500: * Pivot not found: set params and repeat
! 501: *
! 502: P = IMAX
! 503: COLMAX = ROWMAX
! 504: IMAX = JMAX
! 505: *
! 506: * Copy updated JMAXth (next IMAXth) column to Kth of W
! 507: *
! 508: CALL ZCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
! 509: *
! 510: END IF
! 511: *
! 512: *
! 513: * END pivot search loop body
! 514: *
! 515: IF( .NOT.DONE ) GOTO 12
! 516: *
! 517: END IF
! 518: *
! 519: * END pivot search
! 520: *
! 521: * ============================================================
! 522: *
! 523: * KK is the column of A where pivoting step stopped
! 524: *
! 525: KK = K - KSTEP + 1
! 526: *
! 527: * KKW is the column of W which corresponds to column KK of A
! 528: *
! 529: KKW = NB + KK - N
! 530: *
! 531: * Interchange rows and columns P and K.
! 532: * Updated column P is already stored in column KW of W.
! 533: *
! 534: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 535: *
! 536: * Copy non-updated column K to column P of submatrix A
! 537: * at step K. No need to copy element into columns
! 538: * K and K-1 of A for 2-by-2 pivot, since these columns
! 539: * will be later overwritten.
! 540: *
! 541: A( P, P ) = DBLE( A( K, K ) )
! 542: CALL ZCOPY( K-1-P, A( P+1, K ), 1, A( P, P+1 ),
! 543: $ LDA )
! 544: CALL ZLACGV( K-1-P, A( P, P+1 ), LDA )
! 545: IF( P.GT.1 )
! 546: $ CALL ZCOPY( P-1, A( 1, K ), 1, A( 1, P ), 1 )
! 547: *
! 548: * Interchange rows K and P in the last K+1 to N columns of A
! 549: * (columns K and K-1 of A for 2-by-2 pivot will be
! 550: * later overwritten). Interchange rows K and P
! 551: * in last KKW to NB columns of W.
! 552: *
! 553: IF( K.LT.N )
! 554: $ CALL ZSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ),
! 555: $ LDA )
! 556: CALL ZSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ),
! 557: $ LDW )
! 558: END IF
! 559: *
! 560: * Interchange rows and columns KP and KK.
! 561: * Updated column KP is already stored in column KKW of W.
! 562: *
! 563: IF( KP.NE.KK ) THEN
! 564: *
! 565: * Copy non-updated column KK to column KP of submatrix A
! 566: * at step K. No need to copy element into column K
! 567: * (or K and K-1 for 2-by-2 pivot) of A, since these columns
! 568: * will be later overwritten.
! 569: *
! 570: A( KP, KP ) = DBLE( A( KK, KK ) )
! 571: CALL ZCOPY( KK-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
! 572: $ LDA )
! 573: CALL ZLACGV( KK-1-KP, A( KP, KP+1 ), LDA )
! 574: IF( KP.GT.1 )
! 575: $ CALL ZCOPY( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
! 576: *
! 577: * Interchange rows KK and KP in last K+1 to N columns of A
! 578: * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
! 579: * later overwritten). Interchange rows KK and KP
! 580: * in last KKW to NB columns of W.
! 581: *
! 582: IF( K.LT.N )
! 583: $ CALL ZSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
! 584: $ LDA )
! 585: CALL ZSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
! 586: $ LDW )
! 587: END IF
! 588: *
! 589: IF( KSTEP.EQ.1 ) THEN
! 590: *
! 591: * 1-by-1 pivot block D(k): column kw of W now holds
! 592: *
! 593: * W(kw) = U(k)*D(k),
! 594: *
! 595: * where U(k) is the k-th column of U
! 596: *
! 597: * (1) Store subdiag. elements of column U(k)
! 598: * and 1-by-1 block D(k) in column k of A.
! 599: * (NOTE: Diagonal element U(k,k) is a UNIT element
! 600: * and not stored)
! 601: * A(k,k) := D(k,k) = W(k,kw)
! 602: * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
! 603: *
! 604: * (NOTE: No need to use for Hermitian matrix
! 605: * A( K, K ) = REAL( W( K, K) ) to separately copy diagonal
! 606: * element D(k,k) from W (potentially saves only one load))
! 607: CALL ZCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
! 608: IF( K.GT.1 ) THEN
! 609: *
! 610: * (NOTE: No need to check if A(k,k) is NOT ZERO,
! 611: * since that was ensured earlier in pivot search:
! 612: * case A(k,k) = 0 falls into 2x2 pivot case(3))
! 613: *
! 614: * Handle division by a small number
! 615: *
! 616: T = DBLE( A( K, K ) )
! 617: IF( ABS( T ).GE.SFMIN ) THEN
! 618: R1 = ONE / T
! 619: CALL ZDSCAL( K-1, R1, A( 1, K ), 1 )
! 620: ELSE
! 621: DO 14 II = 1, K-1
! 622: A( II, K ) = A( II, K ) / T
! 623: 14 CONTINUE
! 624: END IF
! 625: *
! 626: * (2) Conjugate column W(kw)
! 627: *
! 628: CALL ZLACGV( K-1, W( 1, KW ), 1 )
! 629: *
! 630: * Store the superdiagonal element of D in array E
! 631: *
! 632: E( K ) = CZERO
! 633: *
! 634: END IF
! 635: *
! 636: ELSE
! 637: *
! 638: * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
! 639: *
! 640: * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
! 641: *
! 642: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
! 643: * of U
! 644: *
! 645: * (1) Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
! 646: * block D(k-1:k,k-1:k) in columns k-1 and k of A.
! 647: * (NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
! 648: * block and not stored)
! 649: * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
! 650: * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
! 651: * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
! 652: *
! 653: IF( K.GT.2 ) THEN
! 654: *
! 655: * Factor out the columns of the inverse of 2-by-2 pivot
! 656: * block D, so that each column contains 1, to reduce the
! 657: * number of FLOPS when we multiply panel
! 658: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
! 659: *
! 660: * D**(-1) = ( d11 cj(d21) )**(-1) =
! 661: * ( d21 d22 )
! 662: *
! 663: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
! 664: * ( (-d21) ( d11 ) )
! 665: *
! 666: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
! 667: *
! 668: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
! 669: * ( ( -1 ) ( d11/conj(d21) ) )
! 670: *
! 671: * = 1/(|d21|**2) * 1/(D22*D11-1) *
! 672: *
! 673: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
! 674: * ( ( -1 ) ( D22 ) )
! 675: *
! 676: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
! 677: * ( ( -1 ) ( D22 ) )
! 678: *
! 679: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
! 680: * ( ( -1 ) ( D22 ) )
! 681: *
! 682: * Handle division by a small number. (NOTE: order of
! 683: * operations is important)
! 684: *
! 685: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
! 686: * ( (( -1 ) ) (( D22 ) ) ),
! 687: *
! 688: * where D11 = d22/d21,
! 689: * D22 = d11/conj(d21),
! 690: * D21 = d21,
! 691: * T = 1/(D22*D11-1).
! 692: *
! 693: * (NOTE: No need to check for division by ZERO,
! 694: * since that was ensured earlier in pivot search:
! 695: * (a) d21 != 0 in 2x2 pivot case(4),
! 696: * since |d21| should be larger than |d11| and |d22|;
! 697: * (b) (D22*D11 - 1) != 0, since from (a),
! 698: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
! 699: *
! 700: D21 = W( K-1, KW )
! 701: D11 = W( K, KW ) / DCONJG( D21 )
! 702: D22 = W( K-1, KW-1 ) / D21
! 703: T = ONE / ( DBLE( D11*D22 )-ONE )
! 704: *
! 705: * Update elements in columns A(k-1) and A(k) as
! 706: * dot products of rows of ( W(kw-1) W(kw) ) and columns
! 707: * of D**(-1)
! 708: *
! 709: DO 20 J = 1, K - 2
! 710: A( J, K-1 ) = T*( ( D11*W( J, KW-1 )-W( J, KW ) ) /
! 711: $ D21 )
! 712: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
! 713: $ DCONJG( D21 ) )
! 714: 20 CONTINUE
! 715: END IF
! 716: *
! 717: * Copy diagonal elements of D(K) to A,
! 718: * copy superdiagonal element of D(K) to E(K) and
! 719: * ZERO out superdiagonal entry of A
! 720: *
! 721: A( K-1, K-1 ) = W( K-1, KW-1 )
! 722: A( K-1, K ) = CZERO
! 723: A( K, K ) = W( K, KW )
! 724: E( K ) = W( K-1, KW )
! 725: E( K-1 ) = CZERO
! 726: *
! 727: * (2) Conjugate columns W(kw) and W(kw-1)
! 728: *
! 729: CALL ZLACGV( K-1, W( 1, KW ), 1 )
! 730: CALL ZLACGV( K-2, W( 1, KW-1 ), 1 )
! 731: *
! 732: END IF
! 733: *
! 734: * End column K is nonsingular
! 735: *
! 736: END IF
! 737: *
! 738: * Store details of the interchanges in IPIV
! 739: *
! 740: IF( KSTEP.EQ.1 ) THEN
! 741: IPIV( K ) = KP
! 742: ELSE
! 743: IPIV( K ) = -P
! 744: IPIV( K-1 ) = -KP
! 745: END IF
! 746: *
! 747: * Decrease K and return to the start of the main loop
! 748: *
! 749: K = K - KSTEP
! 750: GO TO 10
! 751: *
! 752: 30 CONTINUE
! 753: *
! 754: * Update the upper triangle of A11 (= A(1:k,1:k)) as
! 755: *
! 756: * A11 := A11 - U12*D*U12**H = A11 - U12*W**H
! 757: *
! 758: * computing blocks of NB columns at a time (note that conjg(W) is
! 759: * actually stored)
! 760: *
! 761: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
! 762: JB = MIN( NB, K-J+1 )
! 763: *
! 764: * Update the upper triangle of the diagonal block
! 765: *
! 766: DO 40 JJ = J, J + JB - 1
! 767: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 768: CALL ZGEMV( 'No transpose', JJ-J+1, N-K, -CONE,
! 769: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, CONE,
! 770: $ A( J, JJ ), 1 )
! 771: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 772: 40 CONTINUE
! 773: *
! 774: * Update the rectangular superdiagonal block
! 775: *
! 776: IF( J.GE.2 )
! 777: $ CALL ZGEMM( 'No transpose', 'Transpose', J-1, JB, N-K,
! 778: $ -CONE, A( 1, K+1 ), LDA, W( J, KW+1 ), LDW,
! 779: $ CONE, A( 1, J ), LDA )
! 780: 50 CONTINUE
! 781: *
! 782: * Set KB to the number of columns factorized
! 783: *
! 784: KB = N - K
! 785: *
! 786: ELSE
! 787: *
! 788: * Factorize the leading columns of A using the lower triangle
! 789: * of A and working forwards, and compute the matrix W = L21*D
! 790: * for use in updating A22 (note that conjg(W) is actually stored)
! 791: *
! 792: * Initilize the unused last entry of the subdiagonal array E.
! 793: *
! 794: E( N ) = CZERO
! 795: *
! 796: * K is the main loop index, increasing from 1 in steps of 1 or 2
! 797: *
! 798: K = 1
! 799: 70 CONTINUE
! 800: *
! 801: * Exit from loop
! 802: *
! 803: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
! 804: $ GO TO 90
! 805: *
! 806: KSTEP = 1
! 807: P = K
! 808: *
! 809: * Copy column K of A to column K of W and update column K of W
! 810: *
! 811: W( K, K ) = DBLE( A( K, K ) )
! 812: IF( K.LT.N )
! 813: $ CALL ZCOPY( N-K, A( K+1, K ), 1, W( K+1, K ), 1 )
! 814: IF( K.GT.1 ) THEN
! 815: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE, A( K, 1 ),
! 816: $ LDA, W( K, 1 ), LDW, CONE, W( K, K ), 1 )
! 817: W( K, K ) = DBLE( W( K, K ) )
! 818: END IF
! 819: *
! 820: * Determine rows and columns to be interchanged and whether
! 821: * a 1-by-1 or 2-by-2 pivot block will be used
! 822: *
! 823: ABSAKK = ABS( DBLE( W( K, K ) ) )
! 824: *
! 825: * IMAX is the row-index of the largest off-diagonal element in
! 826: * column K, and COLMAX is its absolute value.
! 827: * Determine both COLMAX and IMAX.
! 828: *
! 829: IF( K.LT.N ) THEN
! 830: IMAX = K + IZAMAX( N-K, W( K+1, K ), 1 )
! 831: COLMAX = CABS1( W( IMAX, K ) )
! 832: ELSE
! 833: COLMAX = ZERO
! 834: END IF
! 835: *
! 836: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
! 837: *
! 838: * Column K is zero or underflow: set INFO and continue
! 839: *
! 840: IF( INFO.EQ.0 )
! 841: $ INFO = K
! 842: KP = K
! 843: A( K, K ) = DBLE( W( K, K ) )
! 844: IF( K.LT.N )
! 845: $ CALL ZCOPY( N-K, W( K+1, K ), 1, A( K+1, K ), 1 )
! 846: *
! 847: * Set E( K ) to zero
! 848: *
! 849: IF( K.LT.N )
! 850: $ E( K ) = CZERO
! 851: *
! 852: ELSE
! 853: *
! 854: * ============================================================
! 855: *
! 856: * BEGIN pivot search
! 857: *
! 858: * Case(1)
! 859: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
! 860: * (used to handle NaN and Inf)
! 861: *
! 862: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
! 863: *
! 864: * no interchange, use 1-by-1 pivot block
! 865: *
! 866: KP = K
! 867: *
! 868: ELSE
! 869: *
! 870: DONE = .FALSE.
! 871: *
! 872: * Loop until pivot found
! 873: *
! 874: 72 CONTINUE
! 875: *
! 876: * BEGIN pivot search loop body
! 877: *
! 878: *
! 879: * Copy column IMAX to column k+1 of W and update it
! 880: *
! 881: CALL ZCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
! 882: CALL ZLACGV( IMAX-K, W( K, K+1 ), 1 )
! 883: W( IMAX, K+1 ) = DBLE( A( IMAX, IMAX ) )
! 884: *
! 885: IF( IMAX.LT.N )
! 886: $ CALL ZCOPY( N-IMAX, A( IMAX+1, IMAX ), 1,
! 887: $ W( IMAX+1, K+1 ), 1 )
! 888: *
! 889: IF( K.GT.1 ) THEN
! 890: CALL ZGEMV( 'No transpose', N-K+1, K-1, -CONE,
! 891: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
! 892: $ CONE, W( K, K+1 ), 1 )
! 893: W( IMAX, K+1 ) = DBLE( W( IMAX, K+1 ) )
! 894: END IF
! 895: *
! 896: * JMAX is the column-index of the largest off-diagonal
! 897: * element in row IMAX, and ROWMAX is its absolute value.
! 898: * Determine both ROWMAX and JMAX.
! 899: *
! 900: IF( IMAX.NE.K ) THEN
! 901: JMAX = K - 1 + IZAMAX( IMAX-K, W( K, K+1 ), 1 )
! 902: ROWMAX = CABS1( W( JMAX, K+1 ) )
! 903: ELSE
! 904: ROWMAX = ZERO
! 905: END IF
! 906: *
! 907: IF( IMAX.LT.N ) THEN
! 908: ITEMP = IMAX + IZAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
! 909: DTEMP = CABS1( W( ITEMP, K+1 ) )
! 910: IF( DTEMP.GT.ROWMAX ) THEN
! 911: ROWMAX = DTEMP
! 912: JMAX = ITEMP
! 913: END IF
! 914: END IF
! 915: *
! 916: * Case(2)
! 917: * Equivalent to testing for
! 918: * ABS( REAL( W( IMAX,K+1 ) ) ).GE.ALPHA*ROWMAX
! 919: * (used to handle NaN and Inf)
! 920: *
! 921: IF( .NOT.( ABS( DBLE( W( IMAX,K+1 ) ) )
! 922: $ .LT.ALPHA*ROWMAX ) ) THEN
! 923: *
! 924: * interchange rows and columns K and IMAX,
! 925: * use 1-by-1 pivot block
! 926: *
! 927: KP = IMAX
! 928: *
! 929: * copy column K+1 of W to column K of W
! 930: *
! 931: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
! 932: *
! 933: DONE = .TRUE.
! 934: *
! 935: * Case(3)
! 936: * Equivalent to testing for ROWMAX.EQ.COLMAX,
! 937: * (used to handle NaN and Inf)
! 938: *
! 939: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
! 940: $ THEN
! 941: *
! 942: * interchange rows and columns K+1 and IMAX,
! 943: * use 2-by-2 pivot block
! 944: *
! 945: KP = IMAX
! 946: KSTEP = 2
! 947: DONE = .TRUE.
! 948: *
! 949: * Case(4)
! 950: ELSE
! 951: *
! 952: * Pivot not found: set params and repeat
! 953: *
! 954: P = IMAX
! 955: COLMAX = ROWMAX
! 956: IMAX = JMAX
! 957: *
! 958: * Copy updated JMAXth (next IMAXth) column to Kth of W
! 959: *
! 960: CALL ZCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
! 961: *
! 962: END IF
! 963: *
! 964: *
! 965: * End pivot search loop body
! 966: *
! 967: IF( .NOT.DONE ) GOTO 72
! 968: *
! 969: END IF
! 970: *
! 971: * END pivot search
! 972: *
! 973: * ============================================================
! 974: *
! 975: * KK is the column of A where pivoting step stopped
! 976: *
! 977: KK = K + KSTEP - 1
! 978: *
! 979: * Interchange rows and columns P and K (only for 2-by-2 pivot).
! 980: * Updated column P is already stored in column K of W.
! 981: *
! 982: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
! 983: *
! 984: * Copy non-updated column KK-1 to column P of submatrix A
! 985: * at step K. No need to copy element into columns
! 986: * K and K+1 of A for 2-by-2 pivot, since these columns
! 987: * will be later overwritten.
! 988: *
! 989: A( P, P ) = DBLE( A( K, K ) )
! 990: CALL ZCOPY( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
! 991: CALL ZLACGV( P-K-1, A( P, K+1 ), LDA )
! 992: IF( P.LT.N )
! 993: $ CALL ZCOPY( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
! 994: *
! 995: * Interchange rows K and P in first K-1 columns of A
! 996: * (columns K and K+1 of A for 2-by-2 pivot will be
! 997: * later overwritten). Interchange rows K and P
! 998: * in first KK columns of W.
! 999: *
! 1000: IF( K.GT.1 )
! 1001: $ CALL ZSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
! 1002: CALL ZSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
! 1003: END IF
! 1004: *
! 1005: * Interchange rows and columns KP and KK.
! 1006: * Updated column KP is already stored in column KK of W.
! 1007: *
! 1008: IF( KP.NE.KK ) THEN
! 1009: *
! 1010: * Copy non-updated column KK to column KP of submatrix A
! 1011: * at step K. No need to copy element into column K
! 1012: * (or K and K+1 for 2-by-2 pivot) of A, since these columns
! 1013: * will be later overwritten.
! 1014: *
! 1015: A( KP, KP ) = DBLE( A( KK, KK ) )
! 1016: CALL ZCOPY( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
! 1017: $ LDA )
! 1018: CALL ZLACGV( KP-KK-1, A( KP, KK+1 ), LDA )
! 1019: IF( KP.LT.N )
! 1020: $ CALL ZCOPY( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
! 1021: *
! 1022: * Interchange rows KK and KP in first K-1 columns of A
! 1023: * (column K (or K and K+1 for 2-by-2 pivot) of A will be
! 1024: * later overwritten). Interchange rows KK and KP
! 1025: * in first KK columns of W.
! 1026: *
! 1027: IF( K.GT.1 )
! 1028: $ CALL ZSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
! 1029: CALL ZSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
! 1030: END IF
! 1031: *
! 1032: IF( KSTEP.EQ.1 ) THEN
! 1033: *
! 1034: * 1-by-1 pivot block D(k): column k of W now holds
! 1035: *
! 1036: * W(k) = L(k)*D(k),
! 1037: *
! 1038: * where L(k) is the k-th column of L
! 1039: *
! 1040: * (1) Store subdiag. elements of column L(k)
! 1041: * and 1-by-1 block D(k) in column k of A.
! 1042: * (NOTE: Diagonal element L(k,k) is a UNIT element
! 1043: * and not stored)
! 1044: * A(k,k) := D(k,k) = W(k,k)
! 1045: * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
! 1046: *
! 1047: * (NOTE: No need to use for Hermitian matrix
! 1048: * A( K, K ) = REAL( W( K, K) ) to separately copy diagonal
! 1049: * element D(k,k) from W (potentially saves only one load))
! 1050: CALL ZCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
! 1051: IF( K.LT.N ) THEN
! 1052: *
! 1053: * (NOTE: No need to check if A(k,k) is NOT ZERO,
! 1054: * since that was ensured earlier in pivot search:
! 1055: * case A(k,k) = 0 falls into 2x2 pivot case(3))
! 1056: *
! 1057: * Handle division by a small number
! 1058: *
! 1059: T = DBLE( A( K, K ) )
! 1060: IF( ABS( T ).GE.SFMIN ) THEN
! 1061: R1 = ONE / T
! 1062: CALL ZDSCAL( N-K, R1, A( K+1, K ), 1 )
! 1063: ELSE
! 1064: DO 74 II = K + 1, N
! 1065: A( II, K ) = A( II, K ) / T
! 1066: 74 CONTINUE
! 1067: END IF
! 1068: *
! 1069: * (2) Conjugate column W(k)
! 1070: *
! 1071: CALL ZLACGV( N-K, W( K+1, K ), 1 )
! 1072: *
! 1073: * Store the subdiagonal element of D in array E
! 1074: *
! 1075: E( K ) = CZERO
! 1076: *
! 1077: END IF
! 1078: *
! 1079: ELSE
! 1080: *
! 1081: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
! 1082: *
! 1083: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
! 1084: *
! 1085: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
! 1086: * of L
! 1087: *
! 1088: * (1) Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
! 1089: * block D(k:k+1,k:k+1) in columns k and k+1 of A.
! 1090: * NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
! 1091: * block and not stored.
! 1092: * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
! 1093: * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
! 1094: * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
! 1095: *
! 1096: IF( K.LT.N-1 ) THEN
! 1097: *
! 1098: * Factor out the columns of the inverse of 2-by-2 pivot
! 1099: * block D, so that each column contains 1, to reduce the
! 1100: * number of FLOPS when we multiply panel
! 1101: * ( W(kw-1) W(kw) ) by this inverse, i.e. by D**(-1).
! 1102: *
! 1103: * D**(-1) = ( d11 cj(d21) )**(-1) =
! 1104: * ( d21 d22 )
! 1105: *
! 1106: * = 1/(d11*d22-|d21|**2) * ( ( d22) (-cj(d21) ) ) =
! 1107: * ( (-d21) ( d11 ) )
! 1108: *
! 1109: * = 1/(|d21|**2) * 1/((d11/cj(d21))*(d22/d21)-1) *
! 1110: *
! 1111: * * ( d21*( d22/d21 ) conj(d21)*( - 1 ) ) =
! 1112: * ( ( -1 ) ( d11/conj(d21) ) )
! 1113: *
! 1114: * = 1/(|d21|**2) * 1/(D22*D11-1) *
! 1115: *
! 1116: * * ( d21*( D11 ) conj(d21)*( -1 ) ) =
! 1117: * ( ( -1 ) ( D22 ) )
! 1118: *
! 1119: * = (1/|d21|**2) * T * ( d21*( D11 ) conj(d21)*( -1 ) ) =
! 1120: * ( ( -1 ) ( D22 ) )
! 1121: *
! 1122: * = ( (T/conj(d21))*( D11 ) (T/d21)*( -1 ) ) =
! 1123: * ( ( -1 ) ( D22 ) )
! 1124: *
! 1125: * Handle division by a small number. (NOTE: order of
! 1126: * operations is important)
! 1127: *
! 1128: * = ( T*(( D11 )/conj(D21)) T*(( -1 )/D21 ) )
! 1129: * ( (( -1 ) ) (( D22 ) ) ),
! 1130: *
! 1131: * where D11 = d22/d21,
! 1132: * D22 = d11/conj(d21),
! 1133: * D21 = d21,
! 1134: * T = 1/(D22*D11-1).
! 1135: *
! 1136: * (NOTE: No need to check for division by ZERO,
! 1137: * since that was ensured earlier in pivot search:
! 1138: * (a) d21 != 0 in 2x2 pivot case(4),
! 1139: * since |d21| should be larger than |d11| and |d22|;
! 1140: * (b) (D22*D11 - 1) != 0, since from (a),
! 1141: * both |D11| < 1, |D22| < 1, hence |D22*D11| << 1.)
! 1142: *
! 1143: D21 = W( K+1, K )
! 1144: D11 = W( K+1, K+1 ) / D21
! 1145: D22 = W( K, K ) / DCONJG( D21 )
! 1146: T = ONE / ( DBLE( D11*D22 )-ONE )
! 1147: *
! 1148: * Update elements in columns A(k) and A(k+1) as
! 1149: * dot products of rows of ( W(k) W(k+1) ) and columns
! 1150: * of D**(-1)
! 1151: *
! 1152: DO 80 J = K + 2, N
! 1153: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
! 1154: $ DCONJG( D21 ) )
! 1155: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
! 1156: $ D21 )
! 1157: 80 CONTINUE
! 1158: END IF
! 1159: *
! 1160: * Copy diagonal elements of D(K) to A,
! 1161: * copy subdiagonal element of D(K) to E(K) and
! 1162: * ZERO out subdiagonal entry of A
! 1163: *
! 1164: A( K, K ) = W( K, K )
! 1165: A( K+1, K ) = CZERO
! 1166: A( K+1, K+1 ) = W( K+1, K+1 )
! 1167: E( K ) = W( K+1, K )
! 1168: E( K+1 ) = CZERO
! 1169: *
! 1170: * (2) Conjugate columns W(k) and W(k+1)
! 1171: *
! 1172: CALL ZLACGV( N-K, W( K+1, K ), 1 )
! 1173: CALL ZLACGV( N-K-1, W( K+2, K+1 ), 1 )
! 1174: *
! 1175: END IF
! 1176: *
! 1177: * End column K is nonsingular
! 1178: *
! 1179: END IF
! 1180: *
! 1181: * Store details of the interchanges in IPIV
! 1182: *
! 1183: IF( KSTEP.EQ.1 ) THEN
! 1184: IPIV( K ) = KP
! 1185: ELSE
! 1186: IPIV( K ) = -P
! 1187: IPIV( K+1 ) = -KP
! 1188: END IF
! 1189: *
! 1190: * Increase K and return to the start of the main loop
! 1191: *
! 1192: K = K + KSTEP
! 1193: GO TO 70
! 1194: *
! 1195: 90 CONTINUE
! 1196: *
! 1197: * Update the lower triangle of A22 (= A(k:n,k:n)) as
! 1198: *
! 1199: * A22 := A22 - L21*D*L21**H = A22 - L21*W**H
! 1200: *
! 1201: * computing blocks of NB columns at a time (note that conjg(W) is
! 1202: * actually stored)
! 1203: *
! 1204: DO 110 J = K, N, NB
! 1205: JB = MIN( NB, N-J+1 )
! 1206: *
! 1207: * Update the lower triangle of the diagonal block
! 1208: *
! 1209: DO 100 JJ = J, J + JB - 1
! 1210: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 1211: CALL ZGEMV( 'No transpose', J+JB-JJ, K-1, -CONE,
! 1212: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, CONE,
! 1213: $ A( JJ, JJ ), 1 )
! 1214: A( JJ, JJ ) = DBLE( A( JJ, JJ ) )
! 1215: 100 CONTINUE
! 1216: *
! 1217: * Update the rectangular subdiagonal block
! 1218: *
! 1219: IF( J+JB.LE.N )
! 1220: $ CALL ZGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
! 1221: $ K-1, -CONE, A( J+JB, 1 ), LDA, W( J, 1 ),
! 1222: $ LDW, CONE, A( J+JB, J ), LDA )
! 1223: 110 CONTINUE
! 1224: *
! 1225: * Set KB to the number of columns factorized
! 1226: *
! 1227: KB = K - 1
! 1228: *
! 1229: END IF
! 1230: RETURN
! 1231: *
! 1232: * End of ZLAHEF_RK
! 1233: *
! 1234: END
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