Annotation of rpl/lapack/lapack/zhetrd_he2hb.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b ZHETRD_HE2HB
! 2: *
! 3: * @precisions fortran z -> s d c
! 4: *
! 5: * =========== DOCUMENTATION ===========
! 6: *
! 7: * Online html documentation available at
! 8: * http://www.netlib.org/lapack/explore-html/
! 9: *
! 10: *> \htmlonly
! 11: *> Download ZHETRD_HE2HB + dependencies
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetrd.f">
! 13: *> [TGZ]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetrd.f">
! 15: *> [ZIP]</a>
! 16: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetrd.f">
! 17: *> [TXT]</a>
! 18: *> \endhtmlonly
! 19: *
! 20: * Definition:
! 21: * ===========
! 22: *
! 23: * SUBROUTINE ZHETRD_HE2HB( UPLO, N, KD, A, LDA, AB, LDAB, TAU,
! 24: * WORK, LWORK, INFO )
! 25: *
! 26: * IMPLICIT NONE
! 27: *
! 28: * .. Scalar Arguments ..
! 29: * CHARACTER UPLO
! 30: * INTEGER INFO, LDA, LDAB, LWORK, N, KD
! 31: * ..
! 32: * .. Array Arguments ..
! 33: * COMPLEX*16 A( LDA, * ), AB( LDAB, * ),
! 34: * TAU( * ), WORK( * )
! 35: * ..
! 36: *
! 37: *
! 38: *> \par Purpose:
! 39: * =============
! 40: *>
! 41: *> \verbatim
! 42: *>
! 43: *> ZHETRD_HE2HB reduces a complex Hermitian matrix A to complex Hermitian
! 44: *> band-diagonal form AB by a unitary similarity transformation:
! 45: *> Q**H * A * Q = AB.
! 46: *> \endverbatim
! 47: *
! 48: * Arguments:
! 49: * ==========
! 50: *
! 51: *> \param[in] UPLO
! 52: *> \verbatim
! 53: *> UPLO is CHARACTER*1
! 54: *> = 'U': Upper triangle of A is stored;
! 55: *> = 'L': Lower triangle of A is stored.
! 56: *> \endverbatim
! 57: *>
! 58: *> \param[in] N
! 59: *> \verbatim
! 60: *> N is INTEGER
! 61: *> The order of the matrix A. N >= 0.
! 62: *> \endverbatim
! 63: *>
! 64: *> \param[in] KD
! 65: *> \verbatim
! 66: *> KD is INTEGER
! 67: *> The number of superdiagonals of the reduced matrix if UPLO = 'U',
! 68: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 69: *> The reduced matrix is stored in the array AB.
! 70: *> \endverbatim
! 71: *>
! 72: *> \param[in,out] A
! 73: *> \verbatim
! 74: *> A is COMPLEX*16 array, dimension (LDA,N)
! 75: *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
! 76: *> N-by-N upper triangular part of A contains the upper
! 77: *> triangular part of the matrix A, and the strictly lower
! 78: *> triangular part of A is not referenced. If UPLO = 'L', the
! 79: *> leading N-by-N lower triangular part of A contains the lower
! 80: *> triangular part of the matrix A, and the strictly upper
! 81: *> triangular part of A is not referenced.
! 82: *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
! 83: *> of A are overwritten by the corresponding elements of the
! 84: *> tridiagonal matrix T, and the elements above the first
! 85: *> superdiagonal, with the array TAU, represent the unitary
! 86: *> matrix Q as a product of elementary reflectors; if UPLO
! 87: *> = 'L', the diagonal and first subdiagonal of A are over-
! 88: *> written by the corresponding elements of the tridiagonal
! 89: *> matrix T, and the elements below the first subdiagonal, with
! 90: *> the array TAU, represent the unitary matrix Q as a product
! 91: *> of elementary reflectors. See Further Details.
! 92: *> \endverbatim
! 93: *>
! 94: *> \param[in] LDA
! 95: *> \verbatim
! 96: *> LDA is INTEGER
! 97: *> The leading dimension of the array A. LDA >= max(1,N).
! 98: *> \endverbatim
! 99: *>
! 100: *> \param[out] AB
! 101: *> \verbatim
! 102: *> AB is COMPLEX*16 array, dimension (LDAB,N)
! 103: *> On exit, the upper or lower triangle of the Hermitian band
! 104: *> matrix A, stored in the first KD+1 rows of the array. The
! 105: *> j-th column of A is stored in the j-th column of the array AB
! 106: *> as follows:
! 107: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
! 108: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
! 109: *> \endverbatim
! 110: *>
! 111: *> \param[in] LDAB
! 112: *> \verbatim
! 113: *> LDAB is INTEGER
! 114: *> The leading dimension of the array AB. LDAB >= KD+1.
! 115: *> \endverbatim
! 116: *>
! 117: *> \param[out] TAU
! 118: *> \verbatim
! 119: *> TAU is COMPLEX*16 array, dimension (N-KD)
! 120: *> The scalar factors of the elementary reflectors (see Further
! 121: *> Details).
! 122: *> \endverbatim
! 123: *>
! 124: *> \param[out] WORK
! 125: *> \verbatim
! 126: *> WORK is COMPLEX*16 array, dimension LWORK.
! 127: *> On exit, if INFO = 0, or if LWORK=-1,
! 128: *> WORK(1) returns the size of LWORK.
! 129: *> \endverbatim
! 130: *>
! 131: *> \param[in] LWORK
! 132: *> \verbatim
! 133: *> LWORK is INTEGER
! 134: *> The dimension of the array WORK which should be calculated
! 135: * by a workspace query. LWORK = MAX(1, LWORK_QUERY)
! 136: *> If LWORK = -1, then a workspace query is assumed; the routine
! 137: *> only calculates the optimal size of the WORK array, returns
! 138: *> this value as the first entry of the WORK array, and no error
! 139: *> message related to LWORK is issued by XERBLA.
! 140: *> LWORK_QUERY = N*KD + N*max(KD,FACTOPTNB) + 2*KD*KD
! 141: *> where FACTOPTNB is the blocking used by the QR or LQ
! 142: *> algorithm, usually FACTOPTNB=128 is a good choice otherwise
! 143: *> putting LWORK=-1 will provide the size of WORK.
! 144: *> \endverbatim
! 145: *>
! 146: *> \param[out] INFO
! 147: *> \verbatim
! 148: *> INFO is INTEGER
! 149: *> = 0: successful exit
! 150: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 151: *> \endverbatim
! 152: *
! 153: * Authors:
! 154: * ========
! 155: *
! 156: *> \author Univ. of Tennessee
! 157: *> \author Univ. of California Berkeley
! 158: *> \author Univ. of Colorado Denver
! 159: *> \author NAG Ltd.
! 160: *
! 161: *> \date December 2016
! 162: *
! 163: *> \ingroup complex16HEcomputational
! 164: *
! 165: *> \par Further Details:
! 166: * =====================
! 167: *>
! 168: *> \verbatim
! 169: *>
! 170: *> Implemented by Azzam Haidar.
! 171: *>
! 172: *> All details are available on technical report, SC11, SC13 papers.
! 173: *>
! 174: *> Azzam Haidar, Hatem Ltaief, and Jack Dongarra.
! 175: *> Parallel reduction to condensed forms for symmetric eigenvalue problems
! 176: *> using aggregated fine-grained and memory-aware kernels. In Proceedings
! 177: *> of 2011 International Conference for High Performance Computing,
! 178: *> Networking, Storage and Analysis (SC '11), New York, NY, USA,
! 179: *> Article 8 , 11 pages.
! 180: *> http://doi.acm.org/10.1145/2063384.2063394
! 181: *>
! 182: *> A. Haidar, J. Kurzak, P. Luszczek, 2013.
! 183: *> An improved parallel singular value algorithm and its implementation
! 184: *> for multicore hardware, In Proceedings of 2013 International Conference
! 185: *> for High Performance Computing, Networking, Storage and Analysis (SC '13).
! 186: *> Denver, Colorado, USA, 2013.
! 187: *> Article 90, 12 pages.
! 188: *> http://doi.acm.org/10.1145/2503210.2503292
! 189: *>
! 190: *> A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra.
! 191: *> A novel hybrid CPU-GPU generalized eigensolver for electronic structure
! 192: *> calculations based on fine-grained memory aware tasks.
! 193: *> International Journal of High Performance Computing Applications.
! 194: *> Volume 28 Issue 2, Pages 196-209, May 2014.
! 195: *> http://hpc.sagepub.com/content/28/2/196
! 196: *>
! 197: *> \endverbatim
! 198: *>
! 199: *> \verbatim
! 200: *>
! 201: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
! 202: *> reflectors
! 203: *>
! 204: *> Q = H(k)**H . . . H(2)**H H(1)**H, where k = n-kd.
! 205: *>
! 206: *> Each H(i) has the form
! 207: *>
! 208: *> H(i) = I - tau * v * v**H
! 209: *>
! 210: *> where tau is a complex scalar, and v is a complex vector with
! 211: *> v(1:i+kd-1) = 0 and v(i+kd) = 1; conjg(v(i+kd+1:n)) is stored on exit in
! 212: *> A(i,i+kd+1:n), and tau in TAU(i).
! 213: *>
! 214: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
! 215: *> reflectors
! 216: *>
! 217: *> Q = H(1) H(2) . . . H(k), where k = n-kd.
! 218: *>
! 219: *> Each H(i) has the form
! 220: *>
! 221: *> H(i) = I - tau * v * v**H
! 222: *>
! 223: *> where tau is a complex scalar, and v is a complex vector with
! 224: *> v(kd+1:i) = 0 and v(i+kd+1) = 1; v(i+kd+2:n) is stored on exit in
! 225: * A(i+kd+2:n,i), and tau in TAU(i).
! 226: *>
! 227: *> The contents of A on exit are illustrated by the following examples
! 228: *> with n = 5:
! 229: *>
! 230: *> if UPLO = 'U': if UPLO = 'L':
! 231: *>
! 232: *> ( ab ab/v1 v1 v1 v1 ) ( ab )
! 233: *> ( ab ab/v2 v2 v2 ) ( ab/v1 ab )
! 234: *> ( ab ab/v3 v3 ) ( v1 ab/v2 ab )
! 235: *> ( ab ab/v4 ) ( v1 v2 ab/v3 ab )
! 236: *> ( ab ) ( v1 v2 v3 ab/v4 ab )
! 237: *>
! 238: *> where d and e denote diagonal and off-diagonal elements of T, and vi
! 239: *> denotes an element of the vector defining H(i).
! 240: *> \endverbatim
! 241: *>
! 242: * =====================================================================
! 243: SUBROUTINE ZHETRD_HE2HB( UPLO, N, KD, A, LDA, AB, LDAB, TAU,
! 244: $ WORK, LWORK, INFO )
! 245: *
! 246: IMPLICIT NONE
! 247: *
! 248: * -- LAPACK computational routine (version 3.7.0) --
! 249: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 250: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 251: * December 2016
! 252: *
! 253: * .. Scalar Arguments ..
! 254: CHARACTER UPLO
! 255: INTEGER INFO, LDA, LDAB, LWORK, N, KD
! 256: * ..
! 257: * .. Array Arguments ..
! 258: COMPLEX*16 A( LDA, * ), AB( LDAB, * ),
! 259: $ TAU( * ), WORK( * )
! 260: * ..
! 261: *
! 262: * =====================================================================
! 263: *
! 264: * .. Parameters ..
! 265: DOUBLE PRECISION RONE
! 266: COMPLEX*16 ZERO, ONE, HALF
! 267: PARAMETER ( RONE = 1.0D+0,
! 268: $ ZERO = ( 0.0D+0, 0.0D+0 ),
! 269: $ ONE = ( 1.0D+0, 0.0D+0 ),
! 270: $ HALF = ( 0.5D+0, 0.0D+0 ) )
! 271: * ..
! 272: * .. Local Scalars ..
! 273: LOGICAL LQUERY, UPPER
! 274: INTEGER I, J, IINFO, LWMIN, PN, PK, LK,
! 275: $ LDT, LDW, LDS2, LDS1,
! 276: $ LS2, LS1, LW, LT,
! 277: $ TPOS, WPOS, S2POS, S1POS
! 278: * ..
! 279: * .. External Subroutines ..
! 280: EXTERNAL XERBLA, ZHER2K, ZHEMM, ZGEMM,
! 281: $ ZLARFT, ZGELQF, ZGEQRF, ZLASET
! 282: * ..
! 283: * .. Intrinsic Functions ..
! 284: INTRINSIC MIN, MAX
! 285: * ..
! 286: * .. External Functions ..
! 287: LOGICAL LSAME
! 288: INTEGER ILAENV
! 289: EXTERNAL LSAME, ILAENV
! 290: * ..
! 291: * .. Executable Statements ..
! 292: *
! 293: * Determine the minimal workspace size required
! 294: * and test the input parameters
! 295: *
! 296: INFO = 0
! 297: UPPER = LSAME( UPLO, 'U' )
! 298: LQUERY = ( LWORK.EQ.-1 )
! 299: LWMIN = ILAENV( 20, 'ZHETRD_HE2HB', '', N, KD, -1, -1 )
! 300:
! 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( KD.LT.0 ) THEN
! 306: INFO = -3
! 307: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 308: INFO = -5
! 309: ELSE IF( LDAB.LT.MAX( 1, KD+1 ) ) THEN
! 310: INFO = -7
! 311: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
! 312: INFO = -10
! 313: END IF
! 314: *
! 315: IF( INFO.NE.0 ) THEN
! 316: CALL XERBLA( 'ZHETRD_HE2HB', -INFO )
! 317: RETURN
! 318: ELSE IF( LQUERY ) THEN
! 319: WORK( 1 ) = LWMIN
! 320: RETURN
! 321: END IF
! 322: *
! 323: * Quick return if possible
! 324: * Copy the upper/lower portion of A into AB
! 325: *
! 326: IF( N.LE.KD+1 ) THEN
! 327: IF( UPPER ) THEN
! 328: DO 100 I = 1, N
! 329: LK = MIN( KD+1, I )
! 330: CALL ZCOPY( LK, A( I-LK+1, I ), 1,
! 331: $ AB( KD+1-LK+1, I ), 1 )
! 332: 100 CONTINUE
! 333: ELSE
! 334: DO 110 I = 1, N
! 335: LK = MIN( KD+1, N-I+1 )
! 336: CALL ZCOPY( LK, A( I, I ), 1, AB( 1, I ), 1 )
! 337: 110 CONTINUE
! 338: ENDIF
! 339: WORK( 1 ) = 1
! 340: RETURN
! 341: END IF
! 342: *
! 343: * Determine the pointer position for the workspace
! 344: *
! 345: LDT = KD
! 346: LDS1 = KD
! 347: LT = LDT*KD
! 348: LW = N*KD
! 349: LS1 = LDS1*KD
! 350: LS2 = LWMIN - LT - LW - LS1
! 351: * LS2 = N*MAX(KD,FACTOPTNB)
! 352: TPOS = 1
! 353: WPOS = TPOS + LT
! 354: S1POS = WPOS + LW
! 355: S2POS = S1POS + LS1
! 356: IF( UPPER ) THEN
! 357: LDW = KD
! 358: LDS2 = KD
! 359: ELSE
! 360: LDW = N
! 361: LDS2 = N
! 362: ENDIF
! 363: *
! 364: *
! 365: * Set the workspace of the triangular matrix T to zero once such a
! 366: * way everytime T is generated the upper/lower portion will be always zero
! 367: *
! 368: CALL ZLASET( "A", LDT, KD, ZERO, ZERO, WORK( TPOS ), LDT )
! 369: *
! 370: IF( UPPER ) THEN
! 371: DO 10 I = 1, N - KD, KD
! 372: PN = N-I-KD+1
! 373: PK = MIN( N-I-KD+1, KD )
! 374: *
! 375: * Compute the LQ factorization of the current block
! 376: *
! 377: CALL ZGELQF( KD, PN, A( I, I+KD ), LDA,
! 378: $ TAU( I ), WORK( S2POS ), LS2, IINFO )
! 379: *
! 380: * Copy the upper portion of A into AB
! 381: *
! 382: DO 20 J = I, I+PK-1
! 383: LK = MIN( KD, N-J ) + 1
! 384: CALL ZCOPY( LK, A( J, J ), LDA, AB( KD+1, J ), LDAB-1 )
! 385: 20 CONTINUE
! 386: *
! 387: CALL ZLASET( 'Lower', PK, PK, ZERO, ONE,
! 388: $ A( I, I+KD ), LDA )
! 389: *
! 390: * Form the matrix T
! 391: *
! 392: CALL ZLARFT( 'Forward', 'Rowwise', PN, PK,
! 393: $ A( I, I+KD ), LDA, TAU( I ),
! 394: $ WORK( TPOS ), LDT )
! 395: *
! 396: * Compute W:
! 397: *
! 398: CALL ZGEMM( 'Conjugate', 'No transpose', PK, PN, PK,
! 399: $ ONE, WORK( TPOS ), LDT,
! 400: $ A( I, I+KD ), LDA,
! 401: $ ZERO, WORK( S2POS ), LDS2 )
! 402: *
! 403: CALL ZHEMM( 'Right', UPLO, PK, PN,
! 404: $ ONE, A( I+KD, I+KD ), LDA,
! 405: $ WORK( S2POS ), LDS2,
! 406: $ ZERO, WORK( WPOS ), LDW )
! 407: *
! 408: CALL ZGEMM( 'No transpose', 'Conjugate', PK, PK, PN,
! 409: $ ONE, WORK( WPOS ), LDW,
! 410: $ WORK( S2POS ), LDS2,
! 411: $ ZERO, WORK( S1POS ), LDS1 )
! 412: *
! 413: CALL ZGEMM( 'No transpose', 'No transpose', PK, PN, PK,
! 414: $ -HALF, WORK( S1POS ), LDS1,
! 415: $ A( I, I+KD ), LDA,
! 416: $ ONE, WORK( WPOS ), LDW )
! 417: *
! 418: *
! 419: * Update the unreduced submatrix A(i+kd:n,i+kd:n), using
! 420: * an update of the form: A := A - V'*W - W'*V
! 421: *
! 422: CALL ZHER2K( UPLO, 'Conjugate', PN, PK,
! 423: $ -ONE, A( I, I+KD ), LDA,
! 424: $ WORK( WPOS ), LDW,
! 425: $ RONE, A( I+KD, I+KD ), LDA )
! 426: 10 CONTINUE
! 427: *
! 428: * Copy the upper band to AB which is the band storage matrix
! 429: *
! 430: DO 30 J = N-KD+1, N
! 431: LK = MIN(KD, N-J) + 1
! 432: CALL ZCOPY( LK, A( J, J ), LDA, AB( KD+1, J ), LDAB-1 )
! 433: 30 CONTINUE
! 434: *
! 435: ELSE
! 436: *
! 437: * Reduce the lower triangle of A to lower band matrix
! 438: *
! 439: DO 40 I = 1, N - KD, KD
! 440: PN = N-I-KD+1
! 441: PK = MIN( N-I-KD+1, KD )
! 442: *
! 443: * Compute the QR factorization of the current block
! 444: *
! 445: CALL ZGEQRF( PN, KD, A( I+KD, I ), LDA,
! 446: $ TAU( I ), WORK( S2POS ), LS2, IINFO )
! 447: *
! 448: * Copy the upper portion of A into AB
! 449: *
! 450: DO 50 J = I, I+PK-1
! 451: LK = MIN( KD, N-J ) + 1
! 452: CALL ZCOPY( LK, A( J, J ), 1, AB( 1, J ), 1 )
! 453: 50 CONTINUE
! 454: *
! 455: CALL ZLASET( 'Upper', PK, PK, ZERO, ONE,
! 456: $ A( I+KD, I ), LDA )
! 457: *
! 458: * Form the matrix T
! 459: *
! 460: CALL ZLARFT( 'Forward', 'Columnwise', PN, PK,
! 461: $ A( I+KD, I ), LDA, TAU( I ),
! 462: $ WORK( TPOS ), LDT )
! 463: *
! 464: * Compute W:
! 465: *
! 466: CALL ZGEMM( 'No transpose', 'No transpose', PN, PK, PK,
! 467: $ ONE, A( I+KD, I ), LDA,
! 468: $ WORK( TPOS ), LDT,
! 469: $ ZERO, WORK( S2POS ), LDS2 )
! 470: *
! 471: CALL ZHEMM( 'Left', UPLO, PN, PK,
! 472: $ ONE, A( I+KD, I+KD ), LDA,
! 473: $ WORK( S2POS ), LDS2,
! 474: $ ZERO, WORK( WPOS ), LDW )
! 475: *
! 476: CALL ZGEMM( 'Conjugate', 'No transpose', PK, PK, PN,
! 477: $ ONE, WORK( S2POS ), LDS2,
! 478: $ WORK( WPOS ), LDW,
! 479: $ ZERO, WORK( S1POS ), LDS1 )
! 480: *
! 481: CALL ZGEMM( 'No transpose', 'No transpose', PN, PK, PK,
! 482: $ -HALF, A( I+KD, I ), LDA,
! 483: $ WORK( S1POS ), LDS1,
! 484: $ ONE, WORK( WPOS ), LDW )
! 485: *
! 486: *
! 487: * Update the unreduced submatrix A(i+kd:n,i+kd:n), using
! 488: * an update of the form: A := A - V*W' - W*V'
! 489: *
! 490: CALL ZHER2K( UPLO, 'No transpose', PN, PK,
! 491: $ -ONE, A( I+KD, I ), LDA,
! 492: $ WORK( WPOS ), LDW,
! 493: $ RONE, A( I+KD, I+KD ), LDA )
! 494: * ==================================================================
! 495: * RESTORE A FOR COMPARISON AND CHECKING TO BE REMOVED
! 496: * DO 45 J = I, I+PK-1
! 497: * LK = MIN( KD, N-J ) + 1
! 498: * CALL ZCOPY( LK, AB( 1, J ), 1, A( J, J ), 1 )
! 499: * 45 CONTINUE
! 500: * ==================================================================
! 501: 40 CONTINUE
! 502: *
! 503: * Copy the lower band to AB which is the band storage matrix
! 504: *
! 505: DO 60 J = N-KD+1, N
! 506: LK = MIN(KD, N-J) + 1
! 507: CALL ZCOPY( LK, A( J, J ), 1, AB( 1, J ), 1 )
! 508: 60 CONTINUE
! 509:
! 510: END IF
! 511: *
! 512: WORK( 1 ) = LWMIN
! 513: RETURN
! 514: *
! 515: * End of ZHETRD_HE2HB
! 516: *
! 517: END
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