Annotation of rpl/lapack/lapack/dorbdb3.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b DORBDB3
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DORBDB3 + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dorbdb3.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorbdb3.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorbdb3.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DORBDB3( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
! 22: * TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * DOUBLE PRECISION PHI(*), THETA(*)
! 29: * DOUBLE PRECISION TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
! 30: * $ X11(LDX11,*), X21(LDX21,*)
! 31: * ..
! 32: *
! 33: *
! 34: *> \par Purpose:
! 35: *> =============
! 36: *>
! 37: *>\verbatim
! 38: *>
! 39: *> DORBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny
! 40: *> matrix X with orthonomal columns:
! 41: *>
! 42: *> [ B11 ]
! 43: *> [ X11 ] [ P1 | ] [ 0 ]
! 44: *> [-----] = [---------] [-----] Q1**T .
! 45: *> [ X21 ] [ | P2 ] [ B21 ]
! 46: *> [ 0 ]
! 47: *>
! 48: *> X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P,
! 49: *> Q, or M-Q. Routines DORBDB1, DORBDB2, and DORBDB4 handle cases in
! 50: *> which M-P is not the minimum dimension.
! 51: *>
! 52: *> The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
! 53: *> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
! 54: *> Householder vectors.
! 55: *>
! 56: *> B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented
! 57: *> implicitly by angles THETA, PHI.
! 58: *>
! 59: *>\endverbatim
! 60: *
! 61: * Arguments:
! 62: * ==========
! 63: *
! 64: *> \param[in] M
! 65: *> \verbatim
! 66: *> M is INTEGER
! 67: *> The number of rows X11 plus the number of rows in X21.
! 68: *> \endverbatim
! 69: *>
! 70: *> \param[in] P
! 71: *> \verbatim
! 72: *> P is INTEGER
! 73: *> The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).
! 74: *> \endverbatim
! 75: *>
! 76: *> \param[in] Q
! 77: *> \verbatim
! 78: *> Q is INTEGER
! 79: *> The number of columns in X11 and X21. 0 <= Q <= M.
! 80: *> \endverbatim
! 81: *>
! 82: *> \param[in,out] X11
! 83: *> \verbatim
! 84: *> X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
! 85: *> On entry, the top block of the matrix X to be reduced. On
! 86: *> exit, the columns of tril(X11) specify reflectors for P1 and
! 87: *> the rows of triu(X11,1) specify reflectors for Q1.
! 88: *> \endverbatim
! 89: *>
! 90: *> \param[in] LDX11
! 91: *> \verbatim
! 92: *> LDX11 is INTEGER
! 93: *> The leading dimension of X11. LDX11 >= P.
! 94: *> \endverbatim
! 95: *>
! 96: *> \param[in,out] X21
! 97: *> \verbatim
! 98: *> X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
! 99: *> On entry, the bottom block of the matrix X to be reduced. On
! 100: *> exit, the columns of tril(X21) specify reflectors for P2.
! 101: *> \endverbatim
! 102: *>
! 103: *> \param[in] LDX21
! 104: *> \verbatim
! 105: *> LDX21 is INTEGER
! 106: *> The leading dimension of X21. LDX21 >= M-P.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[out] THETA
! 110: *> \verbatim
! 111: *> THETA is DOUBLE PRECISION array, dimension (Q)
! 112: *> The entries of the bidiagonal blocks B11, B21 are defined by
! 113: *> THETA and PHI. See Further Details.
! 114: *> \endverbatim
! 115: *>
! 116: *> \param[out] PHI
! 117: *> \verbatim
! 118: *> PHI is DOUBLE PRECISION array, dimension (Q-1)
! 119: *> The entries of the bidiagonal blocks B11, B21 are defined by
! 120: *> THETA and PHI. See Further Details.
! 121: *> \endverbatim
! 122: *>
! 123: *> \param[out] TAUP1
! 124: *> \verbatim
! 125: *> TAUP1 is DOUBLE PRECISION array, dimension (P)
! 126: *> The scalar factors of the elementary reflectors that define
! 127: *> P1.
! 128: *> \endverbatim
! 129: *>
! 130: *> \param[out] TAUP2
! 131: *> \verbatim
! 132: *> TAUP2 is DOUBLE PRECISION array, dimension (M-P)
! 133: *> The scalar factors of the elementary reflectors that define
! 134: *> P2.
! 135: *> \endverbatim
! 136: *>
! 137: *> \param[out] TAUQ1
! 138: *> \verbatim
! 139: *> TAUQ1 is DOUBLE PRECISION array, dimension (Q)
! 140: *> The scalar factors of the elementary reflectors that define
! 141: *> Q1.
! 142: *> \endverbatim
! 143: *>
! 144: *> \param[out] WORK
! 145: *> \verbatim
! 146: *> WORK is DOUBLE PRECISION array, dimension (LWORK)
! 147: *> \endverbatim
! 148: *>
! 149: *> \param[in] LWORK
! 150: *> \verbatim
! 151: *> LWORK is INTEGER
! 152: *> The dimension of the array WORK. LWORK >= M-Q.
! 153: *>
! 154: *> If LWORK = -1, then a workspace query is assumed; the routine
! 155: *> only calculates the optimal size of the WORK array, returns
! 156: *> this value as the first entry of the WORK array, and no error
! 157: *> message related to LWORK is issued by XERBLA.
! 158: *> \endverbatim
! 159: *>
! 160: *> \param[out] INFO
! 161: *> \verbatim
! 162: *> INFO is INTEGER
! 163: *> = 0: successful exit.
! 164: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 165: *> \endverbatim
! 166: *
! 167: * Authors:
! 168: * ========
! 169: *
! 170: *> \author Univ. of Tennessee
! 171: *> \author Univ. of California Berkeley
! 172: *> \author Univ. of Colorado Denver
! 173: *> \author NAG Ltd.
! 174: *
! 175: *> \date July 2012
! 176: *
! 177: *> \ingroup doubleOTHERcomputational
! 178: *
! 179: *> \par Further Details:
! 180: * =====================
! 181: *>
! 182: *> \verbatim
! 183: *>
! 184: *> The upper-bidiagonal blocks B11, B21 are represented implicitly by
! 185: *> angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
! 186: *> in each bidiagonal band is a product of a sine or cosine of a THETA
! 187: *> with a sine or cosine of a PHI. See [1] or DORCSD for details.
! 188: *>
! 189: *> P1, P2, and Q1 are represented as products of elementary reflectors.
! 190: *> See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
! 191: *> and DORGLQ.
! 192: *> \endverbatim
! 193: *
! 194: *> \par References:
! 195: * ================
! 196: *>
! 197: *> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
! 198: *> Algorithms, 50(1):33-65, 2009.
! 199: *>
! 200: * =====================================================================
! 201: SUBROUTINE DORBDB3( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
! 202: $ TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
! 203: *
! 204: * -- LAPACK computational routine (version 3.5.0) --
! 205: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 206: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 207: * July 2012
! 208: *
! 209: * .. Scalar Arguments ..
! 210: INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
! 211: * ..
! 212: * .. Array Arguments ..
! 213: DOUBLE PRECISION PHI(*), THETA(*)
! 214: DOUBLE PRECISION TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
! 215: $ X11(LDX11,*), X21(LDX21,*)
! 216: * ..
! 217: *
! 218: * ====================================================================
! 219: *
! 220: * .. Parameters ..
! 221: DOUBLE PRECISION ONE
! 222: PARAMETER ( ONE = 1.0D0 )
! 223: * ..
! 224: * .. Local Scalars ..
! 225: DOUBLE PRECISION C, S
! 226: INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
! 227: $ LWORKMIN, LWORKOPT
! 228: LOGICAL LQUERY
! 229: * ..
! 230: * .. External Subroutines ..
! 231: EXTERNAL DLARF, DLARFGP, DORBDB5, DROT, XERBLA
! 232: * ..
! 233: * .. External Functions ..
! 234: DOUBLE PRECISION DNRM2
! 235: EXTERNAL DNRM2
! 236: * ..
! 237: * .. Intrinsic Function ..
! 238: INTRINSIC ATAN2, COS, MAX, SIN, SQRT
! 239: * ..
! 240: * .. Executable Statements ..
! 241: *
! 242: * Test input arguments
! 243: *
! 244: INFO = 0
! 245: LQUERY = LWORK .EQ. -1
! 246: *
! 247: IF( M .LT. 0 ) THEN
! 248: INFO = -1
! 249: ELSE IF( 2*P .LT. M .OR. P .GT. M ) THEN
! 250: INFO = -2
! 251: ELSE IF( Q .LT. M-P .OR. M-Q .LT. M-P ) THEN
! 252: INFO = -3
! 253: ELSE IF( LDX11 .LT. MAX( 1, P ) ) THEN
! 254: INFO = -5
! 255: ELSE IF( LDX21 .LT. MAX( 1, M-P ) ) THEN
! 256: INFO = -7
! 257: END IF
! 258: *
! 259: * Compute workspace
! 260: *
! 261: IF( INFO .EQ. 0 ) THEN
! 262: ILARF = 2
! 263: LLARF = MAX( P, M-P-1, Q-1 )
! 264: IORBDB5 = 2
! 265: LORBDB5 = Q-1
! 266: LWORKOPT = MAX( ILARF+LLARF-1, IORBDB5+LORBDB5-1 )
! 267: LWORKMIN = LWORKOPT
! 268: WORK(1) = LWORKOPT
! 269: IF( LWORK .LT. LWORKMIN .AND. .NOT.LQUERY ) THEN
! 270: INFO = -14
! 271: END IF
! 272: END IF
! 273: IF( INFO .NE. 0 ) THEN
! 274: CALL XERBLA( 'DORBDB3', -INFO )
! 275: RETURN
! 276: ELSE IF( LQUERY ) THEN
! 277: RETURN
! 278: END IF
! 279: *
! 280: * Reduce rows 1, ..., M-P of X11 and X21
! 281: *
! 282: DO I = 1, M-P
! 283: *
! 284: IF( I .GT. 1 ) THEN
! 285: CALL DROT( Q-I+1, X11(I-1,I), LDX11, X21(I,I), LDX11, C, S )
! 286: END IF
! 287: *
! 288: CALL DLARFGP( Q-I+1, X21(I,I), X21(I,I+1), LDX21, TAUQ1(I) )
! 289: S = X21(I,I)
! 290: X21(I,I) = ONE
! 291: CALL DLARF( 'R', P-I+1, Q-I+1, X21(I,I), LDX21, TAUQ1(I),
! 292: $ X11(I,I), LDX11, WORK(ILARF) )
! 293: CALL DLARF( 'R', M-P-I, Q-I+1, X21(I,I), LDX21, TAUQ1(I),
! 294: $ X21(I+1,I), LDX21, WORK(ILARF) )
! 295: C = SQRT( DNRM2( P-I+1, X11(I,I), 1, X11(I,I),
! 296: $ 1 )**2 + DNRM2( M-P-I, X21(I+1,I), 1, X21(I+1,I), 1 )**2 )
! 297: THETA(I) = ATAN2( S, C )
! 298: *
! 299: CALL DORBDB5( P-I+1, M-P-I, Q-I, X11(I,I), 1, X21(I+1,I), 1,
! 300: $ X11(I,I+1), LDX11, X21(I+1,I+1), LDX21,
! 301: $ WORK(IORBDB5), LORBDB5, CHILDINFO )
! 302: CALL DLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
! 303: IF( I .LT. M-P ) THEN
! 304: CALL DLARFGP( M-P-I, X21(I+1,I), X21(I+2,I), 1, TAUP2(I) )
! 305: PHI(I) = ATAN2( X21(I+1,I), X11(I,I) )
! 306: C = COS( PHI(I) )
! 307: S = SIN( PHI(I) )
! 308: X21(I+1,I) = ONE
! 309: CALL DLARF( 'L', M-P-I, Q-I, X21(I+1,I), 1, TAUP2(I),
! 310: $ X21(I+1,I+1), LDX21, WORK(ILARF) )
! 311: END IF
! 312: X11(I,I) = ONE
! 313: CALL DLARF( 'L', P-I+1, Q-I, X11(I,I), 1, TAUP1(I), X11(I,I+1),
! 314: $ LDX11, WORK(ILARF) )
! 315: *
! 316: END DO
! 317: *
! 318: * Reduce the bottom-right portion of X11 to the identity matrix
! 319: *
! 320: DO I = M-P + 1, Q
! 321: CALL DLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
! 322: X11(I,I) = ONE
! 323: CALL DLARF( 'L', P-I+1, Q-I, X11(I,I), 1, TAUP1(I), X11(I,I+1),
! 324: $ LDX11, WORK(ILARF) )
! 325: END DO
! 326: *
! 327: RETURN
! 328: *
! 329: * End of DORBDB3
! 330: *
! 331: END
! 332:
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