Annotation of rpl/lapack/lapack/dlaed2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLAED2( K, N, N1, D, Q, LDQ, INDXQ, RHO, Z, DLAMDA, W,
! 2: $ Q2, INDX, INDXC, INDXP, COLTYP, INFO )
! 3: *
! 4: * -- LAPACK routine (version 3.2) --
! 5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 7: * November 2006
! 8: *
! 9: * .. Scalar Arguments ..
! 10: INTEGER INFO, K, LDQ, N, N1
! 11: DOUBLE PRECISION RHO
! 12: * ..
! 13: * .. Array Arguments ..
! 14: INTEGER COLTYP( * ), INDX( * ), INDXC( * ), INDXP( * ),
! 15: $ INDXQ( * )
! 16: DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), Q2( * ),
! 17: $ W( * ), Z( * )
! 18: * ..
! 19: *
! 20: * Purpose
! 21: * =======
! 22: *
! 23: * DLAED2 merges the two sets of eigenvalues together into a single
! 24: * sorted set. Then it tries to deflate the size of the problem.
! 25: * There are two ways in which deflation can occur: when two or more
! 26: * eigenvalues are close together or if there is a tiny entry in the
! 27: * Z vector. For each such occurrence the order of the related secular
! 28: * equation problem is reduced by one.
! 29: *
! 30: * Arguments
! 31: * =========
! 32: *
! 33: * K (output) INTEGER
! 34: * The number of non-deflated eigenvalues, and the order of the
! 35: * related secular equation. 0 <= K <=N.
! 36: *
! 37: * N (input) INTEGER
! 38: * The dimension of the symmetric tridiagonal matrix. N >= 0.
! 39: *
! 40: * N1 (input) INTEGER
! 41: * The location of the last eigenvalue in the leading sub-matrix.
! 42: * min(1,N) <= N1 <= N/2.
! 43: *
! 44: * D (input/output) DOUBLE PRECISION array, dimension (N)
! 45: * On entry, D contains the eigenvalues of the two submatrices to
! 46: * be combined.
! 47: * On exit, D contains the trailing (N-K) updated eigenvalues
! 48: * (those which were deflated) sorted into increasing order.
! 49: *
! 50: * Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
! 51: * On entry, Q contains the eigenvectors of two submatrices in
! 52: * the two square blocks with corners at (1,1), (N1,N1)
! 53: * and (N1+1, N1+1), (N,N).
! 54: * On exit, Q contains the trailing (N-K) updated eigenvectors
! 55: * (those which were deflated) in its last N-K columns.
! 56: *
! 57: * LDQ (input) INTEGER
! 58: * The leading dimension of the array Q. LDQ >= max(1,N).
! 59: *
! 60: * INDXQ (input/output) INTEGER array, dimension (N)
! 61: * The permutation which separately sorts the two sub-problems
! 62: * in D into ascending order. Note that elements in the second
! 63: * half of this permutation must first have N1 added to their
! 64: * values. Destroyed on exit.
! 65: *
! 66: * RHO (input/output) DOUBLE PRECISION
! 67: * On entry, the off-diagonal element associated with the rank-1
! 68: * cut which originally split the two submatrices which are now
! 69: * being recombined.
! 70: * On exit, RHO has been modified to the value required by
! 71: * DLAED3.
! 72: *
! 73: * Z (input) DOUBLE PRECISION array, dimension (N)
! 74: * On entry, Z contains the updating vector (the last
! 75: * row of the first sub-eigenvector matrix and the first row of
! 76: * the second sub-eigenvector matrix).
! 77: * On exit, the contents of Z have been destroyed by the updating
! 78: * process.
! 79: *
! 80: * DLAMDA (output) DOUBLE PRECISION array, dimension (N)
! 81: * A copy of the first K eigenvalues which will be used by
! 82: * DLAED3 to form the secular equation.
! 83: *
! 84: * W (output) DOUBLE PRECISION array, dimension (N)
! 85: * The first k values of the final deflation-altered z-vector
! 86: * which will be passed to DLAED3.
! 87: *
! 88: * Q2 (output) DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
! 89: * A copy of the first K eigenvectors which will be used by
! 90: * DLAED3 in a matrix multiply (DGEMM) to solve for the new
! 91: * eigenvectors.
! 92: *
! 93: * INDX (workspace) INTEGER array, dimension (N)
! 94: * The permutation used to sort the contents of DLAMDA into
! 95: * ascending order.
! 96: *
! 97: * INDXC (output) INTEGER array, dimension (N)
! 98: * The permutation used to arrange the columns of the deflated
! 99: * Q matrix into three groups: the first group contains non-zero
! 100: * elements only at and above N1, the second contains
! 101: * non-zero elements only below N1, and the third is dense.
! 102: *
! 103: * INDXP (workspace) INTEGER array, dimension (N)
! 104: * The permutation used to place deflated values of D at the end
! 105: * of the array. INDXP(1:K) points to the nondeflated D-values
! 106: * and INDXP(K+1:N) points to the deflated eigenvalues.
! 107: *
! 108: * COLTYP (workspace/output) INTEGER array, dimension (N)
! 109: * During execution, a label which will indicate which of the
! 110: * following types a column in the Q2 matrix is:
! 111: * 1 : non-zero in the upper half only;
! 112: * 2 : dense;
! 113: * 3 : non-zero in the lower half only;
! 114: * 4 : deflated.
! 115: * On exit, COLTYP(i) is the number of columns of type i,
! 116: * for i=1 to 4 only.
! 117: *
! 118: * INFO (output) INTEGER
! 119: * = 0: successful exit.
! 120: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 121: *
! 122: * Further Details
! 123: * ===============
! 124: *
! 125: * Based on contributions by
! 126: * Jeff Rutter, Computer Science Division, University of California
! 127: * at Berkeley, USA
! 128: * Modified by Francoise Tisseur, University of Tennessee.
! 129: *
! 130: * =====================================================================
! 131: *
! 132: * .. Parameters ..
! 133: DOUBLE PRECISION MONE, ZERO, ONE, TWO, EIGHT
! 134: PARAMETER ( MONE = -1.0D0, ZERO = 0.0D0, ONE = 1.0D0,
! 135: $ TWO = 2.0D0, EIGHT = 8.0D0 )
! 136: * ..
! 137: * .. Local Arrays ..
! 138: INTEGER CTOT( 4 ), PSM( 4 )
! 139: * ..
! 140: * .. Local Scalars ..
! 141: INTEGER CT, I, IMAX, IQ1, IQ2, J, JMAX, JS, K2, N1P1,
! 142: $ N2, NJ, PJ
! 143: DOUBLE PRECISION C, EPS, S, T, TAU, TOL
! 144: * ..
! 145: * .. External Functions ..
! 146: INTEGER IDAMAX
! 147: DOUBLE PRECISION DLAMCH, DLAPY2
! 148: EXTERNAL IDAMAX, DLAMCH, DLAPY2
! 149: * ..
! 150: * .. External Subroutines ..
! 151: EXTERNAL DCOPY, DLACPY, DLAMRG, DROT, DSCAL, XERBLA
! 152: * ..
! 153: * .. Intrinsic Functions ..
! 154: INTRINSIC ABS, MAX, MIN, SQRT
! 155: * ..
! 156: * .. Executable Statements ..
! 157: *
! 158: * Test the input parameters.
! 159: *
! 160: INFO = 0
! 161: *
! 162: IF( N.LT.0 ) THEN
! 163: INFO = -2
! 164: ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN
! 165: INFO = -6
! 166: ELSE IF( MIN( 1, ( N / 2 ) ).GT.N1 .OR. ( N / 2 ).LT.N1 ) THEN
! 167: INFO = -3
! 168: END IF
! 169: IF( INFO.NE.0 ) THEN
! 170: CALL XERBLA( 'DLAED2', -INFO )
! 171: RETURN
! 172: END IF
! 173: *
! 174: * Quick return if possible
! 175: *
! 176: IF( N.EQ.0 )
! 177: $ RETURN
! 178: *
! 179: N2 = N - N1
! 180: N1P1 = N1 + 1
! 181: *
! 182: IF( RHO.LT.ZERO ) THEN
! 183: CALL DSCAL( N2, MONE, Z( N1P1 ), 1 )
! 184: END IF
! 185: *
! 186: * Normalize z so that norm(z) = 1. Since z is the concatenation of
! 187: * two normalized vectors, norm2(z) = sqrt(2).
! 188: *
! 189: T = ONE / SQRT( TWO )
! 190: CALL DSCAL( N, T, Z, 1 )
! 191: *
! 192: * RHO = ABS( norm(z)**2 * RHO )
! 193: *
! 194: RHO = ABS( TWO*RHO )
! 195: *
! 196: * Sort the eigenvalues into increasing order
! 197: *
! 198: DO 10 I = N1P1, N
! 199: INDXQ( I ) = INDXQ( I ) + N1
! 200: 10 CONTINUE
! 201: *
! 202: * re-integrate the deflated parts from the last pass
! 203: *
! 204: DO 20 I = 1, N
! 205: DLAMDA( I ) = D( INDXQ( I ) )
! 206: 20 CONTINUE
! 207: CALL DLAMRG( N1, N2, DLAMDA, 1, 1, INDXC )
! 208: DO 30 I = 1, N
! 209: INDX( I ) = INDXQ( INDXC( I ) )
! 210: 30 CONTINUE
! 211: *
! 212: * Calculate the allowable deflation tolerance
! 213: *
! 214: IMAX = IDAMAX( N, Z, 1 )
! 215: JMAX = IDAMAX( N, D, 1 )
! 216: EPS = DLAMCH( 'Epsilon' )
! 217: TOL = EIGHT*EPS*MAX( ABS( D( JMAX ) ), ABS( Z( IMAX ) ) )
! 218: *
! 219: * If the rank-1 modifier is small enough, no more needs to be done
! 220: * except to reorganize Q so that its columns correspond with the
! 221: * elements in D.
! 222: *
! 223: IF( RHO*ABS( Z( IMAX ) ).LE.TOL ) THEN
! 224: K = 0
! 225: IQ2 = 1
! 226: DO 40 J = 1, N
! 227: I = INDX( J )
! 228: CALL DCOPY( N, Q( 1, I ), 1, Q2( IQ2 ), 1 )
! 229: DLAMDA( J ) = D( I )
! 230: IQ2 = IQ2 + N
! 231: 40 CONTINUE
! 232: CALL DLACPY( 'A', N, N, Q2, N, Q, LDQ )
! 233: CALL DCOPY( N, DLAMDA, 1, D, 1 )
! 234: GO TO 190
! 235: END IF
! 236: *
! 237: * If there are multiple eigenvalues then the problem deflates. Here
! 238: * the number of equal eigenvalues are found. As each equal
! 239: * eigenvalue is found, an elementary reflector is computed to rotate
! 240: * the corresponding eigensubspace so that the corresponding
! 241: * components of Z are zero in this new basis.
! 242: *
! 243: DO 50 I = 1, N1
! 244: COLTYP( I ) = 1
! 245: 50 CONTINUE
! 246: DO 60 I = N1P1, N
! 247: COLTYP( I ) = 3
! 248: 60 CONTINUE
! 249: *
! 250: *
! 251: K = 0
! 252: K2 = N + 1
! 253: DO 70 J = 1, N
! 254: NJ = INDX( J )
! 255: IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
! 256: *
! 257: * Deflate due to small z component.
! 258: *
! 259: K2 = K2 - 1
! 260: COLTYP( NJ ) = 4
! 261: INDXP( K2 ) = NJ
! 262: IF( J.EQ.N )
! 263: $ GO TO 100
! 264: ELSE
! 265: PJ = NJ
! 266: GO TO 80
! 267: END IF
! 268: 70 CONTINUE
! 269: 80 CONTINUE
! 270: J = J + 1
! 271: NJ = INDX( J )
! 272: IF( J.GT.N )
! 273: $ GO TO 100
! 274: IF( RHO*ABS( Z( NJ ) ).LE.TOL ) THEN
! 275: *
! 276: * Deflate due to small z component.
! 277: *
! 278: K2 = K2 - 1
! 279: COLTYP( NJ ) = 4
! 280: INDXP( K2 ) = NJ
! 281: ELSE
! 282: *
! 283: * Check if eigenvalues are close enough to allow deflation.
! 284: *
! 285: S = Z( PJ )
! 286: C = Z( NJ )
! 287: *
! 288: * Find sqrt(a**2+b**2) without overflow or
! 289: * destructive underflow.
! 290: *
! 291: TAU = DLAPY2( C, S )
! 292: T = D( NJ ) - D( PJ )
! 293: C = C / TAU
! 294: S = -S / TAU
! 295: IF( ABS( T*C*S ).LE.TOL ) THEN
! 296: *
! 297: * Deflation is possible.
! 298: *
! 299: Z( NJ ) = TAU
! 300: Z( PJ ) = ZERO
! 301: IF( COLTYP( NJ ).NE.COLTYP( PJ ) )
! 302: $ COLTYP( NJ ) = 2
! 303: COLTYP( PJ ) = 4
! 304: CALL DROT( N, Q( 1, PJ ), 1, Q( 1, NJ ), 1, C, S )
! 305: T = D( PJ )*C**2 + D( NJ )*S**2
! 306: D( NJ ) = D( PJ )*S**2 + D( NJ )*C**2
! 307: D( PJ ) = T
! 308: K2 = K2 - 1
! 309: I = 1
! 310: 90 CONTINUE
! 311: IF( K2+I.LE.N ) THEN
! 312: IF( D( PJ ).LT.D( INDXP( K2+I ) ) ) THEN
! 313: INDXP( K2+I-1 ) = INDXP( K2+I )
! 314: INDXP( K2+I ) = PJ
! 315: I = I + 1
! 316: GO TO 90
! 317: ELSE
! 318: INDXP( K2+I-1 ) = PJ
! 319: END IF
! 320: ELSE
! 321: INDXP( K2+I-1 ) = PJ
! 322: END IF
! 323: PJ = NJ
! 324: ELSE
! 325: K = K + 1
! 326: DLAMDA( K ) = D( PJ )
! 327: W( K ) = Z( PJ )
! 328: INDXP( K ) = PJ
! 329: PJ = NJ
! 330: END IF
! 331: END IF
! 332: GO TO 80
! 333: 100 CONTINUE
! 334: *
! 335: * Record the last eigenvalue.
! 336: *
! 337: K = K + 1
! 338: DLAMDA( K ) = D( PJ )
! 339: W( K ) = Z( PJ )
! 340: INDXP( K ) = PJ
! 341: *
! 342: * Count up the total number of the various types of columns, then
! 343: * form a permutation which positions the four column types into
! 344: * four uniform groups (although one or more of these groups may be
! 345: * empty).
! 346: *
! 347: DO 110 J = 1, 4
! 348: CTOT( J ) = 0
! 349: 110 CONTINUE
! 350: DO 120 J = 1, N
! 351: CT = COLTYP( J )
! 352: CTOT( CT ) = CTOT( CT ) + 1
! 353: 120 CONTINUE
! 354: *
! 355: * PSM(*) = Position in SubMatrix (of types 1 through 4)
! 356: *
! 357: PSM( 1 ) = 1
! 358: PSM( 2 ) = 1 + CTOT( 1 )
! 359: PSM( 3 ) = PSM( 2 ) + CTOT( 2 )
! 360: PSM( 4 ) = PSM( 3 ) + CTOT( 3 )
! 361: K = N - CTOT( 4 )
! 362: *
! 363: * Fill out the INDXC array so that the permutation which it induces
! 364: * will place all type-1 columns first, all type-2 columns next,
! 365: * then all type-3's, and finally all type-4's.
! 366: *
! 367: DO 130 J = 1, N
! 368: JS = INDXP( J )
! 369: CT = COLTYP( JS )
! 370: INDX( PSM( CT ) ) = JS
! 371: INDXC( PSM( CT ) ) = J
! 372: PSM( CT ) = PSM( CT ) + 1
! 373: 130 CONTINUE
! 374: *
! 375: * Sort the eigenvalues and corresponding eigenvectors into DLAMDA
! 376: * and Q2 respectively. The eigenvalues/vectors which were not
! 377: * deflated go into the first K slots of DLAMDA and Q2 respectively,
! 378: * while those which were deflated go into the last N - K slots.
! 379: *
! 380: I = 1
! 381: IQ1 = 1
! 382: IQ2 = 1 + ( CTOT( 1 )+CTOT( 2 ) )*N1
! 383: DO 140 J = 1, CTOT( 1 )
! 384: JS = INDX( I )
! 385: CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
! 386: Z( I ) = D( JS )
! 387: I = I + 1
! 388: IQ1 = IQ1 + N1
! 389: 140 CONTINUE
! 390: *
! 391: DO 150 J = 1, CTOT( 2 )
! 392: JS = INDX( I )
! 393: CALL DCOPY( N1, Q( 1, JS ), 1, Q2( IQ1 ), 1 )
! 394: CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
! 395: Z( I ) = D( JS )
! 396: I = I + 1
! 397: IQ1 = IQ1 + N1
! 398: IQ2 = IQ2 + N2
! 399: 150 CONTINUE
! 400: *
! 401: DO 160 J = 1, CTOT( 3 )
! 402: JS = INDX( I )
! 403: CALL DCOPY( N2, Q( N1+1, JS ), 1, Q2( IQ2 ), 1 )
! 404: Z( I ) = D( JS )
! 405: I = I + 1
! 406: IQ2 = IQ2 + N2
! 407: 160 CONTINUE
! 408: *
! 409: IQ1 = IQ2
! 410: DO 170 J = 1, CTOT( 4 )
! 411: JS = INDX( I )
! 412: CALL DCOPY( N, Q( 1, JS ), 1, Q2( IQ2 ), 1 )
! 413: IQ2 = IQ2 + N
! 414: Z( I ) = D( JS )
! 415: I = I + 1
! 416: 170 CONTINUE
! 417: *
! 418: * The deflated eigenvalues and their corresponding vectors go back
! 419: * into the last N - K slots of D and Q respectively.
! 420: *
! 421: CALL DLACPY( 'A', N, CTOT( 4 ), Q2( IQ1 ), N, Q( 1, K+1 ), LDQ )
! 422: CALL DCOPY( N-K, Z( K+1 ), 1, D( K+1 ), 1 )
! 423: *
! 424: * Copy CTOT into COLTYP for referencing in DLAED3.
! 425: *
! 426: DO 180 J = 1, 4
! 427: COLTYP( J ) = CTOT( J )
! 428: 180 CONTINUE
! 429: *
! 430: 190 CONTINUE
! 431: RETURN
! 432: *
! 433: * End of DLAED2
! 434: *
! 435: END
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