Annotation of rpl/lapack/lapack/dstein.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
! 2: $ IWORK, IFAIL, 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, LDZ, M, N
! 11: * ..
! 12: * .. Array Arguments ..
! 13: INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
! 14: $ IWORK( * )
! 15: DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
! 16: * ..
! 17: *
! 18: * Purpose
! 19: * =======
! 20: *
! 21: * DSTEIN computes the eigenvectors of a real symmetric tridiagonal
! 22: * matrix T corresponding to specified eigenvalues, using inverse
! 23: * iteration.
! 24: *
! 25: * The maximum number of iterations allowed for each eigenvector is
! 26: * specified by an internal parameter MAXITS (currently set to 5).
! 27: *
! 28: * Arguments
! 29: * =========
! 30: *
! 31: * N (input) INTEGER
! 32: * The order of the matrix. N >= 0.
! 33: *
! 34: * D (input) DOUBLE PRECISION array, dimension (N)
! 35: * The n diagonal elements of the tridiagonal matrix T.
! 36: *
! 37: * E (input) DOUBLE PRECISION array, dimension (N-1)
! 38: * The (n-1) subdiagonal elements of the tridiagonal matrix
! 39: * T, in elements 1 to N-1.
! 40: *
! 41: * M (input) INTEGER
! 42: * The number of eigenvectors to be found. 0 <= M <= N.
! 43: *
! 44: * W (input) DOUBLE PRECISION array, dimension (N)
! 45: * The first M elements of W contain the eigenvalues for
! 46: * which eigenvectors are to be computed. The eigenvalues
! 47: * should be grouped by split-off block and ordered from
! 48: * smallest to largest within the block. ( The output array
! 49: * W from DSTEBZ with ORDER = 'B' is expected here. )
! 50: *
! 51: * IBLOCK (input) INTEGER array, dimension (N)
! 52: * The submatrix indices associated with the corresponding
! 53: * eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
! 54: * the first submatrix from the top, =2 if W(i) belongs to
! 55: * the second submatrix, etc. ( The output array IBLOCK
! 56: * from DSTEBZ is expected here. )
! 57: *
! 58: * ISPLIT (input) INTEGER array, dimension (N)
! 59: * The splitting points, at which T breaks up into submatrices.
! 60: * The first submatrix consists of rows/columns 1 to
! 61: * ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
! 62: * through ISPLIT( 2 ), etc.
! 63: * ( The output array ISPLIT from DSTEBZ is expected here. )
! 64: *
! 65: * Z (output) DOUBLE PRECISION array, dimension (LDZ, M)
! 66: * The computed eigenvectors. The eigenvector associated
! 67: * with the eigenvalue W(i) is stored in the i-th column of
! 68: * Z. Any vector which fails to converge is set to its current
! 69: * iterate after MAXITS iterations.
! 70: *
! 71: * LDZ (input) INTEGER
! 72: * The leading dimension of the array Z. LDZ >= max(1,N).
! 73: *
! 74: * WORK (workspace) DOUBLE PRECISION array, dimension (5*N)
! 75: *
! 76: * IWORK (workspace) INTEGER array, dimension (N)
! 77: *
! 78: * IFAIL (output) INTEGER array, dimension (M)
! 79: * On normal exit, all elements of IFAIL are zero.
! 80: * If one or more eigenvectors fail to converge after
! 81: * MAXITS iterations, then their indices are stored in
! 82: * array IFAIL.
! 83: *
! 84: * INFO (output) INTEGER
! 85: * = 0: successful exit.
! 86: * < 0: if INFO = -i, the i-th argument had an illegal value
! 87: * > 0: if INFO = i, then i eigenvectors failed to converge
! 88: * in MAXITS iterations. Their indices are stored in
! 89: * array IFAIL.
! 90: *
! 91: * Internal Parameters
! 92: * ===================
! 93: *
! 94: * MAXITS INTEGER, default = 5
! 95: * The maximum number of iterations performed.
! 96: *
! 97: * EXTRA INTEGER, default = 2
! 98: * The number of iterations performed after norm growth
! 99: * criterion is satisfied, should be at least 1.
! 100: *
! 101: * =====================================================================
! 102: *
! 103: * .. Parameters ..
! 104: DOUBLE PRECISION ZERO, ONE, TEN, ODM3, ODM1
! 105: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 1.0D+1,
! 106: $ ODM3 = 1.0D-3, ODM1 = 1.0D-1 )
! 107: INTEGER MAXITS, EXTRA
! 108: PARAMETER ( MAXITS = 5, EXTRA = 2 )
! 109: * ..
! 110: * .. Local Scalars ..
! 111: INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
! 112: $ INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1,
! 113: $ JBLK, JMAX, NBLK, NRMCHK
! 114: DOUBLE PRECISION DTPCRT, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
! 115: $ SCL, SEP, TOL, XJ, XJM, ZTR
! 116: * ..
! 117: * .. Local Arrays ..
! 118: INTEGER ISEED( 4 )
! 119: * ..
! 120: * .. External Functions ..
! 121: INTEGER IDAMAX
! 122: DOUBLE PRECISION DASUM, DDOT, DLAMCH, DNRM2
! 123: EXTERNAL IDAMAX, DASUM, DDOT, DLAMCH, DNRM2
! 124: * ..
! 125: * .. External Subroutines ..
! 126: EXTERNAL DAXPY, DCOPY, DLAGTF, DLAGTS, DLARNV, DSCAL,
! 127: $ XERBLA
! 128: * ..
! 129: * .. Intrinsic Functions ..
! 130: INTRINSIC ABS, MAX, SQRT
! 131: * ..
! 132: * .. Executable Statements ..
! 133: *
! 134: * Test the input parameters.
! 135: *
! 136: INFO = 0
! 137: DO 10 I = 1, M
! 138: IFAIL( I ) = 0
! 139: 10 CONTINUE
! 140: *
! 141: IF( N.LT.0 ) THEN
! 142: INFO = -1
! 143: ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
! 144: INFO = -4
! 145: ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN
! 146: INFO = -9
! 147: ELSE
! 148: DO 20 J = 2, M
! 149: IF( IBLOCK( J ).LT.IBLOCK( J-1 ) ) THEN
! 150: INFO = -6
! 151: GO TO 30
! 152: END IF
! 153: IF( IBLOCK( J ).EQ.IBLOCK( J-1 ) .AND. W( J ).LT.W( J-1 ) )
! 154: $ THEN
! 155: INFO = -5
! 156: GO TO 30
! 157: END IF
! 158: 20 CONTINUE
! 159: 30 CONTINUE
! 160: END IF
! 161: *
! 162: IF( INFO.NE.0 ) THEN
! 163: CALL XERBLA( 'DSTEIN', -INFO )
! 164: RETURN
! 165: END IF
! 166: *
! 167: * Quick return if possible
! 168: *
! 169: IF( N.EQ.0 .OR. M.EQ.0 ) THEN
! 170: RETURN
! 171: ELSE IF( N.EQ.1 ) THEN
! 172: Z( 1, 1 ) = ONE
! 173: RETURN
! 174: END IF
! 175: *
! 176: * Get machine constants.
! 177: *
! 178: EPS = DLAMCH( 'Precision' )
! 179: *
! 180: * Initialize seed for random number generator DLARNV.
! 181: *
! 182: DO 40 I = 1, 4
! 183: ISEED( I ) = 1
! 184: 40 CONTINUE
! 185: *
! 186: * Initialize pointers.
! 187: *
! 188: INDRV1 = 0
! 189: INDRV2 = INDRV1 + N
! 190: INDRV3 = INDRV2 + N
! 191: INDRV4 = INDRV3 + N
! 192: INDRV5 = INDRV4 + N
! 193: *
! 194: * Compute eigenvectors of matrix blocks.
! 195: *
! 196: J1 = 1
! 197: DO 160 NBLK = 1, IBLOCK( M )
! 198: *
! 199: * Find starting and ending indices of block nblk.
! 200: *
! 201: IF( NBLK.EQ.1 ) THEN
! 202: B1 = 1
! 203: ELSE
! 204: B1 = ISPLIT( NBLK-1 ) + 1
! 205: END IF
! 206: BN = ISPLIT( NBLK )
! 207: BLKSIZ = BN - B1 + 1
! 208: IF( BLKSIZ.EQ.1 )
! 209: $ GO TO 60
! 210: GPIND = B1
! 211: *
! 212: * Compute reorthogonalization criterion and stopping criterion.
! 213: *
! 214: ONENRM = ABS( D( B1 ) ) + ABS( E( B1 ) )
! 215: ONENRM = MAX( ONENRM, ABS( D( BN ) )+ABS( E( BN-1 ) ) )
! 216: DO 50 I = B1 + 1, BN - 1
! 217: ONENRM = MAX( ONENRM, ABS( D( I ) )+ABS( E( I-1 ) )+
! 218: $ ABS( E( I ) ) )
! 219: 50 CONTINUE
! 220: ORTOL = ODM3*ONENRM
! 221: *
! 222: DTPCRT = SQRT( ODM1 / BLKSIZ )
! 223: *
! 224: * Loop through eigenvalues of block nblk.
! 225: *
! 226: 60 CONTINUE
! 227: JBLK = 0
! 228: DO 150 J = J1, M
! 229: IF( IBLOCK( J ).NE.NBLK ) THEN
! 230: J1 = J
! 231: GO TO 160
! 232: END IF
! 233: JBLK = JBLK + 1
! 234: XJ = W( J )
! 235: *
! 236: * Skip all the work if the block size is one.
! 237: *
! 238: IF( BLKSIZ.EQ.1 ) THEN
! 239: WORK( INDRV1+1 ) = ONE
! 240: GO TO 120
! 241: END IF
! 242: *
! 243: * If eigenvalues j and j-1 are too close, add a relatively
! 244: * small perturbation.
! 245: *
! 246: IF( JBLK.GT.1 ) THEN
! 247: EPS1 = ABS( EPS*XJ )
! 248: PERTOL = TEN*EPS1
! 249: SEP = XJ - XJM
! 250: IF( SEP.LT.PERTOL )
! 251: $ XJ = XJM + PERTOL
! 252: END IF
! 253: *
! 254: ITS = 0
! 255: NRMCHK = 0
! 256: *
! 257: * Get random starting vector.
! 258: *
! 259: CALL DLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) )
! 260: *
! 261: * Copy the matrix T so it won't be destroyed in factorization.
! 262: *
! 263: CALL DCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 )
! 264: CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 )
! 265: CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 )
! 266: *
! 267: * Compute LU factors with partial pivoting ( PT = LU )
! 268: *
! 269: TOL = ZERO
! 270: CALL DLAGTF( BLKSIZ, WORK( INDRV4+1 ), XJ, WORK( INDRV2+2 ),
! 271: $ WORK( INDRV3+1 ), TOL, WORK( INDRV5+1 ), IWORK,
! 272: $ IINFO )
! 273: *
! 274: * Update iteration count.
! 275: *
! 276: 70 CONTINUE
! 277: ITS = ITS + 1
! 278: IF( ITS.GT.MAXITS )
! 279: $ GO TO 100
! 280: *
! 281: * Normalize and scale the righthand side vector Pb.
! 282: *
! 283: SCL = BLKSIZ*ONENRM*MAX( EPS,
! 284: $ ABS( WORK( INDRV4+BLKSIZ ) ) ) /
! 285: $ DASUM( BLKSIZ, WORK( INDRV1+1 ), 1 )
! 286: CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
! 287: *
! 288: * Solve the system LU = Pb.
! 289: *
! 290: CALL DLAGTS( -1, BLKSIZ, WORK( INDRV4+1 ), WORK( INDRV2+2 ),
! 291: $ WORK( INDRV3+1 ), WORK( INDRV5+1 ), IWORK,
! 292: $ WORK( INDRV1+1 ), TOL, IINFO )
! 293: *
! 294: * Reorthogonalize by modified Gram-Schmidt if eigenvalues are
! 295: * close enough.
! 296: *
! 297: IF( JBLK.EQ.1 )
! 298: $ GO TO 90
! 299: IF( ABS( XJ-XJM ).GT.ORTOL )
! 300: $ GPIND = J
! 301: IF( GPIND.NE.J ) THEN
! 302: DO 80 I = GPIND, J - 1
! 303: ZTR = -DDOT( BLKSIZ, WORK( INDRV1+1 ), 1, Z( B1, I ),
! 304: $ 1 )
! 305: CALL DAXPY( BLKSIZ, ZTR, Z( B1, I ), 1,
! 306: $ WORK( INDRV1+1 ), 1 )
! 307: 80 CONTINUE
! 308: END IF
! 309: *
! 310: * Check the infinity norm of the iterate.
! 311: *
! 312: 90 CONTINUE
! 313: JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
! 314: NRM = ABS( WORK( INDRV1+JMAX ) )
! 315: *
! 316: * Continue for additional iterations after norm reaches
! 317: * stopping criterion.
! 318: *
! 319: IF( NRM.LT.DTPCRT )
! 320: $ GO TO 70
! 321: NRMCHK = NRMCHK + 1
! 322: IF( NRMCHK.LT.EXTRA+1 )
! 323: $ GO TO 70
! 324: *
! 325: GO TO 110
! 326: *
! 327: * If stopping criterion was not satisfied, update info and
! 328: * store eigenvector number in array ifail.
! 329: *
! 330: 100 CONTINUE
! 331: INFO = INFO + 1
! 332: IFAIL( INFO ) = J
! 333: *
! 334: * Accept iterate as jth eigenvector.
! 335: *
! 336: 110 CONTINUE
! 337: SCL = ONE / DNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 )
! 338: JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
! 339: IF( WORK( INDRV1+JMAX ).LT.ZERO )
! 340: $ SCL = -SCL
! 341: CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
! 342: 120 CONTINUE
! 343: DO 130 I = 1, N
! 344: Z( I, J ) = ZERO
! 345: 130 CONTINUE
! 346: DO 140 I = 1, BLKSIZ
! 347: Z( B1+I-1, J ) = WORK( INDRV1+I )
! 348: 140 CONTINUE
! 349: *
! 350: * Save the shift to check eigenvalue spacing at next
! 351: * iteration.
! 352: *
! 353: XJM = XJ
! 354: *
! 355: 150 CONTINUE
! 356: 160 CONTINUE
! 357: *
! 358: RETURN
! 359: *
! 360: * End of DSTEIN
! 361: *
! 362: END
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