Annotation of rpl/lapack/lapack/zstein.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZSTEIN( 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( * )
! 16: COMPLEX*16 Z( LDZ, * )
! 17: * ..
! 18: *
! 19: * Purpose
! 20: * =======
! 21: *
! 22: * ZSTEIN computes the eigenvectors of a real symmetric tridiagonal
! 23: * matrix T corresponding to specified eigenvalues, using inverse
! 24: * iteration.
! 25: *
! 26: * The maximum number of iterations allowed for each eigenvector is
! 27: * specified by an internal parameter MAXITS (currently set to 5).
! 28: *
! 29: * Although the eigenvectors are real, they are stored in a complex
! 30: * array, which may be passed to ZUNMTR or ZUPMTR for back
! 31: * transformation to the eigenvectors of a complex Hermitian matrix
! 32: * which was reduced to tridiagonal form.
! 33: *
! 34: *
! 35: * Arguments
! 36: * =========
! 37: *
! 38: * N (input) INTEGER
! 39: * The order of the matrix. N >= 0.
! 40: *
! 41: * D (input) DOUBLE PRECISION array, dimension (N)
! 42: * The n diagonal elements of the tridiagonal matrix T.
! 43: *
! 44: * E (input) DOUBLE PRECISION array, dimension (N-1)
! 45: * The (n-1) subdiagonal elements of the tridiagonal matrix
! 46: * T, stored in elements 1 to N-1.
! 47: *
! 48: * M (input) INTEGER
! 49: * The number of eigenvectors to be found. 0 <= M <= N.
! 50: *
! 51: * W (input) DOUBLE PRECISION array, dimension (N)
! 52: * The first M elements of W contain the eigenvalues for
! 53: * which eigenvectors are to be computed. The eigenvalues
! 54: * should be grouped by split-off block and ordered from
! 55: * smallest to largest within the block. ( The output array
! 56: * W from DSTEBZ with ORDER = 'B' is expected here. )
! 57: *
! 58: * IBLOCK (input) INTEGER array, dimension (N)
! 59: * The submatrix indices associated with the corresponding
! 60: * eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
! 61: * the first submatrix from the top, =2 if W(i) belongs to
! 62: * the second submatrix, etc. ( The output array IBLOCK
! 63: * from DSTEBZ is expected here. )
! 64: *
! 65: * ISPLIT (input) INTEGER array, dimension (N)
! 66: * The splitting points, at which T breaks up into submatrices.
! 67: * The first submatrix consists of rows/columns 1 to
! 68: * ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
! 69: * through ISPLIT( 2 ), etc.
! 70: * ( The output array ISPLIT from DSTEBZ is expected here. )
! 71: *
! 72: * Z (output) COMPLEX*16 array, dimension (LDZ, M)
! 73: * The computed eigenvectors. The eigenvector associated
! 74: * with the eigenvalue W(i) is stored in the i-th column of
! 75: * Z. Any vector which fails to converge is set to its current
! 76: * iterate after MAXITS iterations.
! 77: * The imaginary parts of the eigenvectors are set to zero.
! 78: *
! 79: * LDZ (input) INTEGER
! 80: * The leading dimension of the array Z. LDZ >= max(1,N).
! 81: *
! 82: * WORK (workspace) DOUBLE PRECISION array, dimension (5*N)
! 83: *
! 84: * IWORK (workspace) INTEGER array, dimension (N)
! 85: *
! 86: * IFAIL (output) INTEGER array, dimension (M)
! 87: * On normal exit, all elements of IFAIL are zero.
! 88: * If one or more eigenvectors fail to converge after
! 89: * MAXITS iterations, then their indices are stored in
! 90: * array IFAIL.
! 91: *
! 92: * INFO (output) INTEGER
! 93: * = 0: successful exit
! 94: * < 0: if INFO = -i, the i-th argument had an illegal value
! 95: * > 0: if INFO = i, then i eigenvectors failed to converge
! 96: * in MAXITS iterations. Their indices are stored in
! 97: * array IFAIL.
! 98: *
! 99: * Internal Parameters
! 100: * ===================
! 101: *
! 102: * MAXITS INTEGER, default = 5
! 103: * The maximum number of iterations performed.
! 104: *
! 105: * EXTRA INTEGER, default = 2
! 106: * The number of iterations performed after norm growth
! 107: * criterion is satisfied, should be at least 1.
! 108: *
! 109: * =====================================================================
! 110: *
! 111: * .. Parameters ..
! 112: COMPLEX*16 CZERO, CONE
! 113: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
! 114: $ CONE = ( 1.0D+0, 0.0D+0 ) )
! 115: DOUBLE PRECISION ZERO, ONE, TEN, ODM3, ODM1
! 116: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 1.0D+1,
! 117: $ ODM3 = 1.0D-3, ODM1 = 1.0D-1 )
! 118: INTEGER MAXITS, EXTRA
! 119: PARAMETER ( MAXITS = 5, EXTRA = 2 )
! 120: * ..
! 121: * .. Local Scalars ..
! 122: INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
! 123: $ INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1,
! 124: $ JBLK, JMAX, JR, NBLK, NRMCHK
! 125: DOUBLE PRECISION DTPCRT, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
! 126: $ SCL, SEP, TOL, XJ, XJM, ZTR
! 127: * ..
! 128: * .. Local Arrays ..
! 129: INTEGER ISEED( 4 )
! 130: * ..
! 131: * .. External Functions ..
! 132: INTEGER IDAMAX
! 133: DOUBLE PRECISION DASUM, DLAMCH, DNRM2
! 134: EXTERNAL IDAMAX, DASUM, DLAMCH, DNRM2
! 135: * ..
! 136: * .. External Subroutines ..
! 137: EXTERNAL DCOPY, DLAGTF, DLAGTS, DLARNV, DSCAL, XERBLA
! 138: * ..
! 139: * .. Intrinsic Functions ..
! 140: INTRINSIC ABS, DBLE, DCMPLX, MAX, SQRT
! 141: * ..
! 142: * .. Executable Statements ..
! 143: *
! 144: * Test the input parameters.
! 145: *
! 146: INFO = 0
! 147: DO 10 I = 1, M
! 148: IFAIL( I ) = 0
! 149: 10 CONTINUE
! 150: *
! 151: IF( N.LT.0 ) THEN
! 152: INFO = -1
! 153: ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
! 154: INFO = -4
! 155: ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN
! 156: INFO = -9
! 157: ELSE
! 158: DO 20 J = 2, M
! 159: IF( IBLOCK( J ).LT.IBLOCK( J-1 ) ) THEN
! 160: INFO = -6
! 161: GO TO 30
! 162: END IF
! 163: IF( IBLOCK( J ).EQ.IBLOCK( J-1 ) .AND. W( J ).LT.W( J-1 ) )
! 164: $ THEN
! 165: INFO = -5
! 166: GO TO 30
! 167: END IF
! 168: 20 CONTINUE
! 169: 30 CONTINUE
! 170: END IF
! 171: *
! 172: IF( INFO.NE.0 ) THEN
! 173: CALL XERBLA( 'ZSTEIN', -INFO )
! 174: RETURN
! 175: END IF
! 176: *
! 177: * Quick return if possible
! 178: *
! 179: IF( N.EQ.0 .OR. M.EQ.0 ) THEN
! 180: RETURN
! 181: ELSE IF( N.EQ.1 ) THEN
! 182: Z( 1, 1 ) = CONE
! 183: RETURN
! 184: END IF
! 185: *
! 186: * Get machine constants.
! 187: *
! 188: EPS = DLAMCH( 'Precision' )
! 189: *
! 190: * Initialize seed for random number generator DLARNV.
! 191: *
! 192: DO 40 I = 1, 4
! 193: ISEED( I ) = 1
! 194: 40 CONTINUE
! 195: *
! 196: * Initialize pointers.
! 197: *
! 198: INDRV1 = 0
! 199: INDRV2 = INDRV1 + N
! 200: INDRV3 = INDRV2 + N
! 201: INDRV4 = INDRV3 + N
! 202: INDRV5 = INDRV4 + N
! 203: *
! 204: * Compute eigenvectors of matrix blocks.
! 205: *
! 206: J1 = 1
! 207: DO 180 NBLK = 1, IBLOCK( M )
! 208: *
! 209: * Find starting and ending indices of block nblk.
! 210: *
! 211: IF( NBLK.EQ.1 ) THEN
! 212: B1 = 1
! 213: ELSE
! 214: B1 = ISPLIT( NBLK-1 ) + 1
! 215: END IF
! 216: BN = ISPLIT( NBLK )
! 217: BLKSIZ = BN - B1 + 1
! 218: IF( BLKSIZ.EQ.1 )
! 219: $ GO TO 60
! 220: GPIND = B1
! 221: *
! 222: * Compute reorthogonalization criterion and stopping criterion.
! 223: *
! 224: ONENRM = ABS( D( B1 ) ) + ABS( E( B1 ) )
! 225: ONENRM = MAX( ONENRM, ABS( D( BN ) )+ABS( E( BN-1 ) ) )
! 226: DO 50 I = B1 + 1, BN - 1
! 227: ONENRM = MAX( ONENRM, ABS( D( I ) )+ABS( E( I-1 ) )+
! 228: $ ABS( E( I ) ) )
! 229: 50 CONTINUE
! 230: ORTOL = ODM3*ONENRM
! 231: *
! 232: DTPCRT = SQRT( ODM1 / BLKSIZ )
! 233: *
! 234: * Loop through eigenvalues of block nblk.
! 235: *
! 236: 60 CONTINUE
! 237: JBLK = 0
! 238: DO 170 J = J1, M
! 239: IF( IBLOCK( J ).NE.NBLK ) THEN
! 240: J1 = J
! 241: GO TO 180
! 242: END IF
! 243: JBLK = JBLK + 1
! 244: XJ = W( J )
! 245: *
! 246: * Skip all the work if the block size is one.
! 247: *
! 248: IF( BLKSIZ.EQ.1 ) THEN
! 249: WORK( INDRV1+1 ) = ONE
! 250: GO TO 140
! 251: END IF
! 252: *
! 253: * If eigenvalues j and j-1 are too close, add a relatively
! 254: * small perturbation.
! 255: *
! 256: IF( JBLK.GT.1 ) THEN
! 257: EPS1 = ABS( EPS*XJ )
! 258: PERTOL = TEN*EPS1
! 259: SEP = XJ - XJM
! 260: IF( SEP.LT.PERTOL )
! 261: $ XJ = XJM + PERTOL
! 262: END IF
! 263: *
! 264: ITS = 0
! 265: NRMCHK = 0
! 266: *
! 267: * Get random starting vector.
! 268: *
! 269: CALL DLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) )
! 270: *
! 271: * Copy the matrix T so it won't be destroyed in factorization.
! 272: *
! 273: CALL DCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 )
! 274: CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 )
! 275: CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 )
! 276: *
! 277: * Compute LU factors with partial pivoting ( PT = LU )
! 278: *
! 279: TOL = ZERO
! 280: CALL DLAGTF( BLKSIZ, WORK( INDRV4+1 ), XJ, WORK( INDRV2+2 ),
! 281: $ WORK( INDRV3+1 ), TOL, WORK( INDRV5+1 ), IWORK,
! 282: $ IINFO )
! 283: *
! 284: * Update iteration count.
! 285: *
! 286: 70 CONTINUE
! 287: ITS = ITS + 1
! 288: IF( ITS.GT.MAXITS )
! 289: $ GO TO 120
! 290: *
! 291: * Normalize and scale the righthand side vector Pb.
! 292: *
! 293: SCL = BLKSIZ*ONENRM*MAX( EPS,
! 294: $ ABS( WORK( INDRV4+BLKSIZ ) ) ) /
! 295: $ DASUM( BLKSIZ, WORK( INDRV1+1 ), 1 )
! 296: CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
! 297: *
! 298: * Solve the system LU = Pb.
! 299: *
! 300: CALL DLAGTS( -1, BLKSIZ, WORK( INDRV4+1 ), WORK( INDRV2+2 ),
! 301: $ WORK( INDRV3+1 ), WORK( INDRV5+1 ), IWORK,
! 302: $ WORK( INDRV1+1 ), TOL, IINFO )
! 303: *
! 304: * Reorthogonalize by modified Gram-Schmidt if eigenvalues are
! 305: * close enough.
! 306: *
! 307: IF( JBLK.EQ.1 )
! 308: $ GO TO 110
! 309: IF( ABS( XJ-XJM ).GT.ORTOL )
! 310: $ GPIND = J
! 311: IF( GPIND.NE.J ) THEN
! 312: DO 100 I = GPIND, J - 1
! 313: ZTR = ZERO
! 314: DO 80 JR = 1, BLKSIZ
! 315: ZTR = ZTR + WORK( INDRV1+JR )*
! 316: $ DBLE( Z( B1-1+JR, I ) )
! 317: 80 CONTINUE
! 318: DO 90 JR = 1, BLKSIZ
! 319: WORK( INDRV1+JR ) = WORK( INDRV1+JR ) -
! 320: $ ZTR*DBLE( Z( B1-1+JR, I ) )
! 321: 90 CONTINUE
! 322: 100 CONTINUE
! 323: END IF
! 324: *
! 325: * Check the infinity norm of the iterate.
! 326: *
! 327: 110 CONTINUE
! 328: JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
! 329: NRM = ABS( WORK( INDRV1+JMAX ) )
! 330: *
! 331: * Continue for additional iterations after norm reaches
! 332: * stopping criterion.
! 333: *
! 334: IF( NRM.LT.DTPCRT )
! 335: $ GO TO 70
! 336: NRMCHK = NRMCHK + 1
! 337: IF( NRMCHK.LT.EXTRA+1 )
! 338: $ GO TO 70
! 339: *
! 340: GO TO 130
! 341: *
! 342: * If stopping criterion was not satisfied, update info and
! 343: * store eigenvector number in array ifail.
! 344: *
! 345: 120 CONTINUE
! 346: INFO = INFO + 1
! 347: IFAIL( INFO ) = J
! 348: *
! 349: * Accept iterate as jth eigenvector.
! 350: *
! 351: 130 CONTINUE
! 352: SCL = ONE / DNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 )
! 353: JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
! 354: IF( WORK( INDRV1+JMAX ).LT.ZERO )
! 355: $ SCL = -SCL
! 356: CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
! 357: 140 CONTINUE
! 358: DO 150 I = 1, N
! 359: Z( I, J ) = CZERO
! 360: 150 CONTINUE
! 361: DO 160 I = 1, BLKSIZ
! 362: Z( B1+I-1, J ) = DCMPLX( WORK( INDRV1+I ), ZERO )
! 363: 160 CONTINUE
! 364: *
! 365: * Save the shift to check eigenvalue spacing at next
! 366: * iteration.
! 367: *
! 368: XJM = XJ
! 369: *
! 370: 170 CONTINUE
! 371: 180 CONTINUE
! 372: *
! 373: RETURN
! 374: *
! 375: * End of ZSTEIN
! 376: *
! 377: END
CVSweb interface <joel.bertrand@systella.fr>