Annotation of rpl/lapack/lapack/zhbgvx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
! 2: $ LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
! 3: $ LDZ, WORK, RWORK, IWORK, IFAIL, INFO )
! 4: *
! 5: * -- LAPACK driver routine (version 3.2) --
! 6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 8: * November 2006
! 9: *
! 10: * .. Scalar Arguments ..
! 11: CHARACTER JOBZ, RANGE, UPLO
! 12: INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
! 13: $ N
! 14: DOUBLE PRECISION ABSTOL, VL, VU
! 15: * ..
! 16: * .. Array Arguments ..
! 17: INTEGER IFAIL( * ), IWORK( * )
! 18: DOUBLE PRECISION RWORK( * ), W( * )
! 19: COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
! 20: $ WORK( * ), Z( LDZ, * )
! 21: * ..
! 22: *
! 23: * Purpose
! 24: * =======
! 25: *
! 26: * ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors
! 27: * of a complex generalized Hermitian-definite banded eigenproblem, of
! 28: * the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian
! 29: * and banded, and B is also positive definite. Eigenvalues and
! 30: * eigenvectors can be selected by specifying either all eigenvalues,
! 31: * a range of values or a range of indices for the desired eigenvalues.
! 32: *
! 33: * Arguments
! 34: * =========
! 35: *
! 36: * JOBZ (input) CHARACTER*1
! 37: * = 'N': Compute eigenvalues only;
! 38: * = 'V': Compute eigenvalues and eigenvectors.
! 39: *
! 40: * RANGE (input) CHARACTER*1
! 41: * = 'A': all eigenvalues will be found;
! 42: * = 'V': all eigenvalues in the half-open interval (VL,VU]
! 43: * will be found;
! 44: * = 'I': the IL-th through IU-th eigenvalues will be found.
! 45: *
! 46: * UPLO (input) CHARACTER*1
! 47: * = 'U': Upper triangles of A and B are stored;
! 48: * = 'L': Lower triangles of A and B are stored.
! 49: *
! 50: * N (input) INTEGER
! 51: * The order of the matrices A and B. N >= 0.
! 52: *
! 53: * KA (input) INTEGER
! 54: * The number of superdiagonals of the matrix A if UPLO = 'U',
! 55: * or the number of subdiagonals if UPLO = 'L'. KA >= 0.
! 56: *
! 57: * KB (input) INTEGER
! 58: * The number of superdiagonals of the matrix B if UPLO = 'U',
! 59: * or the number of subdiagonals if UPLO = 'L'. KB >= 0.
! 60: *
! 61: * AB (input/output) COMPLEX*16 array, dimension (LDAB, N)
! 62: * On entry, the upper or lower triangle of the Hermitian band
! 63: * matrix A, stored in the first ka+1 rows of the array. The
! 64: * j-th column of A is stored in the j-th column of the array AB
! 65: * as follows:
! 66: * if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
! 67: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
! 68: *
! 69: * On exit, the contents of AB are destroyed.
! 70: *
! 71: * LDAB (input) INTEGER
! 72: * The leading dimension of the array AB. LDAB >= KA+1.
! 73: *
! 74: * BB (input/output) COMPLEX*16 array, dimension (LDBB, N)
! 75: * On entry, the upper or lower triangle of the Hermitian band
! 76: * matrix B, stored in the first kb+1 rows of the array. The
! 77: * j-th column of B is stored in the j-th column of the array BB
! 78: * as follows:
! 79: * if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
! 80: * if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).
! 81: *
! 82: * On exit, the factor S from the split Cholesky factorization
! 83: * B = S**H*S, as returned by ZPBSTF.
! 84: *
! 85: * LDBB (input) INTEGER
! 86: * The leading dimension of the array BB. LDBB >= KB+1.
! 87: *
! 88: * Q (output) COMPLEX*16 array, dimension (LDQ, N)
! 89: * If JOBZ = 'V', the n-by-n matrix used in the reduction of
! 90: * A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
! 91: * and consequently C to tridiagonal form.
! 92: * If JOBZ = 'N', the array Q is not referenced.
! 93: *
! 94: * LDQ (input) INTEGER
! 95: * The leading dimension of the array Q. If JOBZ = 'N',
! 96: * LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
! 97: *
! 98: * VL (input) DOUBLE PRECISION
! 99: * VU (input) DOUBLE PRECISION
! 100: * If RANGE='V', the lower and upper bounds of the interval to
! 101: * be searched for eigenvalues. VL < VU.
! 102: * Not referenced if RANGE = 'A' or 'I'.
! 103: *
! 104: * IL (input) INTEGER
! 105: * IU (input) INTEGER
! 106: * If RANGE='I', the indices (in ascending order) of the
! 107: * smallest and largest eigenvalues to be returned.
! 108: * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 109: * Not referenced if RANGE = 'A' or 'V'.
! 110: *
! 111: * ABSTOL (input) DOUBLE PRECISION
! 112: * The absolute error tolerance for the eigenvalues.
! 113: * An approximate eigenvalue is accepted as converged
! 114: * when it is determined to lie in an interval [a,b]
! 115: * of width less than or equal to
! 116: *
! 117: * ABSTOL + EPS * max( |a|,|b| ) ,
! 118: *
! 119: * where EPS is the machine precision. If ABSTOL is less than
! 120: * or equal to zero, then EPS*|T| will be used in its place,
! 121: * where |T| is the 1-norm of the tridiagonal matrix obtained
! 122: * by reducing AP to tridiagonal form.
! 123: *
! 124: * Eigenvalues will be computed most accurately when ABSTOL is
! 125: * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
! 126: * If this routine returns with INFO>0, indicating that some
! 127: * eigenvectors did not converge, try setting ABSTOL to
! 128: * 2*DLAMCH('S').
! 129: *
! 130: * M (output) INTEGER
! 131: * The total number of eigenvalues found. 0 <= M <= N.
! 132: * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
! 133: *
! 134: * W (output) DOUBLE PRECISION array, dimension (N)
! 135: * If INFO = 0, the eigenvalues in ascending order.
! 136: *
! 137: * Z (output) COMPLEX*16 array, dimension (LDZ, N)
! 138: * If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
! 139: * eigenvectors, with the i-th column of Z holding the
! 140: * eigenvector associated with W(i). The eigenvectors are
! 141: * normalized so that Z**H*B*Z = I.
! 142: * If JOBZ = 'N', then Z is not referenced.
! 143: *
! 144: * LDZ (input) INTEGER
! 145: * The leading dimension of the array Z. LDZ >= 1, and if
! 146: * JOBZ = 'V', LDZ >= N.
! 147: *
! 148: * WORK (workspace) COMPLEX*16 array, dimension (N)
! 149: *
! 150: * RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
! 151: *
! 152: * IWORK (workspace) INTEGER array, dimension (5*N)
! 153: *
! 154: * IFAIL (output) INTEGER array, dimension (N)
! 155: * If JOBZ = 'V', then if INFO = 0, the first M elements of
! 156: * IFAIL are zero. If INFO > 0, then IFAIL contains the
! 157: * indices of the eigenvectors that failed to converge.
! 158: * If JOBZ = 'N', then IFAIL is not referenced.
! 159: *
! 160: * INFO (output) INTEGER
! 161: * = 0: successful exit
! 162: * < 0: if INFO = -i, the i-th argument had an illegal value
! 163: * > 0: if INFO = i, and i is:
! 164: * <= N: then i eigenvectors failed to converge. Their
! 165: * indices are stored in array IFAIL.
! 166: * > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF
! 167: * returned INFO = i: B is not positive definite.
! 168: * The factorization of B could not be completed and
! 169: * no eigenvalues or eigenvectors were computed.
! 170: *
! 171: * Further Details
! 172: * ===============
! 173: *
! 174: * Based on contributions by
! 175: * Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
! 176: *
! 177: * =====================================================================
! 178: *
! 179: * .. Parameters ..
! 180: DOUBLE PRECISION ZERO
! 181: PARAMETER ( ZERO = 0.0D+0 )
! 182: COMPLEX*16 CZERO, CONE
! 183: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
! 184: $ CONE = ( 1.0D+0, 0.0D+0 ) )
! 185: * ..
! 186: * .. Local Scalars ..
! 187: LOGICAL ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
! 188: CHARACTER ORDER, VECT
! 189: INTEGER I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
! 190: $ INDIWK, INDRWK, INDWRK, ITMP1, J, JJ, NSPLIT
! 191: DOUBLE PRECISION TMP1
! 192: * ..
! 193: * .. External Functions ..
! 194: LOGICAL LSAME
! 195: EXTERNAL LSAME
! 196: * ..
! 197: * .. External Subroutines ..
! 198: EXTERNAL DCOPY, DSTEBZ, DSTERF, XERBLA, ZCOPY, ZGEMV,
! 199: $ ZHBGST, ZHBTRD, ZLACPY, ZPBSTF, ZSTEIN, ZSTEQR,
! 200: $ ZSWAP
! 201: * ..
! 202: * .. Intrinsic Functions ..
! 203: INTRINSIC MIN
! 204: * ..
! 205: * .. Executable Statements ..
! 206: *
! 207: * Test the input parameters.
! 208: *
! 209: WANTZ = LSAME( JOBZ, 'V' )
! 210: UPPER = LSAME( UPLO, 'U' )
! 211: ALLEIG = LSAME( RANGE, 'A' )
! 212: VALEIG = LSAME( RANGE, 'V' )
! 213: INDEIG = LSAME( RANGE, 'I' )
! 214: *
! 215: INFO = 0
! 216: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
! 217: INFO = -1
! 218: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
! 219: INFO = -2
! 220: ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
! 221: INFO = -3
! 222: ELSE IF( N.LT.0 ) THEN
! 223: INFO = -4
! 224: ELSE IF( KA.LT.0 ) THEN
! 225: INFO = -5
! 226: ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
! 227: INFO = -6
! 228: ELSE IF( LDAB.LT.KA+1 ) THEN
! 229: INFO = -8
! 230: ELSE IF( LDBB.LT.KB+1 ) THEN
! 231: INFO = -10
! 232: ELSE IF( LDQ.LT.1 .OR. ( WANTZ .AND. LDQ.LT.N ) ) THEN
! 233: INFO = -12
! 234: ELSE
! 235: IF( VALEIG ) THEN
! 236: IF( N.GT.0 .AND. VU.LE.VL )
! 237: $ INFO = -14
! 238: ELSE IF( INDEIG ) THEN
! 239: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
! 240: INFO = -15
! 241: ELSE IF ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
! 242: INFO = -16
! 243: END IF
! 244: END IF
! 245: END IF
! 246: IF( INFO.EQ.0) THEN
! 247: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
! 248: INFO = -21
! 249: END IF
! 250: END IF
! 251: *
! 252: IF( INFO.NE.0 ) THEN
! 253: CALL XERBLA( 'ZHBGVX', -INFO )
! 254: RETURN
! 255: END IF
! 256: *
! 257: * Quick return if possible
! 258: *
! 259: M = 0
! 260: IF( N.EQ.0 )
! 261: $ RETURN
! 262: *
! 263: * Form a split Cholesky factorization of B.
! 264: *
! 265: CALL ZPBSTF( UPLO, N, KB, BB, LDBB, INFO )
! 266: IF( INFO.NE.0 ) THEN
! 267: INFO = N + INFO
! 268: RETURN
! 269: END IF
! 270: *
! 271: * Transform problem to standard eigenvalue problem.
! 272: *
! 273: CALL ZHBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ,
! 274: $ WORK, RWORK, IINFO )
! 275: *
! 276: * Solve the standard eigenvalue problem.
! 277: * Reduce Hermitian band matrix to tridiagonal form.
! 278: *
! 279: INDD = 1
! 280: INDE = INDD + N
! 281: INDRWK = INDE + N
! 282: INDWRK = 1
! 283: IF( WANTZ ) THEN
! 284: VECT = 'U'
! 285: ELSE
! 286: VECT = 'N'
! 287: END IF
! 288: CALL ZHBTRD( VECT, UPLO, N, KA, AB, LDAB, RWORK( INDD ),
! 289: $ RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
! 290: *
! 291: * If all eigenvalues are desired and ABSTOL is less than or equal
! 292: * to zero, then call DSTERF or ZSTEQR. If this fails for some
! 293: * eigenvalue, then try DSTEBZ.
! 294: *
! 295: TEST = .FALSE.
! 296: IF( INDEIG ) THEN
! 297: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
! 298: TEST = .TRUE.
! 299: END IF
! 300: END IF
! 301: IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
! 302: CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
! 303: INDEE = INDRWK + 2*N
! 304: CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
! 305: IF( .NOT.WANTZ ) THEN
! 306: CALL DSTERF( N, W, RWORK( INDEE ), INFO )
! 307: ELSE
! 308: CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
! 309: CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
! 310: $ RWORK( INDRWK ), INFO )
! 311: IF( INFO.EQ.0 ) THEN
! 312: DO 10 I = 1, N
! 313: IFAIL( I ) = 0
! 314: 10 CONTINUE
! 315: END IF
! 316: END IF
! 317: IF( INFO.EQ.0 ) THEN
! 318: M = N
! 319: GO TO 30
! 320: END IF
! 321: INFO = 0
! 322: END IF
! 323: *
! 324: * Otherwise, call DSTEBZ and, if eigenvectors are desired,
! 325: * call ZSTEIN.
! 326: *
! 327: IF( WANTZ ) THEN
! 328: ORDER = 'B'
! 329: ELSE
! 330: ORDER = 'E'
! 331: END IF
! 332: INDIBL = 1
! 333: INDISP = INDIBL + N
! 334: INDIWK = INDISP + N
! 335: CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
! 336: $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
! 337: $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
! 338: $ IWORK( INDIWK ), INFO )
! 339: *
! 340: IF( WANTZ ) THEN
! 341: CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
! 342: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
! 343: $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
! 344: *
! 345: * Apply unitary matrix used in reduction to tridiagonal
! 346: * form to eigenvectors returned by ZSTEIN.
! 347: *
! 348: DO 20 J = 1, M
! 349: CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
! 350: CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO,
! 351: $ Z( 1, J ), 1 )
! 352: 20 CONTINUE
! 353: END IF
! 354: *
! 355: 30 CONTINUE
! 356: *
! 357: * If eigenvalues are not in order, then sort them, along with
! 358: * eigenvectors.
! 359: *
! 360: IF( WANTZ ) THEN
! 361: DO 50 J = 1, M - 1
! 362: I = 0
! 363: TMP1 = W( J )
! 364: DO 40 JJ = J + 1, M
! 365: IF( W( JJ ).LT.TMP1 ) THEN
! 366: I = JJ
! 367: TMP1 = W( JJ )
! 368: END IF
! 369: 40 CONTINUE
! 370: *
! 371: IF( I.NE.0 ) THEN
! 372: ITMP1 = IWORK( INDIBL+I-1 )
! 373: W( I ) = W( J )
! 374: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
! 375: W( J ) = TMP1
! 376: IWORK( INDIBL+J-1 ) = ITMP1
! 377: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
! 378: IF( INFO.NE.0 ) THEN
! 379: ITMP1 = IFAIL( I )
! 380: IFAIL( I ) = IFAIL( J )
! 381: IFAIL( J ) = ITMP1
! 382: END IF
! 383: END IF
! 384: 50 CONTINUE
! 385: END IF
! 386: *
! 387: RETURN
! 388: *
! 389: * End of ZHBGVX
! 390: *
! 391: END
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