Annotation of rpl/lapack/lapack/zgeesx.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
! 2: $ VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
! 3: $ BWORK, 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 JOBVS, SENSE, SORT
! 12: INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
! 13: DOUBLE PRECISION RCONDE, RCONDV
! 14: * ..
! 15: * .. Array Arguments ..
! 16: LOGICAL BWORK( * )
! 17: DOUBLE PRECISION RWORK( * )
! 18: COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
! 19: * ..
! 20: * .. Function Arguments ..
! 21: LOGICAL SELECT
! 22: EXTERNAL SELECT
! 23: * ..
! 24: *
! 25: * Purpose
! 26: * =======
! 27: *
! 28: * ZGEESX computes for an N-by-N complex nonsymmetric matrix A, the
! 29: * eigenvalues, the Schur form T, and, optionally, the matrix of Schur
! 30: * vectors Z. This gives the Schur factorization A = Z*T*(Z**H).
! 31: *
! 32: * Optionally, it also orders the eigenvalues on the diagonal of the
! 33: * Schur form so that selected eigenvalues are at the top left;
! 34: * computes a reciprocal condition number for the average of the
! 35: * selected eigenvalues (RCONDE); and computes a reciprocal condition
! 36: * number for the right invariant subspace corresponding to the
! 37: * selected eigenvalues (RCONDV). The leading columns of Z form an
! 38: * orthonormal basis for this invariant subspace.
! 39: *
! 40: * For further explanation of the reciprocal condition numbers RCONDE
! 41: * and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
! 42: * these quantities are called s and sep respectively).
! 43: *
! 44: * A complex matrix is in Schur form if it is upper triangular.
! 45: *
! 46: * Arguments
! 47: * =========
! 48: *
! 49: * JOBVS (input) CHARACTER*1
! 50: * = 'N': Schur vectors are not computed;
! 51: * = 'V': Schur vectors are computed.
! 52: *
! 53: * SORT (input) CHARACTER*1
! 54: * Specifies whether or not to order the eigenvalues on the
! 55: * diagonal of the Schur form.
! 56: * = 'N': Eigenvalues are not ordered;
! 57: * = 'S': Eigenvalues are ordered (see SELECT).
! 58: *
! 59: * SELECT (external procedure) LOGICAL FUNCTION of one COMPLEX*16 argument
! 60: * SELECT must be declared EXTERNAL in the calling subroutine.
! 61: * If SORT = 'S', SELECT is used to select eigenvalues to order
! 62: * to the top left of the Schur form.
! 63: * If SORT = 'N', SELECT is not referenced.
! 64: * An eigenvalue W(j) is selected if SELECT(W(j)) is true.
! 65: *
! 66: * SENSE (input) CHARACTER*1
! 67: * Determines which reciprocal condition numbers are computed.
! 68: * = 'N': None are computed;
! 69: * = 'E': Computed for average of selected eigenvalues only;
! 70: * = 'V': Computed for selected right invariant subspace only;
! 71: * = 'B': Computed for both.
! 72: * If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
! 73: *
! 74: * N (input) INTEGER
! 75: * The order of the matrix A. N >= 0.
! 76: *
! 77: * A (input/output) COMPLEX*16 array, dimension (LDA, N)
! 78: * On entry, the N-by-N matrix A.
! 79: * On exit, A is overwritten by its Schur form T.
! 80: *
! 81: * LDA (input) INTEGER
! 82: * The leading dimension of the array A. LDA >= max(1,N).
! 83: *
! 84: * SDIM (output) INTEGER
! 85: * If SORT = 'N', SDIM = 0.
! 86: * If SORT = 'S', SDIM = number of eigenvalues for which
! 87: * SELECT is true.
! 88: *
! 89: * W (output) COMPLEX*16 array, dimension (N)
! 90: * W contains the computed eigenvalues, in the same order
! 91: * that they appear on the diagonal of the output Schur form T.
! 92: *
! 93: * VS (output) COMPLEX*16 array, dimension (LDVS,N)
! 94: * If JOBVS = 'V', VS contains the unitary matrix Z of Schur
! 95: * vectors.
! 96: * If JOBVS = 'N', VS is not referenced.
! 97: *
! 98: * LDVS (input) INTEGER
! 99: * The leading dimension of the array VS. LDVS >= 1, and if
! 100: * JOBVS = 'V', LDVS >= N.
! 101: *
! 102: * RCONDE (output) DOUBLE PRECISION
! 103: * If SENSE = 'E' or 'B', RCONDE contains the reciprocal
! 104: * condition number for the average of the selected eigenvalues.
! 105: * Not referenced if SENSE = 'N' or 'V'.
! 106: *
! 107: * RCONDV (output) DOUBLE PRECISION
! 108: * If SENSE = 'V' or 'B', RCONDV contains the reciprocal
! 109: * condition number for the selected right invariant subspace.
! 110: * Not referenced if SENSE = 'N' or 'E'.
! 111: *
! 112: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
! 113: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 114: *
! 115: * LWORK (input) INTEGER
! 116: * The dimension of the array WORK. LWORK >= max(1,2*N).
! 117: * Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
! 118: * where SDIM is the number of selected eigenvalues computed by
! 119: * this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
! 120: * that an error is only returned if LWORK < max(1,2*N), but if
! 121: * SENSE = 'E' or 'V' or 'B' this may not be large enough.
! 122: * For good performance, LWORK must generally be larger.
! 123: *
! 124: * If LWORK = -1, then a workspace query is assumed; the routine
! 125: * only calculates upper bound on the optimal size of the
! 126: * array WORK, returns this value as the first entry of the WORK
! 127: * array, and no error message related to LWORK is issued by
! 128: * XERBLA.
! 129: *
! 130: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
! 131: *
! 132: * BWORK (workspace) LOGICAL array, dimension (N)
! 133: * Not referenced if SORT = 'N'.
! 134: *
! 135: * INFO (output) INTEGER
! 136: * = 0: successful exit
! 137: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 138: * > 0: if INFO = i, and i is
! 139: * <= N: the QR algorithm failed to compute all the
! 140: * eigenvalues; elements 1:ILO-1 and i+1:N of W
! 141: * contain those eigenvalues which have converged; if
! 142: * JOBVS = 'V', VS contains the transformation which
! 143: * reduces A to its partially converged Schur form.
! 144: * = N+1: the eigenvalues could not be reordered because some
! 145: * eigenvalues were too close to separate (the problem
! 146: * is very ill-conditioned);
! 147: * = N+2: after reordering, roundoff changed values of some
! 148: * complex eigenvalues so that leading eigenvalues in
! 149: * the Schur form no longer satisfy SELECT=.TRUE. This
! 150: * could also be caused by underflow due to scaling.
! 151: *
! 152: * =====================================================================
! 153: *
! 154: * .. Parameters ..
! 155: DOUBLE PRECISION ZERO, ONE
! 156: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
! 157: * ..
! 158: * .. Local Scalars ..
! 159: LOGICAL SCALEA, WANTSB, WANTSE, WANTSN, WANTST, WANTSV,
! 160: $ WANTVS
! 161: INTEGER HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO,
! 162: $ ITAU, IWRK, LWRK, MAXWRK, MINWRK
! 163: DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SMLNUM
! 164: * ..
! 165: * .. Local Arrays ..
! 166: DOUBLE PRECISION DUM( 1 )
! 167: * ..
! 168: * .. External Subroutines ..
! 169: EXTERNAL DLABAD, DLASCL, XERBLA, ZCOPY, ZGEBAK, ZGEBAL,
! 170: $ ZGEHRD, ZHSEQR, ZLACPY, ZLASCL, ZTRSEN, ZUNGHR
! 171: * ..
! 172: * .. External Functions ..
! 173: LOGICAL LSAME
! 174: INTEGER ILAENV
! 175: DOUBLE PRECISION DLAMCH, ZLANGE
! 176: EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
! 177: * ..
! 178: * .. Intrinsic Functions ..
! 179: INTRINSIC MAX, SQRT
! 180: * ..
! 181: * .. Executable Statements ..
! 182: *
! 183: * Test the input arguments
! 184: *
! 185: INFO = 0
! 186: WANTVS = LSAME( JOBVS, 'V' )
! 187: WANTST = LSAME( SORT, 'S' )
! 188: WANTSN = LSAME( SENSE, 'N' )
! 189: WANTSE = LSAME( SENSE, 'E' )
! 190: WANTSV = LSAME( SENSE, 'V' )
! 191: WANTSB = LSAME( SENSE, 'B' )
! 192: IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
! 193: INFO = -1
! 194: ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
! 195: INFO = -2
! 196: ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
! 197: $ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
! 198: INFO = -4
! 199: ELSE IF( N.LT.0 ) THEN
! 200: INFO = -5
! 201: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 202: INFO = -7
! 203: ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
! 204: INFO = -11
! 205: END IF
! 206: *
! 207: * Compute workspace
! 208: * (Note: Comments in the code beginning "Workspace:" describe the
! 209: * minimal amount of real workspace needed at that point in the
! 210: * code, as well as the preferred amount for good performance.
! 211: * CWorkspace refers to complex workspace, and RWorkspace to real
! 212: * workspace. NB refers to the optimal block size for the
! 213: * immediately following subroutine, as returned by ILAENV.
! 214: * HSWORK refers to the workspace preferred by ZHSEQR, as
! 215: * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
! 216: * the worst case.
! 217: * If SENSE = 'E', 'V' or 'B', then the amount of workspace needed
! 218: * depends on SDIM, which is computed by the routine ZTRSEN later
! 219: * in the code.)
! 220: *
! 221: IF( INFO.EQ.0 ) THEN
! 222: IF( N.EQ.0 ) THEN
! 223: MINWRK = 1
! 224: LWRK = 1
! 225: ELSE
! 226: MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 )
! 227: MINWRK = 2*N
! 228: *
! 229: CALL ZHSEQR( 'S', JOBVS, N, 1, N, A, LDA, W, VS, LDVS,
! 230: $ WORK, -1, IEVAL )
! 231: HSWORK = WORK( 1 )
! 232: *
! 233: IF( .NOT.WANTVS ) THEN
! 234: MAXWRK = MAX( MAXWRK, HSWORK )
! 235: ELSE
! 236: MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
! 237: $ ' ', N, 1, N, -1 ) )
! 238: MAXWRK = MAX( MAXWRK, HSWORK )
! 239: END IF
! 240: LWRK = MAXWRK
! 241: IF( .NOT.WANTSN )
! 242: $ LWRK = MAX( LWRK, ( N*N )/2 )
! 243: END IF
! 244: WORK( 1 ) = LWRK
! 245: *
! 246: IF( LWORK.LT.MINWRK ) THEN
! 247: INFO = -15
! 248: END IF
! 249: END IF
! 250: *
! 251: IF( INFO.NE.0 ) THEN
! 252: CALL XERBLA( 'ZGEESX', -INFO )
! 253: RETURN
! 254: END IF
! 255: *
! 256: * Quick return if possible
! 257: *
! 258: IF( N.EQ.0 ) THEN
! 259: SDIM = 0
! 260: RETURN
! 261: END IF
! 262: *
! 263: * Get machine constants
! 264: *
! 265: EPS = DLAMCH( 'P' )
! 266: SMLNUM = DLAMCH( 'S' )
! 267: BIGNUM = ONE / SMLNUM
! 268: CALL DLABAD( SMLNUM, BIGNUM )
! 269: SMLNUM = SQRT( SMLNUM ) / EPS
! 270: BIGNUM = ONE / SMLNUM
! 271: *
! 272: * Scale A if max element outside range [SMLNUM,BIGNUM]
! 273: *
! 274: ANRM = ZLANGE( 'M', N, N, A, LDA, DUM )
! 275: SCALEA = .FALSE.
! 276: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
! 277: SCALEA = .TRUE.
! 278: CSCALE = SMLNUM
! 279: ELSE IF( ANRM.GT.BIGNUM ) THEN
! 280: SCALEA = .TRUE.
! 281: CSCALE = BIGNUM
! 282: END IF
! 283: IF( SCALEA )
! 284: $ CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
! 285: *
! 286: *
! 287: * Permute the matrix to make it more nearly triangular
! 288: * (CWorkspace: none)
! 289: * (RWorkspace: need N)
! 290: *
! 291: IBAL = 1
! 292: CALL ZGEBAL( 'P', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
! 293: *
! 294: * Reduce to upper Hessenberg form
! 295: * (CWorkspace: need 2*N, prefer N+N*NB)
! 296: * (RWorkspace: none)
! 297: *
! 298: ITAU = 1
! 299: IWRK = N + ITAU
! 300: CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
! 301: $ LWORK-IWRK+1, IERR )
! 302: *
! 303: IF( WANTVS ) THEN
! 304: *
! 305: * Copy Householder vectors to VS
! 306: *
! 307: CALL ZLACPY( 'L', N, N, A, LDA, VS, LDVS )
! 308: *
! 309: * Generate unitary matrix in VS
! 310: * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
! 311: * (RWorkspace: none)
! 312: *
! 313: CALL ZUNGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ),
! 314: $ LWORK-IWRK+1, IERR )
! 315: END IF
! 316: *
! 317: SDIM = 0
! 318: *
! 319: * Perform QR iteration, accumulating Schur vectors in VS if desired
! 320: * (CWorkspace: need 1, prefer HSWORK (see comments) )
! 321: * (RWorkspace: none)
! 322: *
! 323: IWRK = ITAU
! 324: CALL ZHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, W, VS, LDVS,
! 325: $ WORK( IWRK ), LWORK-IWRK+1, IEVAL )
! 326: IF( IEVAL.GT.0 )
! 327: $ INFO = IEVAL
! 328: *
! 329: * Sort eigenvalues if desired
! 330: *
! 331: IF( WANTST .AND. INFO.EQ.0 ) THEN
! 332: IF( SCALEA )
! 333: $ CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, W, N, IERR )
! 334: DO 10 I = 1, N
! 335: BWORK( I ) = SELECT( W( I ) )
! 336: 10 CONTINUE
! 337: *
! 338: * Reorder eigenvalues, transform Schur vectors, and compute
! 339: * reciprocal condition numbers
! 340: * (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM)
! 341: * otherwise, need none )
! 342: * (RWorkspace: none)
! 343: *
! 344: CALL ZTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM,
! 345: $ RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1,
! 346: $ ICOND )
! 347: IF( .NOT.WANTSN )
! 348: $ MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) )
! 349: IF( ICOND.EQ.-14 ) THEN
! 350: *
! 351: * Not enough complex workspace
! 352: *
! 353: INFO = -15
! 354: END IF
! 355: END IF
! 356: *
! 357: IF( WANTVS ) THEN
! 358: *
! 359: * Undo balancing
! 360: * (CWorkspace: none)
! 361: * (RWorkspace: need N)
! 362: *
! 363: CALL ZGEBAK( 'P', 'R', N, ILO, IHI, RWORK( IBAL ), N, VS, LDVS,
! 364: $ IERR )
! 365: END IF
! 366: *
! 367: IF( SCALEA ) THEN
! 368: *
! 369: * Undo scaling for the Schur form of A
! 370: *
! 371: CALL ZLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
! 372: CALL ZCOPY( N, A, LDA+1, W, 1 )
! 373: IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN
! 374: DUM( 1 ) = RCONDV
! 375: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
! 376: RCONDV = DUM( 1 )
! 377: END IF
! 378: END IF
! 379: *
! 380: WORK( 1 ) = MAXWRK
! 381: RETURN
! 382: *
! 383: * End of ZGEESX
! 384: *
! 385: END
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