Annotation of rpl/lapack/lapack/dtrsen.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
! 2: $ M, S, SEP, WORK, LWORK, IWORK, LIWORK, 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: CHARACTER COMPQ, JOB
! 11: INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
! 12: DOUBLE PRECISION S, SEP
! 13: * ..
! 14: * .. Array Arguments ..
! 15: LOGICAL SELECT( * )
! 16: INTEGER IWORK( * )
! 17: DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
! 18: $ WR( * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * DTRSEN reorders the real Schur factorization of a real matrix
! 25: * A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
! 26: * the leading diagonal blocks of the upper quasi-triangular matrix T,
! 27: * and the leading columns of Q form an orthonormal basis of the
! 28: * corresponding right invariant subspace.
! 29: *
! 30: * Optionally the routine computes the reciprocal condition numbers of
! 31: * the cluster of eigenvalues and/or the invariant subspace.
! 32: *
! 33: * T must be in Schur canonical form (as returned by DHSEQR), that is,
! 34: * block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
! 35: * 2-by-2 diagonal block has its diagonal elemnts equal and its
! 36: * off-diagonal elements of opposite sign.
! 37: *
! 38: * Arguments
! 39: * =========
! 40: *
! 41: * JOB (input) CHARACTER*1
! 42: * Specifies whether condition numbers are required for the
! 43: * cluster of eigenvalues (S) or the invariant subspace (SEP):
! 44: * = 'N': none;
! 45: * = 'E': for eigenvalues only (S);
! 46: * = 'V': for invariant subspace only (SEP);
! 47: * = 'B': for both eigenvalues and invariant subspace (S and
! 48: * SEP).
! 49: *
! 50: * COMPQ (input) CHARACTER*1
! 51: * = 'V': update the matrix Q of Schur vectors;
! 52: * = 'N': do not update Q.
! 53: *
! 54: * SELECT (input) LOGICAL array, dimension (N)
! 55: * SELECT specifies the eigenvalues in the selected cluster. To
! 56: * select a real eigenvalue w(j), SELECT(j) must be set to
! 57: * .TRUE.. To select a complex conjugate pair of eigenvalues
! 58: * w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
! 59: * either SELECT(j) or SELECT(j+1) or both must be set to
! 60: * .TRUE.; a complex conjugate pair of eigenvalues must be
! 61: * either both included in the cluster or both excluded.
! 62: *
! 63: * N (input) INTEGER
! 64: * The order of the matrix T. N >= 0.
! 65: *
! 66: * T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
! 67: * On entry, the upper quasi-triangular matrix T, in Schur
! 68: * canonical form.
! 69: * On exit, T is overwritten by the reordered matrix T, again in
! 70: * Schur canonical form, with the selected eigenvalues in the
! 71: * leading diagonal blocks.
! 72: *
! 73: * LDT (input) INTEGER
! 74: * The leading dimension of the array T. LDT >= max(1,N).
! 75: *
! 76: * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
! 77: * On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
! 78: * On exit, if COMPQ = 'V', Q has been postmultiplied by the
! 79: * orthogonal transformation matrix which reorders T; the
! 80: * leading M columns of Q form an orthonormal basis for the
! 81: * specified invariant subspace.
! 82: * If COMPQ = 'N', Q is not referenced.
! 83: *
! 84: * LDQ (input) INTEGER
! 85: * The leading dimension of the array Q.
! 86: * LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
! 87: *
! 88: * WR (output) DOUBLE PRECISION array, dimension (N)
! 89: * WI (output) DOUBLE PRECISION array, dimension (N)
! 90: * The real and imaginary parts, respectively, of the reordered
! 91: * eigenvalues of T. The eigenvalues are stored in the same
! 92: * order as on the diagonal of T, with WR(i) = T(i,i) and, if
! 93: * T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
! 94: * WI(i+1) = -WI(i). Note that if a complex eigenvalue is
! 95: * sufficiently ill-conditioned, then its value may differ
! 96: * significantly from its value before reordering.
! 97: *
! 98: * M (output) INTEGER
! 99: * The dimension of the specified invariant subspace.
! 100: * 0 < = M <= N.
! 101: *
! 102: * S (output) DOUBLE PRECISION
! 103: * If JOB = 'E' or 'B', S is a lower bound on the reciprocal
! 104: * condition number for the selected cluster of eigenvalues.
! 105: * S cannot underestimate the true reciprocal condition number
! 106: * by more than a factor of sqrt(N). If M = 0 or N, S = 1.
! 107: * If JOB = 'N' or 'V', S is not referenced.
! 108: *
! 109: * SEP (output) DOUBLE PRECISION
! 110: * If JOB = 'V' or 'B', SEP is the estimated reciprocal
! 111: * condition number of the specified invariant subspace. If
! 112: * M = 0 or N, SEP = norm(T).
! 113: * If JOB = 'N' or 'E', SEP is not referenced.
! 114: *
! 115: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 116: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 117: *
! 118: * LWORK (input) INTEGER
! 119: * The dimension of the array WORK.
! 120: * If JOB = 'N', LWORK >= max(1,N);
! 121: * if JOB = 'E', LWORK >= max(1,M*(N-M));
! 122: * if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
! 123: *
! 124: * If LWORK = -1, then a workspace query is assumed; the routine
! 125: * only calculates the optimal size of the WORK array, returns
! 126: * this value as the first entry of the WORK array, and no error
! 127: * message related to LWORK is issued by XERBLA.
! 128: *
! 129: * IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))
! 130: * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
! 131: *
! 132: * LIWORK (input) INTEGER
! 133: * The dimension of the array IWORK.
! 134: * If JOB = 'N' or 'E', LIWORK >= 1;
! 135: * if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
! 136: *
! 137: * If LIWORK = -1, then a workspace query is assumed; the
! 138: * routine only calculates the optimal size of the IWORK array,
! 139: * returns this value as the first entry of the IWORK array, and
! 140: * no error message related to LIWORK is issued by XERBLA.
! 141: *
! 142: * INFO (output) INTEGER
! 143: * = 0: successful exit
! 144: * < 0: if INFO = -i, the i-th argument had an illegal value
! 145: * = 1: reordering of T failed because some eigenvalues are too
! 146: * close to separate (the problem is very ill-conditioned);
! 147: * T may have been partially reordered, and WR and WI
! 148: * contain the eigenvalues in the same order as in T; S and
! 149: * SEP (if requested) are set to zero.
! 150: *
! 151: * Further Details
! 152: * ===============
! 153: *
! 154: * DTRSEN first collects the selected eigenvalues by computing an
! 155: * orthogonal transformation Z to move them to the top left corner of T.
! 156: * In other words, the selected eigenvalues are the eigenvalues of T11
! 157: * in:
! 158: *
! 159: * Z'*T*Z = ( T11 T12 ) n1
! 160: * ( 0 T22 ) n2
! 161: * n1 n2
! 162: *
! 163: * where N = n1+n2 and Z' means the transpose of Z. The first n1 columns
! 164: * of Z span the specified invariant subspace of T.
! 165: *
! 166: * If T has been obtained from the real Schur factorization of a matrix
! 167: * A = Q*T*Q', then the reordered real Schur factorization of A is given
! 168: * by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span
! 169: * the corresponding invariant subspace of A.
! 170: *
! 171: * The reciprocal condition number of the average of the eigenvalues of
! 172: * T11 may be returned in S. S lies between 0 (very badly conditioned)
! 173: * and 1 (very well conditioned). It is computed as follows. First we
! 174: * compute R so that
! 175: *
! 176: * P = ( I R ) n1
! 177: * ( 0 0 ) n2
! 178: * n1 n2
! 179: *
! 180: * is the projector on the invariant subspace associated with T11.
! 181: * R is the solution of the Sylvester equation:
! 182: *
! 183: * T11*R - R*T22 = T12.
! 184: *
! 185: * Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
! 186: * the two-norm of M. Then S is computed as the lower bound
! 187: *
! 188: * (1 + F-norm(R)**2)**(-1/2)
! 189: *
! 190: * on the reciprocal of 2-norm(P), the true reciprocal condition number.
! 191: * S cannot underestimate 1 / 2-norm(P) by more than a factor of
! 192: * sqrt(N).
! 193: *
! 194: * An approximate error bound for the computed average of the
! 195: * eigenvalues of T11 is
! 196: *
! 197: * EPS * norm(T) / S
! 198: *
! 199: * where EPS is the machine precision.
! 200: *
! 201: * The reciprocal condition number of the right invariant subspace
! 202: * spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
! 203: * SEP is defined as the separation of T11 and T22:
! 204: *
! 205: * sep( T11, T22 ) = sigma-min( C )
! 206: *
! 207: * where sigma-min(C) is the smallest singular value of the
! 208: * n1*n2-by-n1*n2 matrix
! 209: *
! 210: * C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
! 211: *
! 212: * I(m) is an m by m identity matrix, and kprod denotes the Kronecker
! 213: * product. We estimate sigma-min(C) by the reciprocal of an estimate of
! 214: * the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
! 215: * cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
! 216: *
! 217: * When SEP is small, small changes in T can cause large changes in
! 218: * the invariant subspace. An approximate bound on the maximum angular
! 219: * error in the computed right invariant subspace is
! 220: *
! 221: * EPS * norm(T) / SEP
! 222: *
! 223: * =====================================================================
! 224: *
! 225: * .. Parameters ..
! 226: DOUBLE PRECISION ZERO, ONE
! 227: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 228: * ..
! 229: * .. Local Scalars ..
! 230: LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
! 231: $ WANTSP
! 232: INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
! 233: $ NN
! 234: DOUBLE PRECISION EST, RNORM, SCALE
! 235: * ..
! 236: * .. Local Arrays ..
! 237: INTEGER ISAVE( 3 )
! 238: * ..
! 239: * .. External Functions ..
! 240: LOGICAL LSAME
! 241: DOUBLE PRECISION DLANGE
! 242: EXTERNAL LSAME, DLANGE
! 243: * ..
! 244: * .. External Subroutines ..
! 245: EXTERNAL DLACN2, DLACPY, DTREXC, DTRSYL, XERBLA
! 246: * ..
! 247: * .. Intrinsic Functions ..
! 248: INTRINSIC ABS, MAX, SQRT
! 249: * ..
! 250: * .. Executable Statements ..
! 251: *
! 252: * Decode and test the input parameters
! 253: *
! 254: WANTBH = LSAME( JOB, 'B' )
! 255: WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
! 256: WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
! 257: WANTQ = LSAME( COMPQ, 'V' )
! 258: *
! 259: INFO = 0
! 260: LQUERY = ( LWORK.EQ.-1 )
! 261: IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
! 262: $ THEN
! 263: INFO = -1
! 264: ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
! 265: INFO = -2
! 266: ELSE IF( N.LT.0 ) THEN
! 267: INFO = -4
! 268: ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
! 269: INFO = -6
! 270: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
! 271: INFO = -8
! 272: ELSE
! 273: *
! 274: * Set M to the dimension of the specified invariant subspace,
! 275: * and test LWORK and LIWORK.
! 276: *
! 277: M = 0
! 278: PAIR = .FALSE.
! 279: DO 10 K = 1, N
! 280: IF( PAIR ) THEN
! 281: PAIR = .FALSE.
! 282: ELSE
! 283: IF( K.LT.N ) THEN
! 284: IF( T( K+1, K ).EQ.ZERO ) THEN
! 285: IF( SELECT( K ) )
! 286: $ M = M + 1
! 287: ELSE
! 288: PAIR = .TRUE.
! 289: IF( SELECT( K ) .OR. SELECT( K+1 ) )
! 290: $ M = M + 2
! 291: END IF
! 292: ELSE
! 293: IF( SELECT( N ) )
! 294: $ M = M + 1
! 295: END IF
! 296: END IF
! 297: 10 CONTINUE
! 298: *
! 299: N1 = M
! 300: N2 = N - M
! 301: NN = N1*N2
! 302: *
! 303: IF( WANTSP ) THEN
! 304: LWMIN = MAX( 1, 2*NN )
! 305: LIWMIN = MAX( 1, NN )
! 306: ELSE IF( LSAME( JOB, 'N' ) ) THEN
! 307: LWMIN = MAX( 1, N )
! 308: LIWMIN = 1
! 309: ELSE IF( LSAME( JOB, 'E' ) ) THEN
! 310: LWMIN = MAX( 1, NN )
! 311: LIWMIN = 1
! 312: END IF
! 313: *
! 314: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
! 315: INFO = -15
! 316: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
! 317: INFO = -17
! 318: END IF
! 319: END IF
! 320: *
! 321: IF( INFO.EQ.0 ) THEN
! 322: WORK( 1 ) = LWMIN
! 323: IWORK( 1 ) = LIWMIN
! 324: END IF
! 325: *
! 326: IF( INFO.NE.0 ) THEN
! 327: CALL XERBLA( 'DTRSEN', -INFO )
! 328: RETURN
! 329: ELSE IF( LQUERY ) THEN
! 330: RETURN
! 331: END IF
! 332: *
! 333: * Quick return if possible.
! 334: *
! 335: IF( M.EQ.N .OR. M.EQ.0 ) THEN
! 336: IF( WANTS )
! 337: $ S = ONE
! 338: IF( WANTSP )
! 339: $ SEP = DLANGE( '1', N, N, T, LDT, WORK )
! 340: GO TO 40
! 341: END IF
! 342: *
! 343: * Collect the selected blocks at the top-left corner of T.
! 344: *
! 345: KS = 0
! 346: PAIR = .FALSE.
! 347: DO 20 K = 1, N
! 348: IF( PAIR ) THEN
! 349: PAIR = .FALSE.
! 350: ELSE
! 351: SWAP = SELECT( K )
! 352: IF( K.LT.N ) THEN
! 353: IF( T( K+1, K ).NE.ZERO ) THEN
! 354: PAIR = .TRUE.
! 355: SWAP = SWAP .OR. SELECT( K+1 )
! 356: END IF
! 357: END IF
! 358: IF( SWAP ) THEN
! 359: KS = KS + 1
! 360: *
! 361: * Swap the K-th block to position KS.
! 362: *
! 363: IERR = 0
! 364: KK = K
! 365: IF( K.NE.KS )
! 366: $ CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK,
! 367: $ IERR )
! 368: IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
! 369: *
! 370: * Blocks too close to swap: exit.
! 371: *
! 372: INFO = 1
! 373: IF( WANTS )
! 374: $ S = ZERO
! 375: IF( WANTSP )
! 376: $ SEP = ZERO
! 377: GO TO 40
! 378: END IF
! 379: IF( PAIR )
! 380: $ KS = KS + 1
! 381: END IF
! 382: END IF
! 383: 20 CONTINUE
! 384: *
! 385: IF( WANTS ) THEN
! 386: *
! 387: * Solve Sylvester equation for R:
! 388: *
! 389: * T11*R - R*T22 = scale*T12
! 390: *
! 391: CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
! 392: CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
! 393: $ LDT, WORK, N1, SCALE, IERR )
! 394: *
! 395: * Estimate the reciprocal of the condition number of the cluster
! 396: * of eigenvalues.
! 397: *
! 398: RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK )
! 399: IF( RNORM.EQ.ZERO ) THEN
! 400: S = ONE
! 401: ELSE
! 402: S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
! 403: $ SQRT( RNORM ) )
! 404: END IF
! 405: END IF
! 406: *
! 407: IF( WANTSP ) THEN
! 408: *
! 409: * Estimate sep(T11,T22).
! 410: *
! 411: EST = ZERO
! 412: KASE = 0
! 413: 30 CONTINUE
! 414: CALL DLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE )
! 415: IF( KASE.NE.0 ) THEN
! 416: IF( KASE.EQ.1 ) THEN
! 417: *
! 418: * Solve T11*R - R*T22 = scale*X.
! 419: *
! 420: CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
! 421: $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
! 422: $ IERR )
! 423: ELSE
! 424: *
! 425: * Solve T11'*R - R*T22' = scale*X.
! 426: *
! 427: CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT,
! 428: $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
! 429: $ IERR )
! 430: END IF
! 431: GO TO 30
! 432: END IF
! 433: *
! 434: SEP = SCALE / EST
! 435: END IF
! 436: *
! 437: 40 CONTINUE
! 438: *
! 439: * Store the output eigenvalues in WR and WI.
! 440: *
! 441: DO 50 K = 1, N
! 442: WR( K ) = T( K, K )
! 443: WI( K ) = ZERO
! 444: 50 CONTINUE
! 445: DO 60 K = 1, N - 1
! 446: IF( T( K+1, K ).NE.ZERO ) THEN
! 447: WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
! 448: $ SQRT( ABS( T( K+1, K ) ) )
! 449: WI( K+1 ) = -WI( K )
! 450: END IF
! 451: 60 CONTINUE
! 452: *
! 453: WORK( 1 ) = LWMIN
! 454: IWORK( 1 ) = LIWMIN
! 455: *
! 456: RETURN
! 457: *
! 458: * End of DTRSEN
! 459: *
! 460: END
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