Annotation of rpl/lapack/lapack/ztrsen.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
! 2: $ SEP, WORK, LWORK, 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: * Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
! 10: *
! 11: * .. Scalar Arguments ..
! 12: CHARACTER COMPQ, JOB
! 13: INTEGER INFO, LDQ, LDT, LWORK, M, N
! 14: DOUBLE PRECISION S, SEP
! 15: * ..
! 16: * .. Array Arguments ..
! 17: LOGICAL SELECT( * )
! 18: COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * ZTRSEN reorders the Schur factorization of a complex matrix
! 25: * A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
! 26: * the leading positions on the diagonal of the upper triangular matrix
! 27: * T, 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: * Arguments
! 34: * =========
! 35: *
! 36: * JOB (input) CHARACTER*1
! 37: * Specifies whether condition numbers are required for the
! 38: * cluster of eigenvalues (S) or the invariant subspace (SEP):
! 39: * = 'N': none;
! 40: * = 'E': for eigenvalues only (S);
! 41: * = 'V': for invariant subspace only (SEP);
! 42: * = 'B': for both eigenvalues and invariant subspace (S and
! 43: * SEP).
! 44: *
! 45: * COMPQ (input) CHARACTER*1
! 46: * = 'V': update the matrix Q of Schur vectors;
! 47: * = 'N': do not update Q.
! 48: *
! 49: * SELECT (input) LOGICAL array, dimension (N)
! 50: * SELECT specifies the eigenvalues in the selected cluster. To
! 51: * select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
! 52: *
! 53: * N (input) INTEGER
! 54: * The order of the matrix T. N >= 0.
! 55: *
! 56: * T (input/output) COMPLEX*16 array, dimension (LDT,N)
! 57: * On entry, the upper triangular matrix T.
! 58: * On exit, T is overwritten by the reordered matrix T, with the
! 59: * selected eigenvalues as the leading diagonal elements.
! 60: *
! 61: * LDT (input) INTEGER
! 62: * The leading dimension of the array T. LDT >= max(1,N).
! 63: *
! 64: * Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
! 65: * On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
! 66: * On exit, if COMPQ = 'V', Q has been postmultiplied by the
! 67: * unitary transformation matrix which reorders T; the leading M
! 68: * columns of Q form an orthonormal basis for the specified
! 69: * invariant subspace.
! 70: * If COMPQ = 'N', Q is not referenced.
! 71: *
! 72: * LDQ (input) INTEGER
! 73: * The leading dimension of the array Q.
! 74: * LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
! 75: *
! 76: * W (output) COMPLEX*16 array, dimension (N)
! 77: * The reordered eigenvalues of T, in the same order as they
! 78: * appear on the diagonal of T.
! 79: *
! 80: * M (output) INTEGER
! 81: * The dimension of the specified invariant subspace.
! 82: * 0 <= M <= N.
! 83: *
! 84: * S (output) DOUBLE PRECISION
! 85: * If JOB = 'E' or 'B', S is a lower bound on the reciprocal
! 86: * condition number for the selected cluster of eigenvalues.
! 87: * S cannot underestimate the true reciprocal condition number
! 88: * by more than a factor of sqrt(N). If M = 0 or N, S = 1.
! 89: * If JOB = 'N' or 'V', S is not referenced.
! 90: *
! 91: * SEP (output) DOUBLE PRECISION
! 92: * If JOB = 'V' or 'B', SEP is the estimated reciprocal
! 93: * condition number of the specified invariant subspace. If
! 94: * M = 0 or N, SEP = norm(T).
! 95: * If JOB = 'N' or 'E', SEP is not referenced.
! 96: *
! 97: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
! 98: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 99: *
! 100: * LWORK (input) INTEGER
! 101: * The dimension of the array WORK.
! 102: * If JOB = 'N', LWORK >= 1;
! 103: * if JOB = 'E', LWORK = max(1,M*(N-M));
! 104: * if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
! 105: *
! 106: * If LWORK = -1, then a workspace query is assumed; the routine
! 107: * only calculates the optimal size of the WORK array, returns
! 108: * this value as the first entry of the WORK array, and no error
! 109: * message related to LWORK is issued by XERBLA.
! 110: *
! 111: * INFO (output) INTEGER
! 112: * = 0: successful exit
! 113: * < 0: if INFO = -i, the i-th argument had an illegal value
! 114: *
! 115: * Further Details
! 116: * ===============
! 117: *
! 118: * ZTRSEN first collects the selected eigenvalues by computing a unitary
! 119: * transformation Z to move them to the top left corner of T. In other
! 120: * words, the selected eigenvalues are the eigenvalues of T11 in:
! 121: *
! 122: * Z'*T*Z = ( T11 T12 ) n1
! 123: * ( 0 T22 ) n2
! 124: * n1 n2
! 125: *
! 126: * where N = n1+n2 and Z' means the conjugate transpose of Z. The first
! 127: * n1 columns of Z span the specified invariant subspace of T.
! 128: *
! 129: * If T has been obtained from the Schur factorization of a matrix
! 130: * A = Q*T*Q', then the reordered Schur factorization of A is given by
! 131: * A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the
! 132: * corresponding invariant subspace of A.
! 133: *
! 134: * The reciprocal condition number of the average of the eigenvalues of
! 135: * T11 may be returned in S. S lies between 0 (very badly conditioned)
! 136: * and 1 (very well conditioned). It is computed as follows. First we
! 137: * compute R so that
! 138: *
! 139: * P = ( I R ) n1
! 140: * ( 0 0 ) n2
! 141: * n1 n2
! 142: *
! 143: * is the projector on the invariant subspace associated with T11.
! 144: * R is the solution of the Sylvester equation:
! 145: *
! 146: * T11*R - R*T22 = T12.
! 147: *
! 148: * Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
! 149: * the two-norm of M. Then S is computed as the lower bound
! 150: *
! 151: * (1 + F-norm(R)**2)**(-1/2)
! 152: *
! 153: * on the reciprocal of 2-norm(P), the true reciprocal condition number.
! 154: * S cannot underestimate 1 / 2-norm(P) by more than a factor of
! 155: * sqrt(N).
! 156: *
! 157: * An approximate error bound for the computed average of the
! 158: * eigenvalues of T11 is
! 159: *
! 160: * EPS * norm(T) / S
! 161: *
! 162: * where EPS is the machine precision.
! 163: *
! 164: * The reciprocal condition number of the right invariant subspace
! 165: * spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
! 166: * SEP is defined as the separation of T11 and T22:
! 167: *
! 168: * sep( T11, T22 ) = sigma-min( C )
! 169: *
! 170: * where sigma-min(C) is the smallest singular value of the
! 171: * n1*n2-by-n1*n2 matrix
! 172: *
! 173: * C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
! 174: *
! 175: * I(m) is an m by m identity matrix, and kprod denotes the Kronecker
! 176: * product. We estimate sigma-min(C) by the reciprocal of an estimate of
! 177: * the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
! 178: * cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
! 179: *
! 180: * When SEP is small, small changes in T can cause large changes in
! 181: * the invariant subspace. An approximate bound on the maximum angular
! 182: * error in the computed right invariant subspace is
! 183: *
! 184: * EPS * norm(T) / SEP
! 185: *
! 186: * =====================================================================
! 187: *
! 188: * .. Parameters ..
! 189: DOUBLE PRECISION ZERO, ONE
! 190: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 191: * ..
! 192: * .. Local Scalars ..
! 193: LOGICAL LQUERY, WANTBH, WANTQ, WANTS, WANTSP
! 194: INTEGER IERR, K, KASE, KS, LWMIN, N1, N2, NN
! 195: DOUBLE PRECISION EST, RNORM, SCALE
! 196: * ..
! 197: * .. Local Arrays ..
! 198: INTEGER ISAVE( 3 )
! 199: DOUBLE PRECISION RWORK( 1 )
! 200: * ..
! 201: * .. External Functions ..
! 202: LOGICAL LSAME
! 203: DOUBLE PRECISION ZLANGE
! 204: EXTERNAL LSAME, ZLANGE
! 205: * ..
! 206: * .. External Subroutines ..
! 207: EXTERNAL XERBLA, ZLACN2, ZLACPY, ZTREXC, ZTRSYL
! 208: * ..
! 209: * .. Intrinsic Functions ..
! 210: INTRINSIC MAX, SQRT
! 211: * ..
! 212: * .. Executable Statements ..
! 213: *
! 214: * Decode and test the input parameters.
! 215: *
! 216: WANTBH = LSAME( JOB, 'B' )
! 217: WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
! 218: WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
! 219: WANTQ = LSAME( COMPQ, 'V' )
! 220: *
! 221: * Set M to the number of selected eigenvalues.
! 222: *
! 223: M = 0
! 224: DO 10 K = 1, N
! 225: IF( SELECT( K ) )
! 226: $ M = M + 1
! 227: 10 CONTINUE
! 228: *
! 229: N1 = M
! 230: N2 = N - M
! 231: NN = N1*N2
! 232: *
! 233: INFO = 0
! 234: LQUERY = ( LWORK.EQ.-1 )
! 235: *
! 236: IF( WANTSP ) THEN
! 237: LWMIN = MAX( 1, 2*NN )
! 238: ELSE IF( LSAME( JOB, 'N' ) ) THEN
! 239: LWMIN = 1
! 240: ELSE IF( LSAME( JOB, 'E' ) ) THEN
! 241: LWMIN = MAX( 1, NN )
! 242: END IF
! 243: *
! 244: IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
! 245: $ THEN
! 246: INFO = -1
! 247: ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
! 248: INFO = -2
! 249: ELSE IF( N.LT.0 ) THEN
! 250: INFO = -4
! 251: ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
! 252: INFO = -6
! 253: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
! 254: INFO = -8
! 255: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
! 256: INFO = -14
! 257: END IF
! 258: *
! 259: IF( INFO.EQ.0 ) THEN
! 260: WORK( 1 ) = LWMIN
! 261: END IF
! 262: *
! 263: IF( INFO.NE.0 ) THEN
! 264: CALL XERBLA( 'ZTRSEN', -INFO )
! 265: RETURN
! 266: ELSE IF( LQUERY ) THEN
! 267: RETURN
! 268: END IF
! 269: *
! 270: * Quick return if possible
! 271: *
! 272: IF( M.EQ.N .OR. M.EQ.0 ) THEN
! 273: IF( WANTS )
! 274: $ S = ONE
! 275: IF( WANTSP )
! 276: $ SEP = ZLANGE( '1', N, N, T, LDT, RWORK )
! 277: GO TO 40
! 278: END IF
! 279: *
! 280: * Collect the selected eigenvalues at the top left corner of T.
! 281: *
! 282: KS = 0
! 283: DO 20 K = 1, N
! 284: IF( SELECT( K ) ) THEN
! 285: KS = KS + 1
! 286: *
! 287: * Swap the K-th eigenvalue to position KS.
! 288: *
! 289: IF( K.NE.KS )
! 290: $ CALL ZTREXC( COMPQ, N, T, LDT, Q, LDQ, K, KS, IERR )
! 291: END IF
! 292: 20 CONTINUE
! 293: *
! 294: IF( WANTS ) THEN
! 295: *
! 296: * Solve the Sylvester equation for R:
! 297: *
! 298: * T11*R - R*T22 = scale*T12
! 299: *
! 300: CALL ZLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
! 301: CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
! 302: $ LDT, WORK, N1, SCALE, IERR )
! 303: *
! 304: * Estimate the reciprocal of the condition number of the cluster
! 305: * of eigenvalues.
! 306: *
! 307: RNORM = ZLANGE( 'F', N1, N2, WORK, N1, RWORK )
! 308: IF( RNORM.EQ.ZERO ) THEN
! 309: S = ONE
! 310: ELSE
! 311: S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
! 312: $ SQRT( RNORM ) )
! 313: END IF
! 314: END IF
! 315: *
! 316: IF( WANTSP ) THEN
! 317: *
! 318: * Estimate sep(T11,T22).
! 319: *
! 320: EST = ZERO
! 321: KASE = 0
! 322: 30 CONTINUE
! 323: CALL ZLACN2( NN, WORK( NN+1 ), WORK, EST, KASE, ISAVE )
! 324: IF( KASE.NE.0 ) THEN
! 325: IF( KASE.EQ.1 ) THEN
! 326: *
! 327: * Solve T11*R - R*T22 = scale*X.
! 328: *
! 329: CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
! 330: $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
! 331: $ IERR )
! 332: ELSE
! 333: *
! 334: * Solve T11'*R - R*T22' = scale*X.
! 335: *
! 336: CALL ZTRSYL( 'C', 'C', -1, N1, N2, T, LDT,
! 337: $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
! 338: $ IERR )
! 339: END IF
! 340: GO TO 30
! 341: END IF
! 342: *
! 343: SEP = SCALE / EST
! 344: END IF
! 345: *
! 346: 40 CONTINUE
! 347: *
! 348: * Copy reordered eigenvalues to W.
! 349: *
! 350: DO 50 K = 1, N
! 351: W( K ) = T( K, K )
! 352: 50 CONTINUE
! 353: *
! 354: WORK( 1 ) = LWMIN
! 355: *
! 356: RETURN
! 357: *
! 358: * End of ZTRSEN
! 359: *
! 360: END
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