Annotation of rpl/lapack/lapack/ztgsna.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
! 2: $ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
! 3: $ IWORK, INFO )
! 4: *
! 5: * -- LAPACK 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 HOWMNY, JOB
! 12: INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
! 13: * ..
! 14: * .. Array Arguments ..
! 15: LOGICAL SELECT( * )
! 16: INTEGER IWORK( * )
! 17: DOUBLE PRECISION DIF( * ), S( * )
! 18: COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
! 19: $ VR( LDVR, * ), WORK( * )
! 20: * ..
! 21: *
! 22: * Purpose
! 23: * =======
! 24: *
! 25: * ZTGSNA estimates reciprocal condition numbers for specified
! 26: * eigenvalues and/or eigenvectors of a matrix pair (A, B).
! 27: *
! 28: * (A, B) must be in generalized Schur canonical form, that is, A and
! 29: * B are both upper triangular.
! 30: *
! 31: * Arguments
! 32: * =========
! 33: *
! 34: * JOB (input) CHARACTER*1
! 35: * Specifies whether condition numbers are required for
! 36: * eigenvalues (S) or eigenvectors (DIF):
! 37: * = 'E': for eigenvalues only (S);
! 38: * = 'V': for eigenvectors only (DIF);
! 39: * = 'B': for both eigenvalues and eigenvectors (S and DIF).
! 40: *
! 41: * HOWMNY (input) CHARACTER*1
! 42: * = 'A': compute condition numbers for all eigenpairs;
! 43: * = 'S': compute condition numbers for selected eigenpairs
! 44: * specified by the array SELECT.
! 45: *
! 46: * SELECT (input) LOGICAL array, dimension (N)
! 47: * If HOWMNY = 'S', SELECT specifies the eigenpairs for which
! 48: * condition numbers are required. To select condition numbers
! 49: * for the corresponding j-th eigenvalue and/or eigenvector,
! 50: * SELECT(j) must be set to .TRUE..
! 51: * If HOWMNY = 'A', SELECT is not referenced.
! 52: *
! 53: * N (input) INTEGER
! 54: * The order of the square matrix pair (A, B). N >= 0.
! 55: *
! 56: * A (input) COMPLEX*16 array, dimension (LDA,N)
! 57: * The upper triangular matrix A in the pair (A,B).
! 58: *
! 59: * LDA (input) INTEGER
! 60: * The leading dimension of the array A. LDA >= max(1,N).
! 61: *
! 62: * B (input) COMPLEX*16 array, dimension (LDB,N)
! 63: * The upper triangular matrix B in the pair (A, B).
! 64: *
! 65: * LDB (input) INTEGER
! 66: * The leading dimension of the array B. LDB >= max(1,N).
! 67: *
! 68: * VL (input) COMPLEX*16 array, dimension (LDVL,M)
! 69: * IF JOB = 'E' or 'B', VL must contain left eigenvectors of
! 70: * (A, B), corresponding to the eigenpairs specified by HOWMNY
! 71: * and SELECT. The eigenvectors must be stored in consecutive
! 72: * columns of VL, as returned by ZTGEVC.
! 73: * If JOB = 'V', VL is not referenced.
! 74: *
! 75: * LDVL (input) INTEGER
! 76: * The leading dimension of the array VL. LDVL >= 1; and
! 77: * If JOB = 'E' or 'B', LDVL >= N.
! 78: *
! 79: * VR (input) COMPLEX*16 array, dimension (LDVR,M)
! 80: * IF JOB = 'E' or 'B', VR must contain right eigenvectors of
! 81: * (A, B), corresponding to the eigenpairs specified by HOWMNY
! 82: * and SELECT. The eigenvectors must be stored in consecutive
! 83: * columns of VR, as returned by ZTGEVC.
! 84: * If JOB = 'V', VR is not referenced.
! 85: *
! 86: * LDVR (input) INTEGER
! 87: * The leading dimension of the array VR. LDVR >= 1;
! 88: * If JOB = 'E' or 'B', LDVR >= N.
! 89: *
! 90: * S (output) DOUBLE PRECISION array, dimension (MM)
! 91: * If JOB = 'E' or 'B', the reciprocal condition numbers of the
! 92: * selected eigenvalues, stored in consecutive elements of the
! 93: * array.
! 94: * If JOB = 'V', S is not referenced.
! 95: *
! 96: * DIF (output) DOUBLE PRECISION array, dimension (MM)
! 97: * If JOB = 'V' or 'B', the estimated reciprocal condition
! 98: * numbers of the selected eigenvectors, stored in consecutive
! 99: * elements of the array.
! 100: * If the eigenvalues cannot be reordered to compute DIF(j),
! 101: * DIF(j) is set to 0; this can only occur when the true value
! 102: * would be very small anyway.
! 103: * For each eigenvalue/vector specified by SELECT, DIF stores
! 104: * a Frobenius norm-based estimate of Difl.
! 105: * If JOB = 'E', DIF is not referenced.
! 106: *
! 107: * MM (input) INTEGER
! 108: * The number of elements in the arrays S and DIF. MM >= M.
! 109: *
! 110: * M (output) INTEGER
! 111: * The number of elements of the arrays S and DIF used to store
! 112: * the specified condition numbers; for each selected eigenvalue
! 113: * one element is used. If HOWMNY = 'A', M is set to N.
! 114: *
! 115: * WORK (workspace/output) COMPLEX*16 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. LWORK >= max(1,N).
! 120: * If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
! 121: *
! 122: * IWORK (workspace) INTEGER array, dimension (N+2)
! 123: * If JOB = 'E', IWORK is not referenced.
! 124: *
! 125: * INFO (output) INTEGER
! 126: * = 0: Successful exit
! 127: * < 0: If INFO = -i, the i-th argument had an illegal value
! 128: *
! 129: * Further Details
! 130: * ===============
! 131: *
! 132: * The reciprocal of the condition number of the i-th generalized
! 133: * eigenvalue w = (a, b) is defined as
! 134: *
! 135: * S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v))
! 136: *
! 137: * where u and v are the right and left eigenvectors of (A, B)
! 138: * corresponding to w; |z| denotes the absolute value of the complex
! 139: * number, and norm(u) denotes the 2-norm of the vector u. The pair
! 140: * (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the
! 141: * matrix pair (A, B). If both a and b equal zero, then (A,B) is
! 142: * singular and S(I) = -1 is returned.
! 143: *
! 144: * An approximate error bound on the chordal distance between the i-th
! 145: * computed generalized eigenvalue w and the corresponding exact
! 146: * eigenvalue lambda is
! 147: *
! 148: * chord(w, lambda) <= EPS * norm(A, B) / S(I),
! 149: *
! 150: * where EPS is the machine precision.
! 151: *
! 152: * The reciprocal of the condition number of the right eigenvector u
! 153: * and left eigenvector v corresponding to the generalized eigenvalue w
! 154: * is defined as follows. Suppose
! 155: *
! 156: * (A, B) = ( a * ) ( b * ) 1
! 157: * ( 0 A22 ),( 0 B22 ) n-1
! 158: * 1 n-1 1 n-1
! 159: *
! 160: * Then the reciprocal condition number DIF(I) is
! 161: *
! 162: * Difl[(a, b), (A22, B22)] = sigma-min( Zl )
! 163: *
! 164: * where sigma-min(Zl) denotes the smallest singular value of
! 165: *
! 166: * Zl = [ kron(a, In-1) -kron(1, A22) ]
! 167: * [ kron(b, In-1) -kron(1, B22) ].
! 168: *
! 169: * Here In-1 is the identity matrix of size n-1 and X' is the conjugate
! 170: * transpose of X. kron(X, Y) is the Kronecker product between the
! 171: * matrices X and Y.
! 172: *
! 173: * We approximate the smallest singular value of Zl with an upper
! 174: * bound. This is done by ZLATDF.
! 175: *
! 176: * An approximate error bound for a computed eigenvector VL(i) or
! 177: * VR(i) is given by
! 178: *
! 179: * EPS * norm(A, B) / DIF(i).
! 180: *
! 181: * See ref. [2-3] for more details and further references.
! 182: *
! 183: * Based on contributions by
! 184: * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
! 185: * Umea University, S-901 87 Umea, Sweden.
! 186: *
! 187: * References
! 188: * ==========
! 189: *
! 190: * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
! 191: * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
! 192: * M.S. Moonen et al (eds), Linear Algebra for Large Scale and
! 193: * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
! 194: *
! 195: * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
! 196: * Eigenvalues of a Regular Matrix Pair (A, B) and Condition
! 197: * Estimation: Theory, Algorithms and Software, Report
! 198: * UMINF - 94.04, Department of Computing Science, Umea University,
! 199: * S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
! 200: * To appear in Numerical Algorithms, 1996.
! 201: *
! 202: * [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
! 203: * for Solving the Generalized Sylvester Equation and Estimating the
! 204: * Separation between Regular Matrix Pairs, Report UMINF - 93.23,
! 205: * Department of Computing Science, Umea University, S-901 87 Umea,
! 206: * Sweden, December 1993, Revised April 1994, Also as LAPACK Working
! 207: * Note 75.
! 208: * To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
! 209: *
! 210: * =====================================================================
! 211: *
! 212: * .. Parameters ..
! 213: DOUBLE PRECISION ZERO, ONE
! 214: INTEGER IDIFJB
! 215: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, IDIFJB = 3 )
! 216: * ..
! 217: * .. Local Scalars ..
! 218: LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS
! 219: INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2
! 220: DOUBLE PRECISION BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM
! 221: COMPLEX*16 YHAX, YHBX
! 222: * ..
! 223: * .. Local Arrays ..
! 224: COMPLEX*16 DUMMY( 1 ), DUMMY1( 1 )
! 225: * ..
! 226: * .. External Functions ..
! 227: LOGICAL LSAME
! 228: DOUBLE PRECISION DLAMCH, DLAPY2, DZNRM2
! 229: COMPLEX*16 ZDOTC
! 230: EXTERNAL LSAME, DLAMCH, DLAPY2, DZNRM2, ZDOTC
! 231: * ..
! 232: * .. External Subroutines ..
! 233: EXTERNAL DLABAD, XERBLA, ZGEMV, ZLACPY, ZTGEXC, ZTGSYL
! 234: * ..
! 235: * .. Intrinsic Functions ..
! 236: INTRINSIC ABS, DCMPLX, MAX
! 237: * ..
! 238: * .. Executable Statements ..
! 239: *
! 240: * Decode and test the input parameters
! 241: *
! 242: WANTBH = LSAME( JOB, 'B' )
! 243: WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
! 244: WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
! 245: *
! 246: SOMCON = LSAME( HOWMNY, 'S' )
! 247: *
! 248: INFO = 0
! 249: LQUERY = ( LWORK.EQ.-1 )
! 250: *
! 251: IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
! 252: INFO = -1
! 253: ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
! 254: INFO = -2
! 255: ELSE IF( N.LT.0 ) THEN
! 256: INFO = -4
! 257: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 258: INFO = -6
! 259: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 260: INFO = -8
! 261: ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
! 262: INFO = -10
! 263: ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
! 264: INFO = -12
! 265: ELSE
! 266: *
! 267: * Set M to the number of eigenpairs for which condition numbers
! 268: * are required, and test MM.
! 269: *
! 270: IF( SOMCON ) THEN
! 271: M = 0
! 272: DO 10 K = 1, N
! 273: IF( SELECT( K ) )
! 274: $ M = M + 1
! 275: 10 CONTINUE
! 276: ELSE
! 277: M = N
! 278: END IF
! 279: *
! 280: IF( N.EQ.0 ) THEN
! 281: LWMIN = 1
! 282: ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
! 283: LWMIN = 2*N*N
! 284: ELSE
! 285: LWMIN = N
! 286: END IF
! 287: WORK( 1 ) = LWMIN
! 288: *
! 289: IF( MM.LT.M ) THEN
! 290: INFO = -15
! 291: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
! 292: INFO = -18
! 293: END IF
! 294: END IF
! 295: *
! 296: IF( INFO.NE.0 ) THEN
! 297: CALL XERBLA( 'ZTGSNA', -INFO )
! 298: RETURN
! 299: ELSE IF( LQUERY ) THEN
! 300: RETURN
! 301: END IF
! 302: *
! 303: * Quick return if possible
! 304: *
! 305: IF( N.EQ.0 )
! 306: $ RETURN
! 307: *
! 308: * Get machine constants
! 309: *
! 310: EPS = DLAMCH( 'P' )
! 311: SMLNUM = DLAMCH( 'S' ) / EPS
! 312: BIGNUM = ONE / SMLNUM
! 313: CALL DLABAD( SMLNUM, BIGNUM )
! 314: KS = 0
! 315: DO 20 K = 1, N
! 316: *
! 317: * Determine whether condition numbers are required for the k-th
! 318: * eigenpair.
! 319: *
! 320: IF( SOMCON ) THEN
! 321: IF( .NOT.SELECT( K ) )
! 322: $ GO TO 20
! 323: END IF
! 324: *
! 325: KS = KS + 1
! 326: *
! 327: IF( WANTS ) THEN
! 328: *
! 329: * Compute the reciprocal condition number of the k-th
! 330: * eigenvalue.
! 331: *
! 332: RNRM = DZNRM2( N, VR( 1, KS ), 1 )
! 333: LNRM = DZNRM2( N, VL( 1, KS ), 1 )
! 334: CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), A, LDA,
! 335: $ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
! 336: YHAX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
! 337: CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), B, LDB,
! 338: $ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
! 339: YHBX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
! 340: COND = DLAPY2( ABS( YHAX ), ABS( YHBX ) )
! 341: IF( COND.EQ.ZERO ) THEN
! 342: S( KS ) = -ONE
! 343: ELSE
! 344: S( KS ) = COND / ( RNRM*LNRM )
! 345: END IF
! 346: END IF
! 347: *
! 348: IF( WANTDF ) THEN
! 349: IF( N.EQ.1 ) THEN
! 350: DIF( KS ) = DLAPY2( ABS( A( 1, 1 ) ), ABS( B( 1, 1 ) ) )
! 351: ELSE
! 352: *
! 353: * Estimate the reciprocal condition number of the k-th
! 354: * eigenvectors.
! 355: *
! 356: * Copy the matrix (A, B) to the array WORK and move the
! 357: * (k,k)th pair to the (1,1) position.
! 358: *
! 359: CALL ZLACPY( 'Full', N, N, A, LDA, WORK, N )
! 360: CALL ZLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
! 361: IFST = K
! 362: ILST = 1
! 363: *
! 364: CALL ZTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ),
! 365: $ N, DUMMY, 1, DUMMY1, 1, IFST, ILST, IERR )
! 366: *
! 367: IF( IERR.GT.0 ) THEN
! 368: *
! 369: * Ill-conditioned problem - swap rejected.
! 370: *
! 371: DIF( KS ) = ZERO
! 372: ELSE
! 373: *
! 374: * Reordering successful, solve generalized Sylvester
! 375: * equation for R and L,
! 376: * A22 * R - L * A11 = A12
! 377: * B22 * R - L * B11 = B12,
! 378: * and compute estimate of Difl[(A11,B11), (A22, B22)].
! 379: *
! 380: N1 = 1
! 381: N2 = N - N1
! 382: I = N*N + 1
! 383: CALL ZTGSYL( 'N', IDIFJB, N2, N1, WORK( N*N1+N1+1 ),
! 384: $ N, WORK, N, WORK( N1+1 ), N,
! 385: $ WORK( N*N1+N1+I ), N, WORK( I ), N,
! 386: $ WORK( N1+I ), N, SCALE, DIF( KS ), DUMMY,
! 387: $ 1, IWORK, IERR )
! 388: END IF
! 389: END IF
! 390: END IF
! 391: *
! 392: 20 CONTINUE
! 393: WORK( 1 ) = LWMIN
! 394: RETURN
! 395: *
! 396: * End of ZTGSNA
! 397: *
! 398: END
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