Annotation of rpl/lapack/lapack/dtgsna.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DTGSNA( 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 A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
! 18: $ VL( LDVL, * ), VR( LDVR, * ), WORK( * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * DTGSNA estimates reciprocal condition numbers for specified
! 25: * eigenvalues and/or eigenvectors of a matrix pair (A, B) in
! 26: * generalized real Schur canonical form (or of any matrix pair
! 27: * (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where
! 28: * Z' denotes the transpose of Z.
! 29: *
! 30: * (A, B) must be in generalized real Schur form (as returned by DGGES),
! 31: * i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
! 32: * blocks. B is upper triangular.
! 33: *
! 34: *
! 35: * Arguments
! 36: * =========
! 37: *
! 38: * JOB (input) CHARACTER*1
! 39: * Specifies whether condition numbers are required for
! 40: * eigenvalues (S) or eigenvectors (DIF):
! 41: * = 'E': for eigenvalues only (S);
! 42: * = 'V': for eigenvectors only (DIF);
! 43: * = 'B': for both eigenvalues and eigenvectors (S and DIF).
! 44: *
! 45: * HOWMNY (input) CHARACTER*1
! 46: * = 'A': compute condition numbers for all eigenpairs;
! 47: * = 'S': compute condition numbers for selected eigenpairs
! 48: * specified by the array SELECT.
! 49: *
! 50: * SELECT (input) LOGICAL array, dimension (N)
! 51: * If HOWMNY = 'S', SELECT specifies the eigenpairs for which
! 52: * condition numbers are required. To select condition numbers
! 53: * for the eigenpair corresponding to a real eigenvalue w(j),
! 54: * SELECT(j) must be set to .TRUE.. To select condition numbers
! 55: * corresponding to a complex conjugate pair of eigenvalues w(j)
! 56: * and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
! 57: * set to .TRUE..
! 58: * If HOWMNY = 'A', SELECT is not referenced.
! 59: *
! 60: * N (input) INTEGER
! 61: * The order of the square matrix pair (A, B). N >= 0.
! 62: *
! 63: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
! 64: * The upper quasi-triangular matrix A in the pair (A,B).
! 65: *
! 66: * LDA (input) INTEGER
! 67: * The leading dimension of the array A. LDA >= max(1,N).
! 68: *
! 69: * B (input) DOUBLE PRECISION array, dimension (LDB,N)
! 70: * The upper triangular matrix B in the pair (A,B).
! 71: *
! 72: * LDB (input) INTEGER
! 73: * The leading dimension of the array B. LDB >= max(1,N).
! 74: *
! 75: * VL (input) DOUBLE PRECISION array, dimension (LDVL,M)
! 76: * If JOB = 'E' or 'B', VL must contain left eigenvectors of
! 77: * (A, B), corresponding to the eigenpairs specified by HOWMNY
! 78: * and SELECT. The eigenvectors must be stored in consecutive
! 79: * columns of VL, as returned by DTGEVC.
! 80: * If JOB = 'V', VL is not referenced.
! 81: *
! 82: * LDVL (input) INTEGER
! 83: * The leading dimension of the array VL. LDVL >= 1.
! 84: * If JOB = 'E' or 'B', LDVL >= N.
! 85: *
! 86: * VR (input) DOUBLE PRECISION array, dimension (LDVR,M)
! 87: * If JOB = 'E' or 'B', VR must contain right eigenvectors of
! 88: * (A, B), corresponding to the eigenpairs specified by HOWMNY
! 89: * and SELECT. The eigenvectors must be stored in consecutive
! 90: * columns ov VR, as returned by DTGEVC.
! 91: * If JOB = 'V', VR is not referenced.
! 92: *
! 93: * LDVR (input) INTEGER
! 94: * The leading dimension of the array VR. LDVR >= 1.
! 95: * If JOB = 'E' or 'B', LDVR >= N.
! 96: *
! 97: * S (output) DOUBLE PRECISION array, dimension (MM)
! 98: * If JOB = 'E' or 'B', the reciprocal condition numbers of the
! 99: * selected eigenvalues, stored in consecutive elements of the
! 100: * array. For a complex conjugate pair of eigenvalues two
! 101: * consecutive elements of S are set to the same value. Thus
! 102: * S(j), DIF(j), and the j-th columns of VL and VR all
! 103: * correspond to the same eigenpair (but not in general the
! 104: * j-th eigenpair, unless all eigenpairs are selected).
! 105: * If JOB = 'V', S is not referenced.
! 106: *
! 107: * DIF (output) DOUBLE PRECISION array, dimension (MM)
! 108: * If JOB = 'V' or 'B', the estimated reciprocal condition
! 109: * numbers of the selected eigenvectors, stored in consecutive
! 110: * elements of the array. For a complex eigenvector two
! 111: * consecutive elements of DIF are set to the same value. If
! 112: * the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
! 113: * is set to 0; this can only occur when the true value would be
! 114: * very small anyway.
! 115: * If JOB = 'E', DIF is not referenced.
! 116: *
! 117: * MM (input) INTEGER
! 118: * The number of elements in the arrays S and DIF. MM >= M.
! 119: *
! 120: * M (output) INTEGER
! 121: * The number of elements of the arrays S and DIF used to store
! 122: * the specified condition numbers; for each selected real
! 123: * eigenvalue one element is used, and for each selected complex
! 124: * conjugate pair of eigenvalues, two elements are used.
! 125: * If HOWMNY = 'A', M is set to N.
! 126: *
! 127: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 128: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 129: *
! 130: * LWORK (input) INTEGER
! 131: * The dimension of the array WORK. LWORK >= max(1,N).
! 132: * If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16.
! 133: *
! 134: * If LWORK = -1, then a workspace query is assumed; the routine
! 135: * only calculates the optimal size of the WORK array, returns
! 136: * this value as the first entry of the WORK array, and no error
! 137: * message related to LWORK is issued by XERBLA.
! 138: *
! 139: * IWORK (workspace) INTEGER array, dimension (N + 6)
! 140: * If JOB = 'E', IWORK is not referenced.
! 141: *
! 142: * INFO (output) INTEGER
! 143: * =0: Successful exit
! 144: * <0: If INFO = -i, the i-th argument had an illegal value
! 145: *
! 146: *
! 147: * Further Details
! 148: * ===============
! 149: *
! 150: * The reciprocal of the condition number of a generalized eigenvalue
! 151: * w = (a, b) is defined as
! 152: *
! 153: * S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v))
! 154: *
! 155: * where u and v are the left and right eigenvectors of (A, B)
! 156: * corresponding to w; |z| denotes the absolute value of the complex
! 157: * number, and norm(u) denotes the 2-norm of the vector u.
! 158: * The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv)
! 159: * of the matrix pair (A, B). If both a and b equal zero, then (A B) is
! 160: * singular and S(I) = -1 is returned.
! 161: *
! 162: * An approximate error bound on the chordal distance between the i-th
! 163: * computed generalized eigenvalue w and the corresponding exact
! 164: * eigenvalue lambda is
! 165: *
! 166: * chord(w, lambda) <= EPS * norm(A, B) / S(I)
! 167: *
! 168: * where EPS is the machine precision.
! 169: *
! 170: * The reciprocal of the condition number DIF(i) of right eigenvector u
! 171: * and left eigenvector v corresponding to the generalized eigenvalue w
! 172: * is defined as follows:
! 173: *
! 174: * a) If the i-th eigenvalue w = (a,b) is real
! 175: *
! 176: * Suppose U and V are orthogonal transformations such that
! 177: *
! 178: * U'*(A, B)*V = (S, T) = ( a * ) ( b * ) 1
! 179: * ( 0 S22 ),( 0 T22 ) n-1
! 180: * 1 n-1 1 n-1
! 181: *
! 182: * Then the reciprocal condition number DIF(i) is
! 183: *
! 184: * Difl((a, b), (S22, T22)) = sigma-min( Zl ),
! 185: *
! 186: * where sigma-min(Zl) denotes the smallest singular value of the
! 187: * 2(n-1)-by-2(n-1) matrix
! 188: *
! 189: * Zl = [ kron(a, In-1) -kron(1, S22) ]
! 190: * [ kron(b, In-1) -kron(1, T22) ] .
! 191: *
! 192: * Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
! 193: * Kronecker product between the matrices X and Y.
! 194: *
! 195: * Note that if the default method for computing DIF(i) is wanted
! 196: * (see DLATDF), then the parameter DIFDRI (see below) should be
! 197: * changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
! 198: * See DTGSYL for more details.
! 199: *
! 200: * b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
! 201: *
! 202: * Suppose U and V are orthogonal transformations such that
! 203: *
! 204: * U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2
! 205: * ( 0 S22 ),( 0 T22) n-2
! 206: * 2 n-2 2 n-2
! 207: *
! 208: * and (S11, T11) corresponds to the complex conjugate eigenvalue
! 209: * pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
! 210: * that
! 211: *
! 212: * U1'*S11*V1 = ( s11 s12 ) and U1'*T11*V1 = ( t11 t12 )
! 213: * ( 0 s22 ) ( 0 t22 )
! 214: *
! 215: * where the generalized eigenvalues w = s11/t11 and
! 216: * conjg(w) = s22/t22.
! 217: *
! 218: * Then the reciprocal condition number DIF(i) is bounded by
! 219: *
! 220: * min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
! 221: *
! 222: * where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
! 223: * Z1 is the complex 2-by-2 matrix
! 224: *
! 225: * Z1 = [ s11 -s22 ]
! 226: * [ t11 -t22 ],
! 227: *
! 228: * This is done by computing (using real arithmetic) the
! 229: * roots of the characteristical polynomial det(Z1' * Z1 - lambda I),
! 230: * where Z1' denotes the conjugate transpose of Z1 and det(X) denotes
! 231: * the determinant of X.
! 232: *
! 233: * and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
! 234: * upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
! 235: *
! 236: * Z2 = [ kron(S11', In-2) -kron(I2, S22) ]
! 237: * [ kron(T11', In-2) -kron(I2, T22) ]
! 238: *
! 239: * Note that if the default method for computing DIF is wanted (see
! 240: * DLATDF), then the parameter DIFDRI (see below) should be changed
! 241: * from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
! 242: * for more details.
! 243: *
! 244: * For each eigenvalue/vector specified by SELECT, DIF stores a
! 245: * Frobenius norm-based estimate of Difl.
! 246: *
! 247: * An approximate error bound for the i-th computed eigenvector VL(i) or
! 248: * VR(i) is given by
! 249: *
! 250: * EPS * norm(A, B) / DIF(i).
! 251: *
! 252: * See ref. [2-3] for more details and further references.
! 253: *
! 254: * Based on contributions by
! 255: * Bo Kagstrom and Peter Poromaa, Department of Computing Science,
! 256: * Umea University, S-901 87 Umea, Sweden.
! 257: *
! 258: * References
! 259: * ==========
! 260: *
! 261: * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
! 262: * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
! 263: * M.S. Moonen et al (eds), Linear Algebra for Large Scale and
! 264: * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
! 265: *
! 266: * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
! 267: * Eigenvalues of a Regular Matrix Pair (A, B) and Condition
! 268: * Estimation: Theory, Algorithms and Software,
! 269: * Report UMINF - 94.04, Department of Computing Science, Umea
! 270: * University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
! 271: * Note 87. To appear in Numerical Algorithms, 1996.
! 272: *
! 273: * [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
! 274: * for Solving the Generalized Sylvester Equation and Estimating the
! 275: * Separation between Regular Matrix Pairs, Report UMINF - 93.23,
! 276: * Department of Computing Science, Umea University, S-901 87 Umea,
! 277: * Sweden, December 1993, Revised April 1994, Also as LAPACK Working
! 278: * Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
! 279: * No 1, 1996.
! 280: *
! 281: * =====================================================================
! 282: *
! 283: * .. Parameters ..
! 284: INTEGER DIFDRI
! 285: PARAMETER ( DIFDRI = 3 )
! 286: DOUBLE PRECISION ZERO, ONE, TWO, FOUR
! 287: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0,
! 288: $ FOUR = 4.0D+0 )
! 289: * ..
! 290: * .. Local Scalars ..
! 291: LOGICAL LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS
! 292: INTEGER I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2
! 293: DOUBLE PRECISION ALPHAI, ALPHAR, ALPRQT, BETA, C1, C2, COND,
! 294: $ EPS, LNRM, RNRM, ROOT1, ROOT2, SCALE, SMLNUM,
! 295: $ TMPII, TMPIR, TMPRI, TMPRR, UHAV, UHAVI, UHBV,
! 296: $ UHBVI
! 297: * ..
! 298: * .. Local Arrays ..
! 299: DOUBLE PRECISION DUMMY( 1 ), DUMMY1( 1 )
! 300: * ..
! 301: * .. External Functions ..
! 302: LOGICAL LSAME
! 303: DOUBLE PRECISION DDOT, DLAMCH, DLAPY2, DNRM2
! 304: EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2
! 305: * ..
! 306: * .. External Subroutines ..
! 307: EXTERNAL DGEMV, DLACPY, DLAG2, DTGEXC, DTGSYL, XERBLA
! 308: * ..
! 309: * .. Intrinsic Functions ..
! 310: INTRINSIC MAX, MIN, SQRT
! 311: * ..
! 312: * .. Executable Statements ..
! 313: *
! 314: * Decode and test the input parameters
! 315: *
! 316: WANTBH = LSAME( JOB, 'B' )
! 317: WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
! 318: WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
! 319: *
! 320: SOMCON = LSAME( HOWMNY, 'S' )
! 321: *
! 322: INFO = 0
! 323: LQUERY = ( LWORK.EQ.-1 )
! 324: *
! 325: IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
! 326: INFO = -1
! 327: ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
! 328: INFO = -2
! 329: ELSE IF( N.LT.0 ) THEN
! 330: INFO = -4
! 331: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 332: INFO = -6
! 333: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
! 334: INFO = -8
! 335: ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
! 336: INFO = -10
! 337: ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
! 338: INFO = -12
! 339: ELSE
! 340: *
! 341: * Set M to the number of eigenpairs for which condition numbers
! 342: * are required, and test MM.
! 343: *
! 344: IF( SOMCON ) THEN
! 345: M = 0
! 346: PAIR = .FALSE.
! 347: DO 10 K = 1, N
! 348: IF( PAIR ) THEN
! 349: PAIR = .FALSE.
! 350: ELSE
! 351: IF( K.LT.N ) THEN
! 352: IF( A( K+1, K ).EQ.ZERO ) THEN
! 353: IF( SELECT( K ) )
! 354: $ M = M + 1
! 355: ELSE
! 356: PAIR = .TRUE.
! 357: IF( SELECT( K ) .OR. SELECT( K+1 ) )
! 358: $ M = M + 2
! 359: END IF
! 360: ELSE
! 361: IF( SELECT( N ) )
! 362: $ M = M + 1
! 363: END IF
! 364: END IF
! 365: 10 CONTINUE
! 366: ELSE
! 367: M = N
! 368: END IF
! 369: *
! 370: IF( N.EQ.0 ) THEN
! 371: LWMIN = 1
! 372: ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
! 373: LWMIN = 2*N*( N + 2 ) + 16
! 374: ELSE
! 375: LWMIN = N
! 376: END IF
! 377: WORK( 1 ) = LWMIN
! 378: *
! 379: IF( MM.LT.M ) THEN
! 380: INFO = -15
! 381: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
! 382: INFO = -18
! 383: END IF
! 384: END IF
! 385: *
! 386: IF( INFO.NE.0 ) THEN
! 387: CALL XERBLA( 'DTGSNA', -INFO )
! 388: RETURN
! 389: ELSE IF( LQUERY ) THEN
! 390: RETURN
! 391: END IF
! 392: *
! 393: * Quick return if possible
! 394: *
! 395: IF( N.EQ.0 )
! 396: $ RETURN
! 397: *
! 398: * Get machine constants
! 399: *
! 400: EPS = DLAMCH( 'P' )
! 401: SMLNUM = DLAMCH( 'S' ) / EPS
! 402: KS = 0
! 403: PAIR = .FALSE.
! 404: *
! 405: DO 20 K = 1, N
! 406: *
! 407: * Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block.
! 408: *
! 409: IF( PAIR ) THEN
! 410: PAIR = .FALSE.
! 411: GO TO 20
! 412: ELSE
! 413: IF( K.LT.N )
! 414: $ PAIR = A( K+1, K ).NE.ZERO
! 415: END IF
! 416: *
! 417: * Determine whether condition numbers are required for the k-th
! 418: * eigenpair.
! 419: *
! 420: IF( SOMCON ) THEN
! 421: IF( PAIR ) THEN
! 422: IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) )
! 423: $ GO TO 20
! 424: ELSE
! 425: IF( .NOT.SELECT( K ) )
! 426: $ GO TO 20
! 427: END IF
! 428: END IF
! 429: *
! 430: KS = KS + 1
! 431: *
! 432: IF( WANTS ) THEN
! 433: *
! 434: * Compute the reciprocal condition number of the k-th
! 435: * eigenvalue.
! 436: *
! 437: IF( PAIR ) THEN
! 438: *
! 439: * Complex eigenvalue pair.
! 440: *
! 441: RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ),
! 442: $ DNRM2( N, VR( 1, KS+1 ), 1 ) )
! 443: LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ),
! 444: $ DNRM2( N, VL( 1, KS+1 ), 1 ) )
! 445: CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO,
! 446: $ WORK, 1 )
! 447: TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
! 448: TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
! 449: CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS+1 ), 1,
! 450: $ ZERO, WORK, 1 )
! 451: TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
! 452: TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
! 453: UHAV = TMPRR + TMPII
! 454: UHAVI = TMPIR - TMPRI
! 455: CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO,
! 456: $ WORK, 1 )
! 457: TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
! 458: TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
! 459: CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS+1 ), 1,
! 460: $ ZERO, WORK, 1 )
! 461: TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 )
! 462: TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
! 463: UHBV = TMPRR + TMPII
! 464: UHBVI = TMPIR - TMPRI
! 465: UHAV = DLAPY2( UHAV, UHAVI )
! 466: UHBV = DLAPY2( UHBV, UHBVI )
! 467: COND = DLAPY2( UHAV, UHBV )
! 468: S( KS ) = COND / ( RNRM*LNRM )
! 469: S( KS+1 ) = S( KS )
! 470: *
! 471: ELSE
! 472: *
! 473: * Real eigenvalue.
! 474: *
! 475: RNRM = DNRM2( N, VR( 1, KS ), 1 )
! 476: LNRM = DNRM2( N, VL( 1, KS ), 1 )
! 477: CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO,
! 478: $ WORK, 1 )
! 479: UHAV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
! 480: CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO,
! 481: $ WORK, 1 )
! 482: UHBV = DDOT( N, WORK, 1, VL( 1, KS ), 1 )
! 483: COND = DLAPY2( UHAV, UHBV )
! 484: IF( COND.EQ.ZERO ) THEN
! 485: S( KS ) = -ONE
! 486: ELSE
! 487: S( KS ) = COND / ( RNRM*LNRM )
! 488: END IF
! 489: END IF
! 490: END IF
! 491: *
! 492: IF( WANTDF ) THEN
! 493: IF( N.EQ.1 ) THEN
! 494: DIF( KS ) = DLAPY2( A( 1, 1 ), B( 1, 1 ) )
! 495: GO TO 20
! 496: END IF
! 497: *
! 498: * Estimate the reciprocal condition number of the k-th
! 499: * eigenvectors.
! 500: IF( PAIR ) THEN
! 501: *
! 502: * Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)).
! 503: * Compute the eigenvalue(s) at position K.
! 504: *
! 505: WORK( 1 ) = A( K, K )
! 506: WORK( 2 ) = A( K+1, K )
! 507: WORK( 3 ) = A( K, K+1 )
! 508: WORK( 4 ) = A( K+1, K+1 )
! 509: WORK( 5 ) = B( K, K )
! 510: WORK( 6 ) = B( K+1, K )
! 511: WORK( 7 ) = B( K, K+1 )
! 512: WORK( 8 ) = B( K+1, K+1 )
! 513: CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA,
! 514: $ DUMMY1( 1 ), ALPHAR, DUMMY( 1 ), ALPHAI )
! 515: ALPRQT = ONE
! 516: C1 = TWO*( ALPHAR*ALPHAR+ALPHAI*ALPHAI+BETA*BETA )
! 517: C2 = FOUR*BETA*BETA*ALPHAI*ALPHAI
! 518: ROOT1 = C1 + SQRT( C1*C1-4.0D0*C2 )
! 519: ROOT2 = C2 / ROOT1
! 520: ROOT1 = ROOT1 / TWO
! 521: COND = MIN( SQRT( ROOT1 ), SQRT( ROOT2 ) )
! 522: END IF
! 523: *
! 524: * Copy the matrix (A, B) to the array WORK and swap the
! 525: * diagonal block beginning at A(k,k) to the (1,1) position.
! 526: *
! 527: CALL DLACPY( 'Full', N, N, A, LDA, WORK, N )
! 528: CALL DLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
! 529: IFST = K
! 530: ILST = 1
! 531: *
! 532: CALL DTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ), N,
! 533: $ DUMMY, 1, DUMMY1, 1, IFST, ILST,
! 534: $ WORK( N*N*2+1 ), LWORK-2*N*N, IERR )
! 535: *
! 536: IF( IERR.GT.0 ) THEN
! 537: *
! 538: * Ill-conditioned problem - swap rejected.
! 539: *
! 540: DIF( KS ) = ZERO
! 541: ELSE
! 542: *
! 543: * Reordering successful, solve generalized Sylvester
! 544: * equation for R and L,
! 545: * A22 * R - L * A11 = A12
! 546: * B22 * R - L * B11 = B12,
! 547: * and compute estimate of Difl((A11,B11), (A22, B22)).
! 548: *
! 549: N1 = 1
! 550: IF( WORK( 2 ).NE.ZERO )
! 551: $ N1 = 2
! 552: N2 = N - N1
! 553: IF( N2.EQ.0 ) THEN
! 554: DIF( KS ) = COND
! 555: ELSE
! 556: I = N*N + 1
! 557: IZ = 2*N*N + 1
! 558: CALL DTGSYL( 'N', DIFDRI, N2, N1, WORK( N*N1+N1+1 ),
! 559: $ N, WORK, N, WORK( N1+1 ), N,
! 560: $ WORK( N*N1+N1+I ), N, WORK( I ), N,
! 561: $ WORK( N1+I ), N, SCALE, DIF( KS ),
! 562: $ WORK( IZ+1 ), LWORK-2*N*N, IWORK, IERR )
! 563: *
! 564: IF( PAIR )
! 565: $ DIF( KS ) = MIN( MAX( ONE, ALPRQT )*DIF( KS ),
! 566: $ COND )
! 567: END IF
! 568: END IF
! 569: IF( PAIR )
! 570: $ DIF( KS+1 ) = DIF( KS )
! 571: END IF
! 572: IF( PAIR )
! 573: $ KS = KS + 1
! 574: *
! 575: 20 CONTINUE
! 576: WORK( 1 ) = LWMIN
! 577: RETURN
! 578: *
! 579: * End of DTGSNA
! 580: *
! 581: END
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