SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, \$ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, \$ IWORK, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER HOWMNY, JOB INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N * .. * .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DIF( * ), S( * ), \$ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) * .. * * Purpose * ======= * * DTGSNA estimates reciprocal condition numbers for specified * eigenvalues and/or eigenvectors of a matrix pair (A, B) in * generalized real Schur canonical form (or of any matrix pair * (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where * Z' denotes the transpose of Z. * * (A, B) must be in generalized real Schur form (as returned by DGGES), * i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal * blocks. B is upper triangular. * * * Arguments * ========= * * JOB (input) CHARACTER*1 * Specifies whether condition numbers are required for * eigenvalues (S) or eigenvectors (DIF): * = 'E': for eigenvalues only (S); * = 'V': for eigenvectors only (DIF); * = 'B': for both eigenvalues and eigenvectors (S and DIF). * * HOWMNY (input) CHARACTER*1 * = 'A': compute condition numbers for all eigenpairs; * = 'S': compute condition numbers for selected eigenpairs * specified by the array SELECT. * * SELECT (input) LOGICAL array, dimension (N) * If HOWMNY = 'S', SELECT specifies the eigenpairs for which * condition numbers are required. To select condition numbers * for the eigenpair corresponding to a real eigenvalue w(j), * SELECT(j) must be set to .TRUE.. To select condition numbers * corresponding to a complex conjugate pair of eigenvalues w(j) * and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be * set to .TRUE.. * If HOWMNY = 'A', SELECT is not referenced. * * N (input) INTEGER * The order of the square matrix pair (A, B). N >= 0. * * A (input) DOUBLE PRECISION array, dimension (LDA,N) * The upper quasi-triangular matrix A in the pair (A,B). * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (input) DOUBLE PRECISION array, dimension (LDB,N) * The upper triangular matrix B in the pair (A,B). * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * VL (input) DOUBLE PRECISION array, dimension (LDVL,M) * If JOB = 'E' or 'B', VL must contain left eigenvectors of * (A, B), corresponding to the eigenpairs specified by HOWMNY * and SELECT. The eigenvectors must be stored in consecutive * columns of VL, as returned by DTGEVC. * If JOB = 'V', VL is not referenced. * * LDVL (input) INTEGER * The leading dimension of the array VL. LDVL >= 1. * If JOB = 'E' or 'B', LDVL >= N. * * VR (input) DOUBLE PRECISION array, dimension (LDVR,M) * If JOB = 'E' or 'B', VR must contain right eigenvectors of * (A, B), corresponding to the eigenpairs specified by HOWMNY * and SELECT. The eigenvectors must be stored in consecutive * columns ov VR, as returned by DTGEVC. * If JOB = 'V', VR is not referenced. * * LDVR (input) INTEGER * The leading dimension of the array VR. LDVR >= 1. * If JOB = 'E' or 'B', LDVR >= N. * * S (output) DOUBLE PRECISION array, dimension (MM) * If JOB = 'E' or 'B', the reciprocal condition numbers of the * selected eigenvalues, stored in consecutive elements of the * array. For a complex conjugate pair of eigenvalues two * consecutive elements of S are set to the same value. Thus * S(j), DIF(j), and the j-th columns of VL and VR all * correspond to the same eigenpair (but not in general the * j-th eigenpair, unless all eigenpairs are selected). * If JOB = 'V', S is not referenced. * * DIF (output) DOUBLE PRECISION array, dimension (MM) * If JOB = 'V' or 'B', the estimated reciprocal condition * numbers of the selected eigenvectors, stored in consecutive * elements of the array. For a complex eigenvector two * consecutive elements of DIF are set to the same value. If * the eigenvalues cannot be reordered to compute DIF(j), DIF(j) * is set to 0; this can only occur when the true value would be * very small anyway. * If JOB = 'E', DIF is not referenced. * * MM (input) INTEGER * The number of elements in the arrays S and DIF. MM >= M. * * M (output) INTEGER * The number of elements of the arrays S and DIF used to store * the specified condition numbers; for each selected real * eigenvalue one element is used, and for each selected complex * conjugate pair of eigenvalues, two elements are used. * If HOWMNY = 'A', M is set to N. * * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= max(1,N). * If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+16. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal size of the WORK array, returns * this value as the first entry of the WORK array, and no error * message related to LWORK is issued by XERBLA. * * IWORK (workspace) INTEGER array, dimension (N + 6) * If JOB = 'E', IWORK is not referenced. * * INFO (output) INTEGER * =0: Successful exit * <0: If INFO = -i, the i-th argument had an illegal value * * * Further Details * =============== * * The reciprocal of the condition number of a generalized eigenvalue * w = (a, b) is defined as * * S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v)) * * where u and v are the left and right eigenvectors of (A, B) * corresponding to w; |z| denotes the absolute value of the complex * number, and norm(u) denotes the 2-norm of the vector u. * The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv) * of the matrix pair (A, B). If both a and b equal zero, then (A B) is * singular and S(I) = -1 is returned. * * An approximate error bound on the chordal distance between the i-th * computed generalized eigenvalue w and the corresponding exact * eigenvalue lambda is * * chord(w, lambda) <= EPS * norm(A, B) / S(I) * * where EPS is the machine precision. * * The reciprocal of the condition number DIF(i) of right eigenvector u * and left eigenvector v corresponding to the generalized eigenvalue w * is defined as follows: * * a) If the i-th eigenvalue w = (a,b) is real * * Suppose U and V are orthogonal transformations such that * * U'*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 * ( 0 S22 ),( 0 T22 ) n-1 * 1 n-1 1 n-1 * * Then the reciprocal condition number DIF(i) is * * Difl((a, b), (S22, T22)) = sigma-min( Zl ), * * where sigma-min(Zl) denotes the smallest singular value of the * 2(n-1)-by-2(n-1) matrix * * Zl = [ kron(a, In-1) -kron(1, S22) ] * [ kron(b, In-1) -kron(1, T22) ] . * * Here In-1 is the identity matrix of size n-1. kron(X, Y) is the * Kronecker product between the matrices X and Y. * * Note that if the default method for computing DIF(i) is wanted * (see DLATDF), then the parameter DIFDRI (see below) should be * changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). * See DTGSYL for more details. * * b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair, * * Suppose U and V are orthogonal transformations such that * * U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 * ( 0 S22 ),( 0 T22) n-2 * 2 n-2 2 n-2 * * and (S11, T11) corresponds to the complex conjugate eigenvalue * pair (w, conjg(w)). There exist unitary matrices U1 and V1 such * that * * U1'*S11*V1 = ( s11 s12 ) and U1'*T11*V1 = ( t11 t12 ) * ( 0 s22 ) ( 0 t22 ) * * where the generalized eigenvalues w = s11/t11 and * conjg(w) = s22/t22. * * Then the reciprocal condition number DIF(i) is bounded by * * min( d1, max( 1, |real(s11)/real(s22)| )*d2 ) * * where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where * Z1 is the complex 2-by-2 matrix * * Z1 = [ s11 -s22 ] * [ t11 -t22 ], * * This is done by computing (using real arithmetic) the * roots of the characteristical polynomial det(Z1' * Z1 - lambda I), * where Z1' denotes the conjugate transpose of Z1 and det(X) denotes * the determinant of X. * * and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an * upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) * * Z2 = [ kron(S11', In-2) -kron(I2, S22) ] * [ kron(T11', In-2) -kron(I2, T22) ] * * Note that if the default method for computing DIF is wanted (see * DLATDF), then the parameter DIFDRI (see below) should be changed * from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL * for more details. * * For each eigenvalue/vector specified by SELECT, DIF stores a * Frobenius norm-based estimate of Difl. * * An approximate error bound for the i-th computed eigenvector VL(i) or * VR(i) is given by * * EPS * norm(A, B) / DIF(i). * * See ref. [2-3] for more details and further references. * * Based on contributions by * Bo Kagstrom and Peter Poromaa, Department of Computing Science, * Umea University, S-901 87 Umea, Sweden. * * References * ========== * * [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the * Generalized Real Schur Form of a Regular Matrix Pair (A, B), in * M.S. Moonen et al (eds), Linear Algebra for Large Scale and * Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. * * [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified * Eigenvalues of a Regular Matrix Pair (A, B) and Condition * Estimation: Theory, Algorithms and Software, * Report UMINF - 94.04, Department of Computing Science, Umea * University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working * Note 87. To appear in Numerical Algorithms, 1996. * * [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software * for Solving the Generalized Sylvester Equation and Estimating the * Separation between Regular Matrix Pairs, Report UMINF - 93.23, * Department of Computing Science, Umea University, S-901 87 Umea, * Sweden, December 1993, Revised April 1994, Also as LAPACK Working * Note 75. To appear in ACM Trans. on Math. Software, Vol 22, * No 1, 1996. * * ===================================================================== * * .. Parameters .. INTEGER DIFDRI PARAMETER ( DIFDRI = 3 ) DOUBLE PRECISION ZERO, ONE, TWO, FOUR PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0, \$ FOUR = 4.0D+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY, PAIR, SOMCON, WANTBH, WANTDF, WANTS INTEGER I, IERR, IFST, ILST, IZ, K, KS, LWMIN, N1, N2 DOUBLE PRECISION ALPHAI, ALPHAR, ALPRQT, BETA, C1, C2, COND, \$ EPS, LNRM, RNRM, ROOT1, ROOT2, SCALE, SMLNUM, \$ TMPII, TMPIR, TMPRI, TMPRR, UHAV, UHAVI, UHBV, \$ UHBVI * .. * .. Local Arrays .. DOUBLE PRECISION DUMMY( 1 ), DUMMY1( 1 ) * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT, DLAMCH, DLAPY2, DNRM2 EXTERNAL LSAME, DDOT, DLAMCH, DLAPY2, DNRM2 * .. * .. External Subroutines .. EXTERNAL DGEMV, DLACPY, DLAG2, DTGEXC, DTGSYL, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT * .. * .. Executable Statements .. * * Decode and test the input parameters * WANTBH = LSAME( JOB, 'B' ) WANTS = LSAME( JOB, 'E' ) .OR. WANTBH WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH * SOMCON = LSAME( HOWMNY, 'S' ) * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) * IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN INFO = -1 ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( WANTS .AND. LDVL.LT.N ) THEN INFO = -10 ELSE IF( WANTS .AND. LDVR.LT.N ) THEN INFO = -12 ELSE * * Set M to the number of eigenpairs for which condition numbers * are required, and test MM. * IF( SOMCON ) THEN M = 0 PAIR = .FALSE. DO 10 K = 1, N IF( PAIR ) THEN PAIR = .FALSE. ELSE IF( K.LT.N ) THEN IF( A( K+1, K ).EQ.ZERO ) THEN IF( SELECT( K ) ) \$ M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( K ) .OR. SELECT( K+1 ) ) \$ M = M + 2 END IF ELSE IF( SELECT( N ) ) \$ M = M + 1 END IF END IF 10 CONTINUE ELSE M = N END IF * IF( N.EQ.0 ) THEN LWMIN = 1 ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN LWMIN = 2*N*( N + 2 ) + 16 ELSE LWMIN = N END IF WORK( 1 ) = LWMIN * IF( MM.LT.M ) THEN INFO = -15 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -18 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGSNA', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * Get machine constants * EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) / EPS KS = 0 PAIR = .FALSE. * DO 20 K = 1, N * * Determine whether A(k,k) begins a 1-by-1 or 2-by-2 block. * IF( PAIR ) THEN PAIR = .FALSE. GO TO 20 ELSE IF( K.LT.N ) \$ PAIR = A( K+1, K ).NE.ZERO END IF * * Determine whether condition numbers are required for the k-th * eigenpair. * IF( SOMCON ) THEN IF( PAIR ) THEN IF( .NOT.SELECT( K ) .AND. .NOT.SELECT( K+1 ) ) \$ GO TO 20 ELSE IF( .NOT.SELECT( K ) ) \$ GO TO 20 END IF END IF * KS = KS + 1 * IF( WANTS ) THEN * * Compute the reciprocal condition number of the k-th * eigenvalue. * IF( PAIR ) THEN * * Complex eigenvalue pair. * RNRM = DLAPY2( DNRM2( N, VR( 1, KS ), 1 ), \$ DNRM2( N, VR( 1, KS+1 ), 1 ) ) LNRM = DLAPY2( DNRM2( N, VL( 1, KS ), 1 ), \$ DNRM2( N, VL( 1, KS+1 ), 1 ) ) CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO, \$ WORK, 1 ) TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 ) CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS+1 ), 1, \$ ZERO, WORK, 1 ) TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 ) TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) UHAV = TMPRR + TMPII UHAVI = TMPIR - TMPRI CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO, \$ WORK, 1 ) TMPRR = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) TMPRI = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 ) CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS+1 ), 1, \$ ZERO, WORK, 1 ) TMPII = DDOT( N, WORK, 1, VL( 1, KS+1 ), 1 ) TMPIR = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) UHBV = TMPRR + TMPII UHBVI = TMPIR - TMPRI UHAV = DLAPY2( UHAV, UHAVI ) UHBV = DLAPY2( UHBV, UHBVI ) COND = DLAPY2( UHAV, UHBV ) S( KS ) = COND / ( RNRM*LNRM ) S( KS+1 ) = S( KS ) * ELSE * * Real eigenvalue. * RNRM = DNRM2( N, VR( 1, KS ), 1 ) LNRM = DNRM2( N, VL( 1, KS ), 1 ) CALL DGEMV( 'N', N, N, ONE, A, LDA, VR( 1, KS ), 1, ZERO, \$ WORK, 1 ) UHAV = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) CALL DGEMV( 'N', N, N, ONE, B, LDB, VR( 1, KS ), 1, ZERO, \$ WORK, 1 ) UHBV = DDOT( N, WORK, 1, VL( 1, KS ), 1 ) COND = DLAPY2( UHAV, UHBV ) IF( COND.EQ.ZERO ) THEN S( KS ) = -ONE ELSE S( KS ) = COND / ( RNRM*LNRM ) END IF END IF END IF * IF( WANTDF ) THEN IF( N.EQ.1 ) THEN DIF( KS ) = DLAPY2( A( 1, 1 ), B( 1, 1 ) ) GO TO 20 END IF * * Estimate the reciprocal condition number of the k-th * eigenvectors. IF( PAIR ) THEN * * Copy the 2-by 2 pencil beginning at (A(k,k), B(k, k)). * Compute the eigenvalue(s) at position K. * WORK( 1 ) = A( K, K ) WORK( 2 ) = A( K+1, K ) WORK( 3 ) = A( K, K+1 ) WORK( 4 ) = A( K+1, K+1 ) WORK( 5 ) = B( K, K ) WORK( 6 ) = B( K+1, K ) WORK( 7 ) = B( K, K+1 ) WORK( 8 ) = B( K+1, K+1 ) CALL DLAG2( WORK, 2, WORK( 5 ), 2, SMLNUM*EPS, BETA, \$ DUMMY1( 1 ), ALPHAR, DUMMY( 1 ), ALPHAI ) ALPRQT = ONE C1 = TWO*( ALPHAR*ALPHAR+ALPHAI*ALPHAI+BETA*BETA ) C2 = FOUR*BETA*BETA*ALPHAI*ALPHAI ROOT1 = C1 + SQRT( C1*C1-4.0D0*C2 ) ROOT2 = C2 / ROOT1 ROOT1 = ROOT1 / TWO COND = MIN( SQRT( ROOT1 ), SQRT( ROOT2 ) ) END IF * * Copy the matrix (A, B) to the array WORK and swap the * diagonal block beginning at A(k,k) to the (1,1) position. * CALL DLACPY( 'Full', N, N, A, LDA, WORK, N ) CALL DLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N ) IFST = K ILST = 1 * CALL DTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ), N, \$ DUMMY, 1, DUMMY1, 1, IFST, ILST, \$ WORK( N*N*2+1 ), LWORK-2*N*N, IERR ) * IF( IERR.GT.0 ) THEN * * Ill-conditioned problem - swap rejected. * DIF( KS ) = ZERO ELSE * * Reordering successful, solve generalized Sylvester * equation for R and L, * A22 * R - L * A11 = A12 * B22 * R - L * B11 = B12, * and compute estimate of Difl((A11,B11), (A22, B22)). * N1 = 1 IF( WORK( 2 ).NE.ZERO ) \$ N1 = 2 N2 = N - N1 IF( N2.EQ.0 ) THEN DIF( KS ) = COND ELSE I = N*N + 1 IZ = 2*N*N + 1 CALL DTGSYL( 'N', DIFDRI, N2, N1, WORK( N*N1+N1+1 ), \$ N, WORK, N, WORK( N1+1 ), N, \$ WORK( N*N1+N1+I ), N, WORK( I ), N, \$ WORK( N1+I ), N, SCALE, DIF( KS ), \$ WORK( IZ+1 ), LWORK-2*N*N, IWORK, IERR ) * IF( PAIR ) \$ DIF( KS ) = MIN( MAX( ONE, ALPRQT )*DIF( KS ), \$ COND ) END IF END IF IF( PAIR ) \$ DIF( KS+1 ) = DIF( KS ) END IF IF( PAIR ) \$ KS = KS + 1 * 20 CONTINUE WORK( 1 ) = LWMIN RETURN * * End of DTGSNA * END