.TH "SRC/dtgsna.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/dtgsna.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBdtgsna\fP (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)" .br .RI "\fBDTGSNA\fP " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine dtgsna (character job, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) dif, integer mm, integer m, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)" .PP \fBDTGSNA\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> 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**T, Q*B*Z**T) with orthogonal matrices Q and Z, where !> Z**T 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\&. !> !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIJOB\fP .PP .nf !> JOB is 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)\&. !> .fi .PP .br \fIHOWMNY\fP .PP .nf !> HOWMNY is CHARACTER*1 !> = 'A': compute condition numbers for all eigenpairs; !> = 'S': compute condition numbers for selected eigenpairs !> specified by the array SELECT\&. !> .fi .PP .br \fISELECT\fP .PP .nf !> SELECT is 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\&. !> .fi .PP .br \fIN\fP .PP .nf !> N is INTEGER !> The order of the square matrix pair (A, B)\&. N >= 0\&. !> .fi .PP .br \fIA\fP .PP .nf !> A is DOUBLE PRECISION array, dimension (LDA,N) !> The upper quasi-triangular matrix A in the pair (A,B)\&. !> .fi .PP .br \fILDA\fP .PP .nf !> LDA is INTEGER !> The leading dimension of the array A\&. LDA >= max(1,N)\&. !> .fi .PP .br \fIB\fP .PP .nf !> B is DOUBLE PRECISION array, dimension (LDB,N) !> The upper triangular matrix B in the pair (A,B)\&. !> .fi .PP .br \fILDB\fP .PP .nf !> LDB is INTEGER !> The leading dimension of the array B\&. LDB >= max(1,N)\&. !> .fi .PP .br \fIVL\fP .PP .nf !> VL is 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\&. !> .fi .PP .br \fILDVL\fP .PP .nf !> LDVL is INTEGER !> The leading dimension of the array VL\&. LDVL >= 1\&. !> If JOB = 'E' or 'B', LDVL >= N\&. !> .fi .PP .br \fIVR\fP .PP .nf !> VR is 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\&. !> .fi .PP .br \fILDVR\fP .PP .nf !> LDVR is INTEGER !> The leading dimension of the array VR\&. LDVR >= 1\&. !> If JOB = 'E' or 'B', LDVR >= N\&. !> .fi .PP .br \fIS\fP .PP .nf !> S is 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\&. !> .fi .PP .br \fIDIF\fP .PP .nf !> DIF is 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\&. !> .fi .PP .br \fIMM\fP .PP .nf !> MM is INTEGER !> The number of elements in the arrays S and DIF\&. MM >= M\&. !> .fi .PP .br \fIM\fP .PP .nf !> M is 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\&. !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&. !> .fi .PP .br \fILWORK\fP .PP .nf !> LWORK is 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\&. !> .fi .PP .br \fIIWORK\fP .PP .nf !> IWORK is INTEGER array, dimension (N + 6) !> If JOB = 'E', IWORK is not referenced\&. !> .fi .PP .br \fIINFO\fP .PP .nf !> INFO is INTEGER !> =0: Successful exit !> <0: If INFO = -i, the i-th argument had an illegal value !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf !> !> The reciprocal of the condition number of a generalized eigenvalue !> w = (a, b) is defined as !> !> S(w) = (|u**TAv|**2 + |u**TBv|**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**TAv/u**TBv) !> 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**T*(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**T*(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**T*S11*V1 = ( s11 s12 ) and U1**T*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**T * Z1 - lambda I), !> where Z1**T denotes the 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**T, In-2) -kron(I2, S22) ] !> [ kron(T11**T, 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\&. !> .fi .PP .RE .PP \fBContributors:\fP .RS 4 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden\&. .RE .PP \fBReferences:\fP .RS 4 .PP .nf !> !> [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\&. !> .fi .PP .RE .PP .PP Definition at line \fB378\fP of file \fBdtgsna\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.