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version 1.7, 2010/12/21 13:53:40
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version 1.8, 2011/07/22 07:38:12
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| $ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, |
$ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, |
| $ IWORK, INFO ) |
$ IWORK, INFO ) |
| * |
* |
| * -- LAPACK routine (version 3.2) -- |
* -- LAPACK routine (version 3.3.1) -- |
| * -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
| * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
| * November 2006 |
* -- April 2011 -- |
| * |
* |
| * .. Scalar Arguments .. |
* .. Scalar Arguments .. |
| CHARACTER HOWMNY, JOB |
CHARACTER HOWMNY, JOB |
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| * DTGSNA estimates reciprocal condition numbers for specified |
* DTGSNA estimates reciprocal condition numbers for specified |
| * eigenvalues and/or eigenvectors of a matrix pair (A, B) in |
* eigenvalues and/or eigenvectors of a matrix pair (A, B) in |
| * generalized real Schur canonical form (or of any matrix pair |
* generalized real Schur canonical form (or of any matrix pair |
| * (Q*A*Z', Q*B*Z') with orthogonal matrices Q and Z, where |
* (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where |
| * Z' denotes the transpose of Z. |
* Z**T denotes the transpose of Z. |
| * |
* |
| * (A, B) must be in generalized real Schur form (as returned by DGGES), |
* (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 |
* i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal |
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| * The reciprocal of the condition number of a generalized eigenvalue |
* The reciprocal of the condition number of a generalized eigenvalue |
| * w = (a, b) is defined as |
* w = (a, b) is defined as |
| * |
* |
| * S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v)) |
* 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) |
* where u and v are the left and right eigenvectors of (A, B) |
| * corresponding to w; |z| denotes the absolute value of the complex |
* corresponding to w; |z| denotes the absolute value of the complex |
| * number, and norm(u) denotes the 2-norm of the vector u. |
* 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) |
* 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 |
* of the matrix pair (A, B). If both a and b equal zero, then (A B) is |
| * singular and S(I) = -1 is returned. |
* singular and S(I) = -1 is returned. |
| * |
* |
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| * |
* |
| * Suppose U and V are orthogonal transformations such that |
* Suppose U and V are orthogonal transformations such that |
| * |
* |
| * U'*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 |
* U**T*(A, B)*V = (S, T) = ( a * ) ( b * ) 1 |
| * ( 0 S22 ),( 0 T22 ) n-1 |
* ( 0 S22 ),( 0 T22 ) n-1 |
| * 1 n-1 1 n-1 |
* 1 n-1 1 n-1 |
| * |
* |
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| * |
* |
| * Suppose U and V are orthogonal transformations such that |
* Suppose U and V are orthogonal transformations such that |
| * |
* |
| * U'*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 |
* U**T*(A, B)*V = (S, T) = ( S11 * ) ( T11 * ) 2 |
| * ( 0 S22 ),( 0 T22) n-2 |
* ( 0 S22 ),( 0 T22) n-2 |
| * 2 n-2 2 n-2 |
* 2 n-2 2 n-2 |
| * |
* |
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| * pair (w, conjg(w)). There exist unitary matrices U1 and V1 such |
* pair (w, conjg(w)). There exist unitary matrices U1 and V1 such |
| * that |
* that |
| * |
* |
| * U1'*S11*V1 = ( s11 s12 ) and U1'*T11*V1 = ( t11 t12 ) |
* U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 ) |
| * ( 0 s22 ) ( 0 t22 ) |
* ( 0 s22 ) ( 0 t22 ) |
| * |
* |
| * where the generalized eigenvalues w = s11/t11 and |
* where the generalized eigenvalues w = s11/t11 and |
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| * [ t11 -t22 ], |
* [ t11 -t22 ], |
| * |
* |
| * This is done by computing (using real arithmetic) the |
* This is done by computing (using real arithmetic) the |
| * roots of the characteristical polynomial det(Z1' * Z1 - lambda I), |
* roots of the characteristical polynomial det(Z1**T * Z1 - lambda I), |
| * where Z1' denotes the conjugate transpose of Z1 and det(X) denotes |
* where Z1**T denotes the transpose of Z1 and det(X) denotes |
| * the determinant of X. |
* the determinant of X. |
| * |
* |
| * and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an |
* 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) |
* upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2) |
| * |
* |
| * Z2 = [ kron(S11', In-2) -kron(I2, S22) ] |
* Z2 = [ kron(S11**T, In-2) -kron(I2, S22) ] |
| * [ kron(T11', In-2) -kron(I2, T22) ] |
* [ kron(T11**T, In-2) -kron(I2, T22) ] |
| * |
* |
| * Note that if the default method for computing DIF is wanted (see |
* Note that if the default method for computing DIF is wanted (see |
| * DLATDF), then the parameter DIFDRI (see below) should be changed |
* DLATDF), then the parameter DIFDRI (see below) should be changed |
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