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author | julie <julielangou@users.noreply.github.com> | 2011-10-06 06:53:11 +0000 |
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committer | julie <julielangou@users.noreply.github.com> | 2011-10-06 06:53:11 +0000 |
commit | e1d39294aee16fa6db9ba079b14442358217db71 (patch) | |
tree | 30e5aa04c1f6596991fda5334f63dfb9b8027849 /SRC/cgeevx.f | |
parent | 5fe0466a14e395641f4f8a300ecc9dcb8058081b (diff) | |
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Integrating Doxygen in comments
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diff --git a/SRC/cgeevx.f b/SRC/cgeevx.f index d7ee57d3..d7e1a782 100644 --- a/SRC/cgeevx.f +++ b/SRC/cgeevx.f @@ -1,11 +1,289 @@ +*> \brief <b> CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b> +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +* Definition +* ========== +* +* SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, +* LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, +* RCONDV, WORK, LWORK, RWORK, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER BALANC, JOBVL, JOBVR, SENSE +* INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N +* REAL ABNRM +* .. +* .. Array Arguments .. +* REAL RCONDE( * ), RCONDV( * ), RWORK( * ), +* $ SCALE( * ) +* COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), +* $ W( * ), WORK( * ) +* .. +* +* Purpose +* ======= +* +*>\details \b Purpose: +*>\verbatim +*> +*> CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the +*> eigenvalues and, optionally, the left and/or right eigenvectors. +*> +*> Optionally also, it computes a balancing transformation to improve +*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI, +*> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues +*> (RCONDE), and reciprocal condition numbers for the right +*> eigenvectors (RCONDV). +*> +*> The right eigenvector v(j) of A satisfies +*> A * v(j) = lambda(j) * v(j) +*> where lambda(j) is its eigenvalue. +*> The left eigenvector u(j) of A satisfies +*> u(j)**H * A = lambda(j) * u(j)**H +*> where u(j)**H denotes the conjugate transpose of u(j). +*> +*> The computed eigenvectors are normalized to have Euclidean norm +*> equal to 1 and largest component real. +*> +*> Balancing a matrix means permuting the rows and columns to make it +*> more nearly upper triangular, and applying a diagonal similarity +*> transformation D * A * D**(-1), where D is a diagonal matrix, to +*> make its rows and columns closer in norm and the condition numbers +*> of its eigenvalues and eigenvectors smaller. The computed +*> reciprocal condition numbers correspond to the balanced matrix. +*> Permuting rows and columns will not change the condition numbers +*> (in exact arithmetic) but diagonal scaling will. For further +*> explanation of balancing, see section 4.10.2 of the LAPACK +*> Users' Guide. +*> +*>\endverbatim +* +* Arguments +* ========= +* +*> \param[in] BALANC +*> \verbatim +*> BALANC is CHARACTER*1 +*> Indicates how the input matrix should be diagonally scaled +*> and/or permuted to improve the conditioning of its +*> eigenvalues. +*> = 'N': Do not diagonally scale or permute; +*> = 'P': Perform permutations to make the matrix more nearly +*> upper triangular. Do not diagonally scale; +*> = 'S': Diagonally scale the matrix, ie. replace A by +*> D*A*D**(-1), where D is a diagonal matrix chosen +*> to make the rows and columns of A more equal in +*> norm. Do not permute; +*> = 'B': Both diagonally scale and permute A. +*> \endverbatim +*> \verbatim +*> Computed reciprocal condition numbers will be for the matrix +*> after balancing and/or permuting. Permuting does not change +*> condition numbers (in exact arithmetic), but balancing does. +*> \endverbatim +*> +*> \param[in] JOBVL +*> \verbatim +*> JOBVL is CHARACTER*1 +*> = 'N': left eigenvectors of A are not computed; +*> = 'V': left eigenvectors of A are computed. +*> If SENSE = 'E' or 'B', JOBVL must = 'V'. +*> \endverbatim +*> +*> \param[in] JOBVR +*> \verbatim +*> JOBVR is CHARACTER*1 +*> = 'N': right eigenvectors of A are not computed; +*> = 'V': right eigenvectors of A are computed. +*> If SENSE = 'E' or 'B', JOBVR must = 'V'. +*> \endverbatim +*> +*> \param[in] SENSE +*> \verbatim +*> SENSE is CHARACTER*1 +*> Determines which reciprocal condition numbers are computed. +*> = 'N': None are computed; +*> = 'E': Computed for eigenvalues only; +*> = 'V': Computed for right eigenvectors only; +*> = 'B': Computed for eigenvalues and right eigenvectors. +*> \endverbatim +*> \verbatim +*> If SENSE = 'E' or 'B', both left and right eigenvectors +*> must also be computed (JOBVL = 'V' and JOBVR = 'V'). +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The order of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in,out] A +*> \verbatim +*> A is COMPLEX array, dimension (LDA,N) +*> On entry, the N-by-N matrix A. +*> On exit, A has been overwritten. If JOBVL = 'V' or +*> JOBVR = 'V', A contains the Schur form of the balanced +*> version of the matrix A. +*> \endverbatim +*> +*> \param[in] LDA +*> \verbatim +*> LDA is INTEGER +*> The leading dimension of the array A. LDA >= max(1,N). +*> \endverbatim +*> +*> \param[out] W +*> \verbatim +*> W is COMPLEX array, dimension (N) +*> W contains the computed eigenvalues. +*> \endverbatim +*> +*> \param[out] VL +*> \verbatim +*> VL is COMPLEX array, dimension (LDVL,N) +*> If JOBVL = 'V', the left eigenvectors u(j) are stored one +*> after another in the columns of VL, in the same order +*> as their eigenvalues. +*> If JOBVL = 'N', VL is not referenced. +*> u(j) = VL(:,j), the j-th column of VL. +*> \endverbatim +*> +*> \param[in] LDVL +*> \verbatim +*> LDVL is INTEGER +*> The leading dimension of the array VL. LDVL >= 1; if +*> JOBVL = 'V', LDVL >= N. +*> \endverbatim +*> +*> \param[out] VR +*> \verbatim +*> VR is COMPLEX array, dimension (LDVR,N) +*> If JOBVR = 'V', the right eigenvectors v(j) are stored one +*> after another in the columns of VR, in the same order +*> as their eigenvalues. +*> If JOBVR = 'N', VR is not referenced. +*> v(j) = VR(:,j), the j-th column of VR. +*> \endverbatim +*> +*> \param[in] LDVR +*> \verbatim +*> LDVR is INTEGER +*> The leading dimension of the array VR. LDVR >= 1; if +*> JOBVR = 'V', LDVR >= N. +*> \endverbatim +*> +*> \param[out] ILO +*> \verbatim +*> ILO is INTEGER +*> \endverbatim +*> +*> \param[out] IHI +*> \verbatim +*> IHI is INTEGER +*> ILO and IHI are integer values determined when A was +*> balanced. The balanced A(i,j) = 0 if I > J and +*> J = 1,...,ILO-1 or I = IHI+1,...,N. +*> \endverbatim +*> +*> \param[out] SCALE +*> \verbatim +*> SCALE is REAL array, dimension (N) +*> Details of the permutations and scaling factors applied +*> when balancing A. If P(j) is the index of the row and column +*> interchanged with row and column j, and D(j) is the scaling +*> factor applied to row and column j, then +*> SCALE(J) = P(J), for J = 1,...,ILO-1 +*> = D(J), for J = ILO,...,IHI +*> = P(J) for J = IHI+1,...,N. +*> The order in which the interchanges are made is N to IHI+1, +*> then 1 to ILO-1. +*> \endverbatim +*> +*> \param[out] ABNRM +*> \verbatim +*> ABNRM is REAL +*> The one-norm of the balanced matrix (the maximum +*> of the sum of absolute values of elements of any column). +*> \endverbatim +*> +*> \param[out] RCONDE +*> \verbatim +*> RCONDE is REAL array, dimension (N) +*> RCONDE(j) is the reciprocal condition number of the j-th +*> eigenvalue. +*> \endverbatim +*> +*> \param[out] RCONDV +*> \verbatim +*> RCONDV is REAL array, dimension (N) +*> RCONDV(j) is the reciprocal condition number of the j-th +*> right eigenvector. +*> \endverbatim +*> +*> \param[out] WORK +*> \verbatim +*> WORK is COMPLEX array, dimension (MAX(1,LWORK)) +*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +*> \endverbatim +*> +*> \param[in] LWORK +*> \verbatim +*> LWORK is INTEGER +*> The dimension of the array WORK. If SENSE = 'N' or 'E', +*> LWORK >= max(1,2*N), and if SENSE = 'V' or 'B', +*> LWORK >= N*N+2*N. +*> For good performance, LWORK must generally be larger. +*> \endverbatim +*> \verbatim +*> 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. +*> \endverbatim +*> +*> \param[out] RWORK +*> \verbatim +*> RWORK is REAL array, dimension (2*N) +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit +*> < 0: if INFO = -i, the i-th argument had an illegal value. +*> > 0: if INFO = i, the QR algorithm failed to compute all the +*> eigenvalues, and no eigenvectors or condition numbers +*> have been computed; elements 1:ILO-1 and i+1:N of W +*> contain eigenvalues which have converged. +*> \endverbatim +*> +* +* Authors +* ======= +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \date November 2011 +* +*> \ingroup complexGEeigen +* +* ===================================================================== SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, $ LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, $ RCONDV, WORK, LWORK, RWORK, INFO ) * -* -- LAPACK driver routine (version 3.2) -- +* -- LAPACK eigen routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* November 2006 +* November 2011 * * .. Scalar Arguments .. CHARACTER BALANC, JOBVL, JOBVR, SENSE @@ -19,170 +297,6 @@ $ W( * ), WORK( * ) * .. * -* Purpose -* ======= -* -* CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the -* eigenvalues and, optionally, the left and/or right eigenvectors. -* -* Optionally also, it computes a balancing transformation to improve -* the conditioning of the eigenvalues and eigenvectors (ILO, IHI, -* SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues -* (RCONDE), and reciprocal condition numbers for the right -* eigenvectors (RCONDV). -* -* The right eigenvector v(j) of A satisfies -* A * v(j) = lambda(j) * v(j) -* where lambda(j) is its eigenvalue. -* The left eigenvector u(j) of A satisfies -* u(j)**H * A = lambda(j) * u(j)**H -* where u(j)**H denotes the conjugate transpose of u(j). -* -* The computed eigenvectors are normalized to have Euclidean norm -* equal to 1 and largest component real. -* -* Balancing a matrix means permuting the rows and columns to make it -* more nearly upper triangular, and applying a diagonal similarity -* transformation D * A * D**(-1), where D is a diagonal matrix, to -* make its rows and columns closer in norm and the condition numbers -* of its eigenvalues and eigenvectors smaller. The computed -* reciprocal condition numbers correspond to the balanced matrix. -* Permuting rows and columns will not change the condition numbers -* (in exact arithmetic) but diagonal scaling will. For further -* explanation of balancing, see section 4.10.2 of the LAPACK -* Users' Guide. -* -* Arguments -* ========= -* -* BALANC (input) CHARACTER*1 -* Indicates how the input matrix should be diagonally scaled -* and/or permuted to improve the conditioning of its -* eigenvalues. -* = 'N': Do not diagonally scale or permute; -* = 'P': Perform permutations to make the matrix more nearly -* upper triangular. Do not diagonally scale; -* = 'S': Diagonally scale the matrix, ie. replace A by -* D*A*D**(-1), where D is a diagonal matrix chosen -* to make the rows and columns of A more equal in -* norm. Do not permute; -* = 'B': Both diagonally scale and permute A. -* -* Computed reciprocal condition numbers will be for the matrix -* after balancing and/or permuting. Permuting does not change -* condition numbers (in exact arithmetic), but balancing does. -* -* JOBVL (input) CHARACTER*1 -* = 'N': left eigenvectors of A are not computed; -* = 'V': left eigenvectors of A are computed. -* If SENSE = 'E' or 'B', JOBVL must = 'V'. -* -* JOBVR (input) CHARACTER*1 -* = 'N': right eigenvectors of A are not computed; -* = 'V': right eigenvectors of A are computed. -* If SENSE = 'E' or 'B', JOBVR must = 'V'. -* -* SENSE (input) CHARACTER*1 -* Determines which reciprocal condition numbers are computed. -* = 'N': None are computed; -* = 'E': Computed for eigenvalues only; -* = 'V': Computed for right eigenvectors only; -* = 'B': Computed for eigenvalues and right eigenvectors. -* -* If SENSE = 'E' or 'B', both left and right eigenvectors -* must also be computed (JOBVL = 'V' and JOBVR = 'V'). -* -* N (input) INTEGER -* The order of the matrix A. N >= 0. -* -* A (input/output) COMPLEX array, dimension (LDA,N) -* On entry, the N-by-N matrix A. -* On exit, A has been overwritten. If JOBVL = 'V' or -* JOBVR = 'V', A contains the Schur form of the balanced -* version of the matrix A. -* -* LDA (input) INTEGER -* The leading dimension of the array A. LDA >= max(1,N). -* -* W (output) COMPLEX array, dimension (N) -* W contains the computed eigenvalues. -* -* VL (output) COMPLEX array, dimension (LDVL,N) -* If JOBVL = 'V', the left eigenvectors u(j) are stored one -* after another in the columns of VL, in the same order -* as their eigenvalues. -* If JOBVL = 'N', VL is not referenced. -* u(j) = VL(:,j), the j-th column of VL. -* -* LDVL (input) INTEGER -* The leading dimension of the array VL. LDVL >= 1; if -* JOBVL = 'V', LDVL >= N. -* -* VR (output) COMPLEX array, dimension (LDVR,N) -* If JOBVR = 'V', the right eigenvectors v(j) are stored one -* after another in the columns of VR, in the same order -* as their eigenvalues. -* If JOBVR = 'N', VR is not referenced. -* v(j) = VR(:,j), the j-th column of VR. -* -* LDVR (input) INTEGER -* The leading dimension of the array VR. LDVR >= 1; if -* JOBVR = 'V', LDVR >= N. -* -* ILO (output) INTEGER -* -* IHI (output) INTEGER -* ILO and IHI are integer values determined when A was -* balanced. The balanced A(i,j) = 0 if I > J and -* J = 1,...,ILO-1 or I = IHI+1,...,N. -* -* SCALE (output) REAL array, dimension (N) -* Details of the permutations and scaling factors applied -* when balancing A. If P(j) is the index of the row and column -* interchanged with row and column j, and D(j) is the scaling -* factor applied to row and column j, then -* SCALE(J) = P(J), for J = 1,...,ILO-1 -* = D(J), for J = ILO,...,IHI -* = P(J) for J = IHI+1,...,N. -* The order in which the interchanges are made is N to IHI+1, -* then 1 to ILO-1. -* -* ABNRM (output) REAL -* The one-norm of the balanced matrix (the maximum -* of the sum of absolute values of elements of any column). -* -* RCONDE (output) REAL array, dimension (N) -* RCONDE(j) is the reciprocal condition number of the j-th -* eigenvalue. -* -* RCONDV (output) REAL array, dimension (N) -* RCONDV(j) is the reciprocal condition number of the j-th -* right eigenvector. -* -* WORK (workspace/output) COMPLEX 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. If SENSE = 'N' or 'E', -* LWORK >= max(1,2*N), and if SENSE = 'V' or 'B', -* LWORK >= N*N+2*N. -* For good performance, LWORK must generally be larger. -* -* 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. -* -* RWORK (workspace) REAL array, dimension (2*N) -* -* INFO (output) INTEGER -* = 0: successful exit -* < 0: if INFO = -i, the i-th argument had an illegal value. -* > 0: if INFO = i, the QR algorithm failed to compute all the -* eigenvalues, and no eigenvectors or condition numbers -* have been computed; elements 1:ILO-1 and i+1:N of W -* contain eigenvalues which have converged. -* * ===================================================================== * * .. Parameters .. |