version 1.8, 2011/07/22 07:38:05
|
version 1.9, 2011/11/21 20:42:52
|
Line 1
|
Line 1
|
|
*> \brief <b> DGGSVD computes the singular value decomposition (SVD) for OTHER matrices</b> |
|
* |
|
* =========== DOCUMENTATION =========== |
|
* |
|
* Online html documentation available at |
|
* http://www.netlib.org/lapack/explore-html/ |
|
* |
|
*> \htmlonly |
|
*> Download DGGSVD + dependencies |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggsvd.f"> |
|
*> [TGZ]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggsvd.f"> |
|
*> [ZIP]</a> |
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvd.f"> |
|
*> [TXT]</a> |
|
*> \endhtmlonly |
|
* |
|
* Definition: |
|
* =========== |
|
* |
|
* SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, |
|
* LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, |
|
* IWORK, INFO ) |
|
* |
|
* .. Scalar Arguments .. |
|
* CHARACTER JOBQ, JOBU, JOBV |
|
* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P |
|
* .. |
|
* .. Array Arguments .. |
|
* INTEGER IWORK( * ) |
|
* DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), |
|
* $ BETA( * ), Q( LDQ, * ), U( LDU, * ), |
|
* $ V( LDV, * ), WORK( * ) |
|
* .. |
|
* |
|
* |
|
*> \par Purpose: |
|
* ============= |
|
*> |
|
*> \verbatim |
|
*> |
|
*> DGGSVD computes the generalized singular value decomposition (GSVD) |
|
*> of an M-by-N real matrix A and P-by-N real matrix B: |
|
*> |
|
*> U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R ) |
|
*> |
|
*> where U, V and Q are orthogonal matrices. |
|
*> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T, |
|
*> then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and |
|
*> D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the |
|
*> following structures, respectively: |
|
*> |
|
*> If M-K-L >= 0, |
|
*> |
|
*> K L |
|
*> D1 = K ( I 0 ) |
|
*> L ( 0 C ) |
|
*> M-K-L ( 0 0 ) |
|
*> |
|
*> K L |
|
*> D2 = L ( 0 S ) |
|
*> P-L ( 0 0 ) |
|
*> |
|
*> N-K-L K L |
|
*> ( 0 R ) = K ( 0 R11 R12 ) |
|
*> L ( 0 0 R22 ) |
|
*> |
|
*> where |
|
*> |
|
*> C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), |
|
*> S = diag( BETA(K+1), ... , BETA(K+L) ), |
|
*> C**2 + S**2 = I. |
|
*> |
|
*> R is stored in A(1:K+L,N-K-L+1:N) on exit. |
|
*> |
|
*> If M-K-L < 0, |
|
*> |
|
*> K M-K K+L-M |
|
*> D1 = K ( I 0 0 ) |
|
*> M-K ( 0 C 0 ) |
|
*> |
|
*> K M-K K+L-M |
|
*> D2 = M-K ( 0 S 0 ) |
|
*> K+L-M ( 0 0 I ) |
|
*> P-L ( 0 0 0 ) |
|
*> |
|
*> N-K-L K M-K K+L-M |
|
*> ( 0 R ) = K ( 0 R11 R12 R13 ) |
|
*> M-K ( 0 0 R22 R23 ) |
|
*> K+L-M ( 0 0 0 R33 ) |
|
*> |
|
*> where |
|
*> |
|
*> C = diag( ALPHA(K+1), ... , ALPHA(M) ), |
|
*> S = diag( BETA(K+1), ... , BETA(M) ), |
|
*> C**2 + S**2 = I. |
|
*> |
|
*> (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored |
|
*> ( 0 R22 R23 ) |
|
*> in B(M-K+1:L,N+M-K-L+1:N) on exit. |
|
*> |
|
*> The routine computes C, S, R, and optionally the orthogonal |
|
*> transformation matrices U, V and Q. |
|
*> |
|
*> In particular, if B is an N-by-N nonsingular matrix, then the GSVD of |
|
*> A and B implicitly gives the SVD of A*inv(B): |
|
*> A*inv(B) = U*(D1*inv(D2))*V**T. |
|
*> If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is |
|
*> also equal to the CS decomposition of A and B. Furthermore, the GSVD |
|
*> can be used to derive the solution of the eigenvalue problem: |
|
*> A**T*A x = lambda* B**T*B x. |
|
*> In some literature, the GSVD of A and B is presented in the form |
|
*> U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 ) |
|
*> where U and V are orthogonal and X is nonsingular, D1 and D2 are |
|
*> ``diagonal''. The former GSVD form can be converted to the latter |
|
*> form by taking the nonsingular matrix X as |
|
*> |
|
*> X = Q*( I 0 ) |
|
*> ( 0 inv(R) ). |
|
*> \endverbatim |
|
* |
|
* Arguments: |
|
* ========== |
|
* |
|
*> \param[in] JOBU |
|
*> \verbatim |
|
*> JOBU is CHARACTER*1 |
|
*> = 'U': Orthogonal matrix U is computed; |
|
*> = 'N': U is not computed. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] JOBV |
|
*> \verbatim |
|
*> JOBV is CHARACTER*1 |
|
*> = 'V': Orthogonal matrix V is computed; |
|
*> = 'N': V is not computed. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] JOBQ |
|
*> \verbatim |
|
*> JOBQ is CHARACTER*1 |
|
*> = 'Q': Orthogonal matrix Q is computed; |
|
*> = 'N': Q is not computed. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] M |
|
*> \verbatim |
|
*> M is INTEGER |
|
*> The number of rows of the matrix A. M >= 0. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] N |
|
*> \verbatim |
|
*> N is INTEGER |
|
*> The number of columns of the matrices A and B. N >= 0. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] P |
|
*> \verbatim |
|
*> P is INTEGER |
|
*> The number of rows of the matrix B. P >= 0. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] K |
|
*> \verbatim |
|
*> K is INTEGER |
|
*> \endverbatim |
|
*> |
|
*> \param[out] L |
|
*> \verbatim |
|
*> L is INTEGER |
|
*> |
|
*> On exit, K and L specify the dimension of the subblocks |
|
*> described in Purpose. |
|
*> K + L = effective numerical rank of (A**T,B**T)**T. |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] A |
|
*> \verbatim |
|
*> A is DOUBLE PRECISION array, dimension (LDA,N) |
|
*> On entry, the M-by-N matrix A. |
|
*> On exit, A contains the triangular matrix R, or part of R. |
|
*> See Purpose for details. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDA |
|
*> \verbatim |
|
*> LDA is INTEGER |
|
*> The leading dimension of the array A. LDA >= max(1,M). |
|
*> \endverbatim |
|
*> |
|
*> \param[in,out] B |
|
*> \verbatim |
|
*> B is DOUBLE PRECISION array, dimension (LDB,N) |
|
*> On entry, the P-by-N matrix B. |
|
*> On exit, B contains the triangular matrix R if M-K-L < 0. |
|
*> See Purpose for details. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDB |
|
*> \verbatim |
|
*> LDB is INTEGER |
|
*> The leading dimension of the array B. LDB >= max(1,P). |
|
*> \endverbatim |
|
*> |
|
*> \param[out] ALPHA |
|
*> \verbatim |
|
*> ALPHA is DOUBLE PRECISION array, dimension (N) |
|
*> \endverbatim |
|
*> |
|
*> \param[out] BETA |
|
*> \verbatim |
|
*> BETA is DOUBLE PRECISION array, dimension (N) |
|
*> |
|
*> On exit, ALPHA and BETA contain the generalized singular |
|
*> value pairs of A and B; |
|
*> ALPHA(1:K) = 1, |
|
*> BETA(1:K) = 0, |
|
*> and if M-K-L >= 0, |
|
*> ALPHA(K+1:K+L) = C, |
|
*> BETA(K+1:K+L) = S, |
|
*> or if M-K-L < 0, |
|
*> ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 |
|
*> BETA(K+1:M) =S, BETA(M+1:K+L) =1 |
|
*> and |
|
*> ALPHA(K+L+1:N) = 0 |
|
*> BETA(K+L+1:N) = 0 |
|
*> \endverbatim |
|
*> |
|
*> \param[out] U |
|
*> \verbatim |
|
*> U is DOUBLE PRECISION array, dimension (LDU,M) |
|
*> If JOBU = 'U', U contains the M-by-M orthogonal matrix U. |
|
*> If JOBU = 'N', U is not referenced. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDU |
|
*> \verbatim |
|
*> LDU is INTEGER |
|
*> The leading dimension of the array U. LDU >= max(1,M) if |
|
*> JOBU = 'U'; LDU >= 1 otherwise. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] V |
|
*> \verbatim |
|
*> V is DOUBLE PRECISION array, dimension (LDV,P) |
|
*> If JOBV = 'V', V contains the P-by-P orthogonal matrix V. |
|
*> If JOBV = 'N', V is not referenced. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDV |
|
*> \verbatim |
|
*> LDV is INTEGER |
|
*> The leading dimension of the array V. LDV >= max(1,P) if |
|
*> JOBV = 'V'; LDV >= 1 otherwise. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] Q |
|
*> \verbatim |
|
*> Q is DOUBLE PRECISION array, dimension (LDQ,N) |
|
*> If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. |
|
*> If JOBQ = 'N', Q is not referenced. |
|
*> \endverbatim |
|
*> |
|
*> \param[in] LDQ |
|
*> \verbatim |
|
*> LDQ is INTEGER |
|
*> The leading dimension of the array Q. LDQ >= max(1,N) if |
|
*> JOBQ = 'Q'; LDQ >= 1 otherwise. |
|
*> \endverbatim |
|
*> |
|
*> \param[out] WORK |
|
*> \verbatim |
|
*> WORK is DOUBLE PRECISION array, |
|
*> dimension (max(3*N,M,P)+N) |
|
*> \endverbatim |
|
*> |
|
*> \param[out] IWORK |
|
*> \verbatim |
|
*> IWORK is INTEGER array, dimension (N) |
|
*> On exit, IWORK stores the sorting information. More |
|
*> precisely, the following loop will sort ALPHA |
|
*> for I = K+1, min(M,K+L) |
|
*> swap ALPHA(I) and ALPHA(IWORK(I)) |
|
*> endfor |
|
*> such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(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 = 1, the Jacobi-type procedure failed to |
|
*> converge. For further details, see subroutine DTGSJA. |
|
*> \endverbatim |
|
* |
|
*> \par Internal Parameters: |
|
* ========================= |
|
*> |
|
*> \verbatim |
|
*> TOLA DOUBLE PRECISION |
|
*> TOLB DOUBLE PRECISION |
|
*> TOLA and TOLB are the thresholds to determine the effective |
|
*> rank of (A',B')**T. Generally, they are set to |
|
*> TOLA = MAX(M,N)*norm(A)*MAZHEPS, |
|
*> TOLB = MAX(P,N)*norm(B)*MAZHEPS. |
|
*> The size of TOLA and TOLB may affect the size of backward |
|
*> errors of the decomposition. |
|
*> \endverbatim |
|
* |
|
* Authors: |
|
* ======== |
|
* |
|
*> \author Univ. of Tennessee |
|
*> \author Univ. of California Berkeley |
|
*> \author Univ. of Colorado Denver |
|
*> \author NAG Ltd. |
|
* |
|
*> \date November 2011 |
|
* |
|
*> \ingroup doubleOTHERsing |
|
* |
|
*> \par Contributors: |
|
* ================== |
|
*> |
|
*> Ming Gu and Huan Ren, Computer Science Division, University of |
|
*> California at Berkeley, USA |
|
*> |
|
* ===================================================================== |
SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, |
SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, |
$ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, |
$ LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, |
$ IWORK, INFO ) |
$ IWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.3.1) -- |
* -- LAPACK driver routine (version 3.4.0) -- |
* -- 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..-- |
* -- April 2011 -- |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBQ, JOBU, JOBV |
CHARACTER JOBQ, JOBU, JOBV |
Line 18
|
Line 348
|
$ V( LDV, * ), WORK( * ) |
$ V( LDV, * ), WORK( * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
|
* DGGSVD computes the generalized singular value decomposition (GSVD) |
|
* of an M-by-N real matrix A and P-by-N real matrix B: |
|
* |
|
* U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R ) |
|
* |
|
* where U, V and Q are orthogonal matrices. |
|
* Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T, |
|
* then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and |
|
* D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the |
|
* following structures, respectively: |
|
* |
|
* If M-K-L >= 0, |
|
* |
|
* K L |
|
* D1 = K ( I 0 ) |
|
* L ( 0 C ) |
|
* M-K-L ( 0 0 ) |
|
* |
|
* K L |
|
* D2 = L ( 0 S ) |
|
* P-L ( 0 0 ) |
|
* |
|
* N-K-L K L |
|
* ( 0 R ) = K ( 0 R11 R12 ) |
|
* L ( 0 0 R22 ) |
|
* |
|
* where |
|
* |
|
* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), |
|
* S = diag( BETA(K+1), ... , BETA(K+L) ), |
|
* C**2 + S**2 = I. |
|
* |
|
* R is stored in A(1:K+L,N-K-L+1:N) on exit. |
|
* |
|
* If M-K-L < 0, |
|
* |
|
* K M-K K+L-M |
|
* D1 = K ( I 0 0 ) |
|
* M-K ( 0 C 0 ) |
|
* |
|
* K M-K K+L-M |
|
* D2 = M-K ( 0 S 0 ) |
|
* K+L-M ( 0 0 I ) |
|
* P-L ( 0 0 0 ) |
|
* |
|
* N-K-L K M-K K+L-M |
|
* ( 0 R ) = K ( 0 R11 R12 R13 ) |
|
* M-K ( 0 0 R22 R23 ) |
|
* K+L-M ( 0 0 0 R33 ) |
|
* |
|
* where |
|
* |
|
* C = diag( ALPHA(K+1), ... , ALPHA(M) ), |
|
* S = diag( BETA(K+1), ... , BETA(M) ), |
|
* C**2 + S**2 = I. |
|
* |
|
* (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored |
|
* ( 0 R22 R23 ) |
|
* in B(M-K+1:L,N+M-K-L+1:N) on exit. |
|
* |
|
* The routine computes C, S, R, and optionally the orthogonal |
|
* transformation matrices U, V and Q. |
|
* |
|
* In particular, if B is an N-by-N nonsingular matrix, then the GSVD of |
|
* A and B implicitly gives the SVD of A*inv(B): |
|
* A*inv(B) = U*(D1*inv(D2))*V**T. |
|
* If ( A**T,B**T)**T has orthonormal columns, then the GSVD of A and B is |
|
* also equal to the CS decomposition of A and B. Furthermore, the GSVD |
|
* can be used to derive the solution of the eigenvalue problem: |
|
* A**T*A x = lambda* B**T*B x. |
|
* In some literature, the GSVD of A and B is presented in the form |
|
* U**T*A*X = ( 0 D1 ), V**T*B*X = ( 0 D2 ) |
|
* where U and V are orthogonal and X is nonsingular, D1 and D2 are |
|
* ``diagonal''. The former GSVD form can be converted to the latter |
|
* form by taking the nonsingular matrix X as |
|
* |
|
* X = Q*( I 0 ) |
|
* ( 0 inv(R) ). |
|
* |
|
* Arguments |
|
* ========= |
|
* |
|
* JOBU (input) CHARACTER*1 |
|
* = 'U': Orthogonal matrix U is computed; |
|
* = 'N': U is not computed. |
|
* |
|
* JOBV (input) CHARACTER*1 |
|
* = 'V': Orthogonal matrix V is computed; |
|
* = 'N': V is not computed. |
|
* |
|
* JOBQ (input) CHARACTER*1 |
|
* = 'Q': Orthogonal matrix Q is computed; |
|
* = 'N': Q is not computed. |
|
* |
|
* M (input) INTEGER |
|
* The number of rows of the matrix A. M >= 0. |
|
* |
|
* N (input) INTEGER |
|
* The number of columns of the matrices A and B. N >= 0. |
|
* |
|
* P (input) INTEGER |
|
* The number of rows of the matrix B. P >= 0. |
|
* |
|
* K (output) INTEGER |
|
* L (output) INTEGER |
|
* On exit, K and L specify the dimension of the subblocks |
|
* described in the Purpose section. |
|
* K + L = effective numerical rank of (A**T,B**T)**T. |
|
* |
|
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
|
* On entry, the M-by-N matrix A. |
|
* On exit, A contains the triangular matrix R, or part of R. |
|
* See Purpose for details. |
|
* |
|
* LDA (input) INTEGER |
|
* The leading dimension of the array A. LDA >= max(1,M). |
|
* |
|
* B (input/output) DOUBLE PRECISION array, dimension (LDB,N) |
|
* On entry, the P-by-N matrix B. |
|
* On exit, B contains the triangular matrix R if M-K-L < 0. |
|
* See Purpose for details. |
|
* |
|
* LDB (input) INTEGER |
|
* The leading dimension of the array B. LDB >= max(1,P). |
|
* |
|
* ALPHA (output) DOUBLE PRECISION array, dimension (N) |
|
* BETA (output) DOUBLE PRECISION array, dimension (N) |
|
* On exit, ALPHA and BETA contain the generalized singular |
|
* value pairs of A and B; |
|
* ALPHA(1:K) = 1, |
|
* BETA(1:K) = 0, |
|
* and if M-K-L >= 0, |
|
* ALPHA(K+1:K+L) = C, |
|
* BETA(K+1:K+L) = S, |
|
* or if M-K-L < 0, |
|
* ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 |
|
* BETA(K+1:M) =S, BETA(M+1:K+L) =1 |
|
* and |
|
* ALPHA(K+L+1:N) = 0 |
|
* BETA(K+L+1:N) = 0 |
|
* |
|
* U (output) DOUBLE PRECISION array, dimension (LDU,M) |
|
* If JOBU = 'U', U contains the M-by-M orthogonal matrix U. |
|
* If JOBU = 'N', U is not referenced. |
|
* |
|
* LDU (input) INTEGER |
|
* The leading dimension of the array U. LDU >= max(1,M) if |
|
* JOBU = 'U'; LDU >= 1 otherwise. |
|
* |
|
* V (output) DOUBLE PRECISION array, dimension (LDV,P) |
|
* If JOBV = 'V', V contains the P-by-P orthogonal matrix V. |
|
* If JOBV = 'N', V is not referenced. |
|
* |
|
* LDV (input) INTEGER |
|
* The leading dimension of the array V. LDV >= max(1,P) if |
|
* JOBV = 'V'; LDV >= 1 otherwise. |
|
* |
|
* Q (output) DOUBLE PRECISION array, dimension (LDQ,N) |
|
* If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. |
|
* If JOBQ = 'N', Q is not referenced. |
|
* |
|
* LDQ (input) INTEGER |
|
* The leading dimension of the array Q. LDQ >= max(1,N) if |
|
* JOBQ = 'Q'; LDQ >= 1 otherwise. |
|
* |
|
* WORK (workspace) DOUBLE PRECISION array, |
|
* dimension (max(3*N,M,P)+N) |
|
* |
|
* IWORK (workspace/output) INTEGER array, dimension (N) |
|
* On exit, IWORK stores the sorting information. More |
|
* precisely, the following loop will sort ALPHA |
|
* for I = K+1, min(M,K+L) |
|
* swap ALPHA(I) and ALPHA(IWORK(I)) |
|
* endfor |
|
* such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). |
|
* |
|
* INFO (output) INTEGER |
|
* = 0: successful exit |
|
* < 0: if INFO = -i, the i-th argument had an illegal value. |
|
* > 0: if INFO = 1, the Jacobi-type procedure failed to |
|
* converge. For further details, see subroutine DTGSJA. |
|
* |
|
* Internal Parameters |
|
* =================== |
|
* |
|
* TOLA DOUBLE PRECISION |
|
* TOLB DOUBLE PRECISION |
|
* TOLA and TOLB are the thresholds to determine the effective |
|
* rank of (A',B')**T. Generally, they are set to |
|
* TOLA = MAX(M,N)*norm(A)*MAZHEPS, |
|
* TOLB = MAX(P,N)*norm(B)*MAZHEPS. |
|
* The size of TOLA and TOLB may affect the size of backward |
|
* errors of the decomposition. |
|
* |
|
* Further Details |
|
* =============== |
|
* |
|
* 2-96 Based on modifications by |
|
* Ming Gu and Huan Ren, Computer Science Division, University of |
|
* California at Berkeley, USA |
|
* |
|
* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Local Scalars .. |
* .. Local Scalars .. |