SUBROUTINE MB03YT( A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, $ CSR, SNR ) C C SLICOT RELEASE 5.0. C C Copyright (c) 2002-2009 NICONET e.V. C C This program is free software: you can redistribute it and/or C modify it under the terms of the GNU General Public License as C published by the Free Software Foundation, either version 2 of C the License, or (at your option) any later version. C C This program is distributed in the hope that it will be useful, C but WITHOUT ANY WARRANTY; without even the implied warranty of C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the C GNU General Public License for more details. C C You should have received a copy of the GNU General Public License C along with this program. If not, see C . C C PURPOSE C C To compute the periodic Schur factorization of a real 2-by-2 C matrix pair (A,B) where B is upper triangular. This routine C computes orthogonal (rotation) matrices given by CSL, SNL and CSR, C SNR such that C C 1) if the pair (A,B) has two real eigenvalues, then C C [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] C [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] C C [ b11 b12 ] := [ CSR SNR ] [ b11 b12 ] [ CSL -SNL ] C [ 0 b22 ] [ -SNR CSR ] [ 0 b22 ] [ SNL CSL ], C C 2) if the pair (A,B) has a pair of complex conjugate eigenvalues, C then C C [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] C [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] C C [ b11 0 ] := [ CSR SNR ] [ b11 b12 ] [ CSL -SNL ] C [ 0 b22 ] [ -SNR CSR ] [ 0 b22 ] [ SNL CSL ]. C C This is a modified version of the LAPACK routine DLAGV2 for C computing the real, generalized Schur decomposition of a C two-by-two matrix pencil. C C ARGUMENTS C C Input/Output Parameters C C A (input/output) DOUBLE PRECISION array, dimension (LDA,2) C On entry, the leading 2-by-2 part of this array must C contain the matrix A. C On exit, the leading 2-by-2 part of this array contains C the matrix A of the pair in periodic Schur form. C C LDA INTEGER C The leading dimension of the array A. LDA >= 2. C C B (input/output) DOUBLE PRECISION array, dimension (LDB,2) C On entry, the leading 2-by-2 part of this array must C contain the upper triangular matrix B. C On exit, the leading 2-by-2 part of this array contains C the matrix B of the pair in periodic Schur form. C C LDB INTEGER C The leading dimension of the array B. LDB >= 2. C C ALPHAR (output) DOUBLE PRECISION array, dimension (2) C ALPHAI (output) DOUBLE PRECISION array, dimension (2) C BETA (output) DOUBLE PRECISION array, dimension (2) C (ALPHAR(k)+i*ALPHAI(k))*BETA(k) are the eigenvalues of the C pair (A,B), k=1,2, i = sqrt(-1). ALPHAI(1) >= 0. C C CSL (output) DOUBLE PRECISION C The cosine of the first rotation matrix. C C SNL (output) DOUBLE PRECISION C The sine of the first rotation matrix. C C CSR (output) DOUBLE PRECISION C The cosine of the second rotation matrix. C C SNR (output) DOUBLE PRECISION C The sine of the second rotation matrix. C C REFERENCES C C [1] Van Loan, C. C Generalized Singular Values with Algorithms and Applications. C Ph. D. Thesis, University of Michigan, 1973. C C CONTRIBUTORS C C D. Kressner, Technical Univ. Berlin, Germany, and C P. Benner, Technical Univ. Chemnitz, Germany, December 2003. C C REVISIONS C C V. Sima, June 2008 (SLICOT version of the HAPACK routine DLAPV2). C V. Sima, July 2008, May 2009. C C KEYWORDS C C Eigenvalue, periodic Schur form C C ****************************************************************** C C .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) C .. Scalar Arguments .. INTEGER LDA, LDB DOUBLE PRECISION CSL, CSR, SNL, SNR C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), ALPHAI(2), ALPHAR(2), B(LDB,*), $ BETA(2) C .. Local Scalars .. DOUBLE PRECISION ANORM, BNORM, H1, H2, H3, QQ, R, RR, SAFMIN, $ SCALE1, SCALE2, T, ULP, WI, WR1, WR2 C .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL DLAMCH, DLAPY2 C .. External Subroutines .. EXTERNAL DLAG2, DLARTG, DLASV2, DROT C .. Intrinsic Functions .. INTRINSIC ABS, MAX C C .. Executable Statements .. C SAFMIN = DLAMCH( 'S' ) ULP = DLAMCH( 'P' ) C C Scale A. C ANORM = MAX( ABS( A(1,1) ) + ABS( A(2,1) ), $ ABS( A(1,2) ) + ABS( A(2,2) ), SAFMIN ) A(1,1) = A(1,1) / ANORM A(1,2) = A(1,2) / ANORM A(2,1) = A(2,1) / ANORM A(2,2) = A(2,2) / ANORM C C Scale B. C BNORM = MAX( ABS( B(1,1) ), ABS( B(1,2) ) + ABS( B(2,2) ), SAFMIN) B(1,1) = B(1,1) / BNORM B(1,2) = B(1,2) / BNORM B(2,2) = B(2,2) / BNORM C C Check if A can be deflated. C IF ( ABS( A(2,1) ).LE.ULP ) THEN CSL = ONE SNL = ZERO CSR = ONE SNR = ZERO WI = ZERO A(2,1) = ZERO B(2,1) = ZERO C C Check if B is singular. C ELSE IF ( ABS( B(1,1) ).LE.ULP ) THEN CALL DLARTG( A(2,2), A(2,1), CSR, SNR, T ) SNR = -SNR CALL DROT( 2, A(1,1), 1, A(1,2), 1, CSR, SNR ) CALL DROT( 2, B(1,1), LDB, B(2,1), LDB, CSR, SNR ) CSL = ONE SNL = ZERO WI = ZERO A(2,1) = ZERO B(1,1) = ZERO B(2,1) = ZERO ELSE IF( ABS( B(2,2) ).LE.ULP ) THEN CALL DLARTG( A(1,1), A(2,1), CSL, SNL, R ) CSR = ONE SNR = ZERO WI = ZERO CALL DROT( 2, A(1,1), LDA, A(2,1), LDA, CSL, SNL ) CALL DROT( 2, B(1,1), 1, B(1,2), 1, CSL, SNL ) A(2,1) = ZERO B(2,1) = ZERO B(2,2) = ZERO ELSE C C B is nonsingular, first compute the eigenvalues of A / adj(B). C R = B(1,1) B(1,1) = B(2,2) B(2,2) = R B(1,2) = -B(1,2) CALL DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, $ WI ) C IF( WI.EQ.ZERO ) THEN C C Two real eigenvalues, compute s*A-w*B. C H1 = SCALE1*A(1,1) - WR1*B(1,1) H2 = SCALE1*A(1,2) - WR1*B(1,2) H3 = SCALE1*A(2,2) - WR1*B(2,2) C RR = DLAPY2( H1, H2 ) QQ = DLAPY2( SCALE1*A(2,1), H3 ) C IF ( RR.GT.QQ ) THEN C C Find right rotation matrix to zero 1,1 element of C (sA - wB). C CALL DLARTG( H2, H1, CSR, SNR, T ) C ELSE C C Find right rotation matrix to zero 2,1 element of C (sA - wB). C CALL DLARTG( H3, SCALE1*A(2,1), CSR, SNR, T ) C END IF C SNR = -SNR CALL DROT( 2, A(1,1), 1, A(1,2), 1, CSR, SNR ) CALL DROT( 2, B(1,1), 1, B(1,2), 1, CSR, SNR ) C C Compute inf norms of A and B. C H1 = MAX( ABS( A(1,1) ) + ABS( A(1,2) ), $ ABS( A(2,1) ) + ABS( A(2,2) ) ) H2 = MAX( ABS( B(1,1) ) + ABS( B(1,2) ), $ ABS( B(2,1) ) + ABS( B(2,2) ) ) C IF( ( SCALE1*H1 ).GE.ABS( WR1 )*H2 ) THEN C C Find left rotation matrix Q to zero out B(2,1). C CALL DLARTG( B(1,1), B(2,1), CSL, SNL, R ) C ELSE C C Find left rotation matrix Q to zero out A(2,1). C CALL DLARTG( A(1,1), A(2,1), CSL, SNL, R ) C END IF C CALL DROT( 2, A(1,1), LDA, A(2,1), LDA, CSL, SNL ) CALL DROT( 2, B(1,1), LDB, B(2,1), LDB, CSL, SNL ) C A(2,1) = ZERO B(2,1) = ZERO C C Re-adjoint B. C R = B(1,1) B(1,1) = B(2,2) B(2,2) = R B(1,2) = -B(1,2) C ELSE C C A pair of complex conjugate eigenvalues: C first compute the SVD of the matrix adj(B). C R = B(1,1) B(1,1) = B(2,2) B(2,2) = R B(1,2) = -B(1,2) CALL DLASV2( B(1,1), B(1,2), B(2,2), R, T, SNL, CSL, $ SNR, CSR ) C C Form (A,B) := Q(A,adj(B))Z' where Q is left rotation matrix C and Z is right rotation matrix computed from DLASV2. C CALL DROT( 2, A(1,1), LDA, A(2,1), LDA, CSL, SNL ) CALL DROT( 2, B(1,1), LDB, B(2,1), LDB, CSR, SNR ) CALL DROT( 2, A(1,1), 1, A(1,2), 1, CSR, SNR ) CALL DROT( 2, B(1,1), 1, B(1,2), 1, CSL, SNL ) C B(2,1) = ZERO B(1,2) = ZERO END IF C END IF C C Unscaling C R = B(1,1) T = B(2,2) A(1,1) = ANORM*A(1,1) A(2,1) = ANORM*A(2,1) A(1,2) = ANORM*A(1,2) A(2,2) = ANORM*A(2,2) B(1,1) = BNORM*B(1,1) B(2,1) = BNORM*B(2,1) B(1,2) = BNORM*B(1,2) B(2,2) = BNORM*B(2,2) C IF( WI.EQ.ZERO ) THEN ALPHAR(1) = A(1,1) ALPHAR(2) = A(2,2) ALPHAI(1) = ZERO ALPHAI(2) = ZERO BETA(1) = B(1,1) BETA(2) = B(2,2) ELSE WR1 = ANORM*WR1 WI = ANORM*WI IF ( ABS( WR1 ).GT.ONE .OR. WI.GT.ONE ) THEN WR1 = WR1*R WI = WI*R R = ONE END IF IF ( ABS( WR1 ).GT.ONE .OR. ABS( WI ).GT.ONE ) THEN WR1 = WR1*T WI = WI*T T = ONE END IF ALPHAR(1) = ( WR1 / SCALE1 )*R*T ALPHAI(1) = ABS( ( WI / SCALE1 )*R*T ) ALPHAR(2) = ALPHAR(1) ALPHAI(2) = -ALPHAI(1) BETA(1) = BNORM BETA(2) = BNORM END IF RETURN C *** Last line of MB03YT *** END