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  • 12/20/2021
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CGELSD Example Program in Fortran

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Any license * under such intellectual property rights must be express and approved by Intel * in writing. * ============================================================================= * * CGELSD Example. * ============== * * Program computes the minimum norm-solution to a complex linear least squares * problem using the singular value decomposition of A, * where A is the coefficient matrix: * * ( 4.55, -0.32) ( -4.36, -4.76) ( 3.99, -6.84) ( 8.03, -6.47) * ( 8.87, -3.11) ( 0.02, 8.43) ( 5.43, -9.30) ( 2.28, 8.94) * ( -0.74, 1.16) ( 3.80, -6.12) ( -7.24, 0.72) ( 2.21, 9.52) * * and B is the right-hand side matrix: * * ( -8.25, 7.98) ( 2.91, -8.81) * ( -5.04, 3.33) ( 6.19, 0.19) * ( 7.98, -4.38) ( -5.96, 7.18) * * Description. * ============ * * The routine computes the minimum-norm solution to a complex linear least * squares problem: minimize ||b - A*x|| using the singular value * decomposition (SVD) of A. A is an m-by-n matrix which may be rank-deficient. * * Several right hand side vectors b and solution vectors x can be handled * in a single call; they are stored as the columns of the m-by-nrhs right * hand side matrix B and the n-by-nrhs solution matrix X. * * The effective rank of A is determined by treating as zero those singular * values which are less than rcond times the largest singular value. * * Example Program Results. * ======================== * * CGELSD Example Program Results * * Minimum norm solution * ( -0.08, 0.09) ( 0.04, 0.16) * ( -0.17, 0.10) ( 0.17, -0.47) * ( -0.92, -0.01) ( 0.71, -0.41) * ( -0.47, -0.26) ( 0.69, 0.02) * * Effective rank = 3 * * Singular values * 20.01 18.21 7.88 * ============================================================================= * * .. Parameters .. INTEGER M, N, NRHS PARAMETER ( M = 3, N = 4, NRHS = 2 ) INTEGER LDA, LDB PARAMETER ( LDA = M, LDB = N ) INTEGER LWMAX PARAMETER ( LWMAX = 1000 ) * * .. Local Scalars .. INTEGER INFO, LWORK, RANK REAL RCOND * * .. Local Arrays .. * IWORK dimension should be at least 3*MIN(M,N)*NLVL + 11*MIN(M,N), * RWORK dimension should be at least 10*MIN(M,N)+2*MIN(M,N)*SMLSIZ+ * +8*MIN(M,N)*NLVL+3*SMLSIZ*NRHS+(SMLSIZ+1)**2, * where NLVL = MAX( 0, INT( LOG_2( MIN(M,N)/(SMLSIZ+1) ) )+1 ) * and SMLSIZ = 25 INTEGER IWORK( 3*M*0+11*M ) REAL S( M ), RWORK( 10*M+2*M*25+8*M*0+3*25*NRHS+26*26 $ ) COMPLEX A( LDA, N ), B( LDB, NRHS ), WORK( LWMAX ) DATA A/ $ ( 4.55,-0.32),( 8.87,-3.11),(-0.74, 1.16), $ (-4.36,-4.76),( 0.02, 8.43),( 3.80,-6.12), $ ( 3.99,-6.84),( 5.43,-9.30),(-7.24, 0.72), $ ( 8.03,-6.47),( 2.28, 8.94),( 2.21, 9.52) $ / DATA B/ $ (-8.25, 7.98),(-5.04, 3.33),( 7.98,-4.38),( 0.00, 0.00), $ ( 2.91,-8.81),( 6.19, 0.19),(-5.96, 7.18),( 0.00, 0.00) $ / * * .. External Subroutines .. EXTERNAL CGELSD EXTERNAL PRINT_MATRIX, PRINT_RMATRIX * * .. Intrinsic Functions .. INTRINSIC INT, MIN * * .. Executable Statements .. WRITE(*,*)'CGELSD Example Program Results' * Negative RCOND means using default (machine precision) value RCOND = -1.0 * * Query the optimal workspace. * LWORK = -1 CALL CGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, $ LWORK, RWORK, IWORK, INFO ) LWORK = MIN( LWMAX, INT( WORK( 1 ) ) ) * * Solve the equations A*X = B. * CALL CGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, $ LWORK, RWORK, IWORK, INFO ) * * Check for convergence. * IF( INFO.GT.0 ) THEN WRITE(*,*)'The algorithm computing SVD failed to converge;' WRITE(*,*)'the least squares solution could not be computed.' STOP END IF * * Print minimum norm solution. * CALL PRINT_MATRIX( 'Minimum norm solution', N, NRHS, B, LDB ) * * Print effective rank. * WRITE(*,'(/A,I6)')' Effective rank = ', RANK * * Print singular values. * CALL PRINT_RMATRIX( 'Singular values', 1, M, S, 1 ) STOP END * * End of CGELSD Example. * * ============================================================================= * * Auxiliary routine: printing a matrix. * SUBROUTINE PRINT_MATRIX( DESC, M, N, A, LDA ) CHARACTER*(*) DESC INTEGER M, N, LDA COMPLEX A( LDA, * ) * INTEGER I, J * WRITE(*,*) WRITE(*,*) DESC DO I = 1, M WRITE(*,9998) ( A( I, J ), J = 1, N ) END DO * 9998 FORMAT( 11(:,1X,'(',F6.2,',',F6.2,')') ) RETURN END * * Auxiliary routine: printing a real matrix. * SUBROUTINE PRINT_RMATRIX( DESC, M, N, A, LDA ) CHARACTER*(*) DESC INTEGER M, N, LDA REAL A( LDA, * ) * INTEGER I, J * WRITE(*,*) WRITE(*,*) DESC DO I = 1, M WRITE(*,9998) ( A( I, J ), J = 1, N ) END DO * 9998 FORMAT( 11(:,1X,F6.2) ) RETURN END

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