PROGRAM f08bffe ! F08BFF Example Program Text ! Mark 23 Release. NAG Copyright 2011. ! .. Use Statements .. USE nag_library, ONLY : dgeqp3, dnrm2, dormqr, dtrsm, nag_wp, x04caf ! .. Implicit None Statement .. IMPLICIT NONE ! .. Parameters .. REAL (KIND=nag_wp), PARAMETER :: one = 1.0E0_nag_wp REAL (KIND=nag_wp), PARAMETER :: zero = 0.0E0_nag_wp INTEGER, PARAMETER :: inc1 = 1, nb = 64, nin = 5, nout = 6 ! .. Local Scalars .. REAL (KIND=nag_wp) :: tol INTEGER :: i, ifail, info, j, k, lda, ldb, & lwork, m, n, nrhs ! .. Local Arrays .. REAL (KIND=nag_wp), ALLOCATABLE :: a(:,:), b(:,:), rnorm(:), tau(:), & work(:) INTEGER, ALLOCATABLE :: jpvt(:) ! .. Intrinsic Functions .. INTRINSIC abs ! .. Executable Statements .. WRITE (nout,*) 'F08BFF Example Program Results' WRITE (nout,*) ! Skip heading in data file READ (nin,*) READ (nin,*) m, n, nrhs lda = m ldb = m lwork = 2*n + (n+1)*nb ALLOCATE (a(lda,n),b(ldb,nrhs),rnorm(n),tau(n),work(lwork),jpvt(n)) ! Read A and B from data file READ (nin,*) (a(i,1:n),i=1,m) READ (nin,*) (b(i,1:nrhs),i=1,m) ! Initialize JPVT to be zero so that all columns are free jpvt(1:n) = 0 ! Compute the QR factorization of A ! The NAG name equivalent of dgeqp3 is f08bff CALL dgeqp3(m,n,a,lda,jpvt,tau,work,lwork,info) ! Compute C = (C1) = (Q**T)*B, storing the result in B ! (C2) ! The NAG name equivalent of dormqr is f08agf CALL dormqr('Left','Transpose',m,nrhs,n,a,lda,tau,b,ldb,work,lwork, & info) ! Choose TOL to reflect the relative accuracy of the input data tol = 0.01_nag_wp ! Determine and print the rank, K, of R relative to TOL LOOP: DO k = 1, n IF (abs(a(k,k))<=tol*abs(a(1,1))) EXIT LOOP END DO LOOP k = k - 1 WRITE (nout,*) 'Tolerance used to estimate the rank of A' WRITE (nout,99999) tol WRITE (nout,*) 'Estimated rank of A' WRITE (nout,99998) k WRITE (nout,*) FLUSH (nout) ! Compute least-squares solutions by backsubstitution in ! R(1:K,1:K)*Y = C1, storing the result in B CALL dtrsm('Left','Upper','No transpose','Non-Unit',k,nrhs,one,a,lda,b, & ldb) ! Compute estimates of the square roots of the residual sums of ! squares (2-norm of each of the columns of C2) ! The NAG name equivalent of dnrm2 is f06ejf DO j = 1, nrhs rnorm(j) = dnrm2(m-k,b(k+1,j),inc1) END DO ! Set the remaining elements of the solutions to zero (to give ! the basic solutions) b(k+1:n,1:nrhs) = zero ! Permute the least-squares solutions stored in B to give X = P*Y DO j = 1, nrhs work(jpvt(1:n)) = b(1:n,j) b(1:n,j) = work(1:n) END DO ! Print least-squares solutions ! ifail: behaviour on error exit ! =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft ifail = 0 CALL x04caf('General',' ',n,nrhs,b,ldb,'Least-squares solution(s)', & ifail) ! Print the square roots of the residual sums of squares WRITE (nout,*) WRITE (nout,*) 'Square root(s) of the residual sum(s) of squares' WRITE (nout,99999) rnorm(1:nrhs) 99999 FORMAT (5X,1P,6E11.2) 99998 FORMAT (1X,I8) END PROGRAM f08bffe