Program f08bhfe ! F08BHF Example Program Text ! Mark 24 Release. NAG Copyright 2012. ! .. Use Statements .. Use nag_library, Only: dgeqp3, dnrm2, dormqr, dormrz, dtrsm, dtzrzf, & 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 Procedures .. Intrinsic :: abs ! .. Executable Statements .. Write (nout,*) 'F08BHF 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 with column pivoting as ! A = Q*(R11 R12)*(P**T) ! ( 0 R22) ! 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 the RZ factorization of the K by K part of R as ! (R11 R12) = (T 0)*Z ! The NAG name equivalent of dtzrzf is f08bhf Call dtzrzf(k,n,a,lda,tau,work,lwork,info) ! Compute least-squares solutions of triangular problems by ! back substitution in T*Y1 = C1, storing the result in B ! The NAG name equivalent of dtrsm is f06yjf 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 minimum-norm solutions), Y2 = 0 b(k+1:n,1:nrhs) = zero ! Form W = (Z**T)*Y ! The NAG name equivalent of dormrz is f08bkf Call dormrz('Left','Transpose',n,nrhs,k,n-k,a,lda,tau,b,ldb,work,lwork, & info) ! Permute the least-squares solutions stored in B to give X = P*W 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 f08bhfe