Program f08bvfe ! F08BVF Example Program Text ! Mark 24 Release. NAG Copyright 2012. ! .. Use Statements .. Use nag_library, Only: dznrm2, nag_wp, x04dbf, zgeqp3, ztrsm, ztzrzf, & zunmqr, zunmrz ! .. Implicit None Statement .. Implicit None ! .. Parameters .. Complex (Kind=nag_wp), Parameter :: one = (1.0_nag_wp,0.0_nag_wp) Complex (Kind=nag_wp), Parameter :: zero = (0.0_nag_wp,0.0_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 .. Complex (Kind=nag_wp), Allocatable :: a(:,:), b(:,:), tau(:), work(:) Real (Kind=nag_wp), Allocatable :: rnorm(:), rwork(:) Integer, Allocatable :: jpvt(:) Character (1) :: clabs(1), rlabs(1) ! .. Intrinsic Procedures .. Intrinsic :: abs ! .. Executable Statements .. Write (nout,*) 'F08BVF Example Program Results' Write (nout,*) ! Skip heading in data file Read (nin,*) Read (nin,*) m, n, nrhs lda = m ldb = m lwork = (n+1)*nb Allocate (a(lda,n),b(ldb,nrhs),tau(n),work(lwork),rnorm(n),rwork(2*n), & 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 zgeqp3 is f08btf Call zgeqp3(m,n,a,lda,jpvt,tau,work,lwork,rwork,info) ! Compute C = (C1) = (Q**H)*B, storing the result in B ! (C2) ! The NAG name equivalent of zunmqr is f08auf Call zunmqr('Left','Conjugate 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 ! (R1 R2) = (T 0)*Z ! The NAG name equivalent of ztzrzf is f08bvf Call ztzrzf(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 ztrsm is f06zjf Call ztrsm('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 dznrm2 is f06jjf Do j = 1, nrhs rnorm(j) = dznrm2(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**H)*Y ! The NAG name equivalent of zunmrz is f08bxf Call zunmrz('Left','Conjugate 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 Do i = 1, n work(jpvt(i)) = b(i,j) End Do 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 x04dbf('General',' ',n,nrhs,b,ldb,'Bracketed','F7.4', & 'Least-squares solution(s)','Integer',rlabs,'Integer',clabs,80,0, & 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 (3X,1P,7E11.2) 99998 Format (1X,I6) End Program f08bvfe