Program dtzrzf_example ! DTZRZF Example Program Text ! Copyright 2017, Numerical Algorithms Group Ltd. http://www.nag.com ! .. Use Statements .. Use blas_interfaces, Only: dnrm2, dtrsm Use lapack_example_aux, Only: nagf_file_print_matrix_real_gen Use lapack_interfaces, Only: dgeqp3, dormqr, dormrz, dtzrzf Use lapack_precision, Only: dp ! .. Implicit None Statement .. Implicit None ! .. Parameters .. Real (Kind=dp), Parameter :: one = 1.0E0_dp Real (Kind=dp), Parameter :: zero = 0.0E0_dp Integer, Parameter :: inc1 = 1, nb = 64, nin = 5, nout = 6 ! .. Local Scalars .. Real (Kind=dp) :: tol Integer :: i, ifail, info, j, k, lda, ldb, lwork, m, n, nrhs ! .. Local Arrays .. Real (Kind=dp), Allocatable :: a(:, :), b(:, :), rnorm(:), tau(:), & work(:) Integer, Allocatable :: jpvt(:) ! .. Intrinsic Procedures .. Intrinsic :: abs ! .. Executable Statements .. Write (nout, *) 'DTZRZF 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) Call dgeqp3(m, n, a, lda, jpvt, tau, work, lwork, info) ! Compute C = (C1) = (Q**T)*B, storing the result in B ! (C2) 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_dp ! 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))) Then Exit loop End If End Do loop k = k - 1 Write (nout, *) 'Tolerance used to estimate the rank of A' Write (nout, 100) tol Write (nout, *) 'Estimated rank of A' Write (nout, 110) k Write (nout, *) Flush (nout) ! Compute the RZ factorization of the K by K part of R as ! (R11 R12) = (T 0)*Z 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 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) 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 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 nagf_file_print_matrix_real_gen('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, 100) rnorm(1:nrhs) 100 Format (5X, 1P, 6E11.2) 110 Format (1X, I8) End Program