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Program f08bhfe
! F08BHF Example Program Text
! Mark 26.1 Release. NAG Copyright 2017.
! .. 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))) Then
Exit loop
End If
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
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