*+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ * File lsqrchk fortran (double precision) * * acheck xcheck * * These routines are for testing algorithms LSQR and CRAIG * (Paige and Saunders, ACM TOMS, 1982). * * acheck tests if products of the form y := y + Ax and x := x + A'y * seem to refer to the same A. * xcheck tests if a solution from LSQR or CRAIG seems to be * correct. * acheck and xcheck may be helpful to all users of LSQR and CRAIG. * * 10 Feb 1992: acheck revised and xcheck implemented. * 27 May 1993: acheck and xcheck kept separate from test problems. * * Michael Saunders, Dept of Operations Research, Stanford University. *+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine acheck( m, n, nout, aprod, eps, \$ leniw, lenrw, iw, rw, \$ v, w, x, y, \$ inform ) implicit none external aprod integer m, n, nout, inform, leniw, lenrw integer iw(leniw) double precision eps double precision v(n), w(m), x(n), y(m), rw(lenrw) * One-liner: acheck checks the two modes of aprod for LSQR and CRAIG. * * Purpose: acheck may be called to test the user-written subroutine * aprod required by LSQR and CRAIG. For some m x n matrix A, * aprod with mode = 1 and 2 supplies LSQR and CRAIG with products * of the form * y := y + Ax and x := x + A'y * respectively, where A' means A(transpose). * acheck tries to verify that A and A' refer to the same matrix. * * Method: We cook up some "unlikely" vectors x and y of unit length * and test if y'(y + Ax) = x'(x + A'y). * * Arguments: * Arg Dim Type I/O/S Description * * m Integer I No. of rows of A. * n Integer I No. of columns of A. * nout Integer I A file number for printed output. * aprod External The routine to be tested. * See LSQR or CRAIG. * eps Double I The machine precision. * leniw Integer I These four parameters are passed * lenrw Integer I to aprod but not otherwise used. * iw leniw Integer I LSQR and CRAIG pass them to aprod * rw lenrw Double I in the same way. * v n Double S * w m Double S * x n Double S * y m Double S * inform Integer O Error indicator. * inform = 0 if aprod seems to be * consistent. * inform = 1 otherwise. * * Error Handling: See inform above. * * Parameter Constants: * Param Type Description * one double 1.0d+0 * power double eps**power is used as the tolerance for judging * whether y'(y + Ax) = x'(x + A'y) * to sufficient accuracy. * power should be in the range (0.25, 0.9) say. * For example, power = 0.75 means that we are happy * if three quarters of the available digits agree. * power = 0.5 seems a reasonable requirement * (asking for half the digits to agree). * * * Files Used: * nout O Screen diagnostics * * Procedures: * dcopy BLAS * ddot BLAS * dnrm2 BLAS * dscal BLAS * * Environment: * FORTRAN 77 with implicit none. * * History: * 04 Sep 1991 Initial design and code. * Michael Saunders, Dept of Operations Research, * Stanford University * 10 Feb 1992 aprod and eps added as parameters. * tol defined via power. C----------------------------------------------------------------------- * Procedures. external dcopy, ddot, dnrm2, dscal double precision ddot, dnrm2 * Local constants. double precision one, power parameter (one = 1.0d+0, power = 0.5d+0) * Local variables. integer i, j double precision alfa, beta, t, test1, test2, test3, tol * Execution. tol = eps**power if (nout .gt. 0) write(nout, 1000) * ================================================================== * Cook up some "unlikely" vectors x and y of unit length. * ================================================================== t = one do 20 j = 1, n t = t + one x(j) = sqrt( t ) 20 continue t = one do 30 i = 1, m t = t + one y(i) = one / sqrt( t ) 30 continue alfa = dnrm2 ( n, x, 1 ) beta = dnrm2 ( m, y, 1 ) call dscal ( n, (one / alfa), x, 1 ) call dscal ( m, (one / beta), y, 1 ) * ================================================================== * Test if y'(y + Ax) = x'(x + A'y). * ================================================================== * First set w = y + Ax, v = x + A'y. call dcopy ( m, y, 1, w, 1 ) call dcopy ( n, x, 1, v, 1 ) call aprod ( 1, m, n, x, w, leniw, lenrw, iw, rw ) call aprod ( 2, m, n, v, y, leniw, lenrw, iw, rw ) * Now set alfa = y'w, beta = x'v. alfa = ddot ( m, y, 1, w, 1 ) beta = ddot ( n, x, 1, v, 1 ) test1 = abs( alfa - beta ) test2 = one + abs( alfa ) + abs( beta ) test3 = test1 / test 2 * See if alfa and beta are essentially the same. if ( test3 .le. tol ) then inform = 0 if (nout .gt. 0) write(nout, 1010) test3 else inform = 1 if (nout .gt. 0) write(nout, 1020) test3 end if return 1000 format(//' Enter acheck. Test of aprod for LSQR and CRAIG') 1010 format( ' aprod seems OK. Relative error =', 1p, e10.1) 1020 format( ' aprod seems incorrect. Relative error =', 1p, e10.1) * end of acheck end *+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine xcheck( m, n, nout, aprod, anorm, damp, eps, \$ leniw, lenrw, iw, rw, \$ b, u, v, w, x, \$ inform, test1, test2, test3 ) implicit none * implicit double precision (a-h,o-z) external aprod integer m, n, nout, inform, leniw, lenrw integer iw(leniw) double precision anorm, damp, eps, test1, test2, test3 double precision rw(lenrw), b(m), u(m), v(n), w(n), x(n) * ------------------------------------------------------------------ * One-liner: xcheck tests if x solves a certain least-squares problem. * * Purpose: xcheck computes residuals and norms associated with the * vector x and the least-squares problem solved by LSQR or CRAIG. * It determines whether x seems to be a solution to any of three * possible systems: 1. Ax = b * 2. min norm(Ax - b) * 3. min norm(Ax - b)^2 + damp^2 * norm(x)^2. * * Arguments: * Arg Dim Type I/O/S Description * * m Integer I The number of rows in A. * n Integer I The number of columns in A. * nout Integer I A file number for printed output. * If nout = 0, nothing is printed. * aprod External The subroutine defining A. * See LSQR or CRAIG. * anorm Double I An estimate of norm(A) or * norm( A, delta*I ) if delta > 0. * Normally this will be available * from LSQR or CRAIG. * damp Double I Possibly defines a damped problem. * eps Double I Machine precision. * leniw Integer I These four parameters are passed * lenrw Integer I to aprod but not otherwise used. * iw leniw Integer I LSQR and CRAIG pass them to aprod * rw lenrw Double I in the same way. * b m Double I The right-hand side of Ax = b etc. * u m Double O On exit, u = r (where r = b - Ax). * v n Double O On exit, v = A'r. * w n Double O On exit, w = A'r - damp^2 x. * x n Double I The given estimate of a solution. * inform Integer O inform = 0 if b = 0 and x = 0. * inform = 1, 2 or 3 if x seems to * solve systems 1 2 or 3 above. * test1 Double O These are dimensionless quantities * test2 Double O that should be "small" if x does * test3 Double O seem to solve one of the systems. * "small" means less than * tol = eps**power, where power is * defined as a parameter below. * * Files Used: * nout O Screen diagnostics * * Procedures: * dcopy BLAS * dnrm2 BLAS * dscal BLAS * * Environment: * FORTRAN 77 with implicit none. * * History: * 07 Feb 1992 Initial design and code. * Michael Saunders, Dept of Operations Research, * Stanford University. * ------------------------------------------------------------------ * Procedures. external dcopy, dnrm2, dscal double precision dnrm2 * Local constants. double precision zero, one, power parameter (zero = 0.0d+0, one = 1.0d+0, power = 0.5d+0) * Local variables. integer j double precision bnorm, dampsq, rho1, rho2, sigma1, sigma2, \$ tol, snorm, xnorm, xsnorm * Execution. dampsq = damp**2 tol = eps**power * Compute u = b - Ax via u = -b + Ax, u = -u. * This is usual residual vector r. call dcopy ( m, b, 1, u, 1 ) call dscal ( m, (-one), u, 1 ) call aprod ( 1, m, n, x, u, leniw, lenrw, iw, rw ) call dscal ( m, (-one), u, 1 ) * Compute v = A'u via v = 0, v = v + A'u. do 100 j = 1, n v(j) = zero 100 continue call aprod ( 2, m, n, v, u, leniw, lenrw, iw, rw ) * Compute w = A'u - damp**2 * x. * This will be close to zero in all cases * if x is close to a solution. call dcopy ( n, v, 1, w, 1 ) if (damp .ne. zero) then do 200 j = 1, n w(j) = w(j) - dampsq * x(j) 200 continue end if * Compute the norms associated with b, x, u, v, w. bnorm = dnrm2 ( m, b, 1 ) xnorm = dnrm2 ( n, x, 1 ) rho1 = dnrm2 ( m, u, 1 ) sigma1 = dnrm2 ( n, v, 1 ) if (nout .gt. 0) write(nout, 2200) damp, xnorm, rho1, sigma1 if (damp .eq. zero) then rho2 = rho1 sigma2 = sigma1 else rho2 = sqrt( rho1**2 + dampsq * xnorm**2 ) sigma2 = dnrm2 ( n, w, 1 ) snorm = rho1 / damp xsnorm = rho2 / damp if (nout .gt. 0) write(nout, 2300) snorm, xsnorm, rho2, sigma2 end if * See if x seems to solve Ax = b or min norm(Ax - b) * or the damped least-squares system. if (bnorm .eq. zero .and. xnorm .eq. zero) then inform = 0 test1 = zero test2 = zero test3 = zero else inform = 4 test1 = rho1 / (bnorm + anorm * xnorm) test2 = zero if (rho1 .gt. zero) test2 = sigma1 / (anorm * rho1) test3 = test2 if (rho2 .gt. zero) test3 = sigma2 / (anorm * rho2) if (test3 .le. tol) inform = 3 if (test2 .le. tol) inform = 2 if (test1 .le. tol) inform = 1 end if if (nout .gt. 0) \$ write(nout, 3000) inform, tol, test1, test2, test3 return 2200 format(1p \$ // ' Enter xcheck. Does x solve Ax = b, etc?' \$ / ' damp =', e10.3 \$ / ' norm(x) =', e10.3 \$ / ' norm(r) =', e15.8, ' = rho1' \$ / ' norm(A''r) =',e10.3, ' = sigma1') 2300 format(1p \$ / ' norm(s) =', e10.3 \$ / ' norm(x,s) =', e10.3 \$ / ' norm(rbar) =', e15.8, ' = rho2' \$ / ' norm(Abar''rbar) =',e10.3, ' = sigma2') 3000 format(1p \$ / ' inform =', i2 \$ / ' tol =', e10.3 \$ / ' test1 =', e10.3, ' (Ax = b)' \$ / ' test2 =', e10.3, ' (least-squares)' \$ / ' test3 =', e10.3, ' (damped least-squares)') * end of xcheck end