From 549356ec1c50e3404e55b964e66e3132127c84a6 Mon Sep 17 00:00:00 2001 From: Antony Lewis Date: Mon, 8 Jun 2020 10:04:57 +0100 Subject: [PATCH] add Powell's NEWUOA for unbounded miminization; fix initial --- camb/cambdll.dll | Bin 2710471 -> 2764829 bytes fortran/DarkEnergyQuintessence.f90 | 54 +- fortran/Makefile_main | 2 +- ...trainedMinimize.f90 => PowellMinimize.f90} | 1667 ++++++++++++++++- fortran/VisualStudio/CAMBLib.vfproj | 2 +- 5 files changed, 1701 insertions(+), 24 deletions(-) rename fortran/{PowellConstrainedMinimize.f90 => PowellMinimize.f90} (59%) diff --git a/camb/cambdll.dll b/camb/cambdll.dll index 7d8659be646873e53505864820b35af5a90e63e5..5bebee30b661dcc7df2cedeb8908d37709b8c3ce 100644 GIT binary patch delta 383655 zcmafc2V9iL^Y`uZc!(n29Y>RP2OL!qu%cK%K@U9?dvCGC*bBC(U=*U4%h^_aG#X>N zscJ|>jT$@l7EMHBqDeH*i49{^EaCmm^Bk7sm;Za8#C>*mW_EUVc6N4lpNEp-582z7 z*0aS6g0|kfI8f6rO7mWvMsr&eZT1p`FwF>Gjpq82_Y!NQMQb!38jYcN$@#(j_GHHH#=~-5@E<%&!y(BzHo7T;wiCq$I6my10Ncy-a-z7;kRveD< 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z0^(+I^b`>`J-iPR_!5(Aa#A}76k>SiF%V-7RRiVA;`dUv#d^(t263;|NyFbku-Hr1zCQ>)3Z#*_Aj!#*s<9lzi@#J( zicvIiCe^}NFRP&zcM&qgW`mE2uxUT*9F{JOOa*bVb>n;xH=EKog3$Rw(&-OL$BH3q zaTg^lV}gK{E4KL z7{OH7Y!I59la024vf&p5zg5>=qyrHx2{Lio%(7ynD%mK18#-ZD{P!S{0SU`ge%_y` zePRd!7{fvG*(e`Rgw=iyh>=(TQm#ipXeLOu*=~>!S~1y)$3P-le4zXW7vN)J;LWw} z1aPwv+y*)&tR60qNo-r`S`c?4JZZSjn~0+hJMWwgo#ly&%`PhSn~fG_h20LN^9d!Y z*#jVfgx8a27gI|c1#BjG2|E1zsvfJl>f%(%$28gPtSl z+U@fEuDyHq>LuzP$jcKKs`8%9>?wcg-mPcX-o3i_5*yxwjqou?bJ69vqm4Ll%rW%7 zzVGJk==*MmMzwd$`MATvx;vVF)gdcQyzmpMF#r1wAJpA3>1>C9V_m(*9W$40$ZJoY o)!WgyeaE&nMYok%Eyb>nJ7kJT&5oI(PNR;O> TEarlyQuintessence_SelfPointer procedure, private :: fdeAta procedure, private :: fde_peak + procedure, private :: check_error procedure :: calc_zc_fde end type TEarlyQuintessence @@ -313,7 +314,7 @@ subroutine TEarlyQuintessence_Init(this, State) real(dl) fzero, xzero integer iflag, iter Type(TTimer) :: Timer - Type(TBOBYQA) :: Minimize + Type(TNEWUOA) :: Minimize real(dl) log_params(2), param_min(2), param_max(2) !Make interpolation table, etc, @@ -326,8 +327,8 @@ subroutine TEarlyQuintessence_Init(this, State) !Find underlying parameters m,f to give specified zc and fde_zc (peak early dark energy fraction) !Input m,f are used as starting values for search, which is done by brute force !(so should generalize easily, but not optimized for this specific potential) - log_params(1) = log(this%m) - log_params(2) = log(this%f) + log_params(1) = log(this%f) + log_params(2) = log(this%m) if (.false.) then ! Can just iterate linear optimizations when nearly orthogonal @@ -348,23 +349,27 @@ subroutine TEarlyQuintessence_Init(this, State) end if if (this%DebugLevel>0) call Timer%Start() !Minimize in log f, log m - param_min(1) = log(0.001_dl) - param_min(2) = log(1d-57) - param_max(1) = log(1e5_dl) - param_max(2) = log(1d-50) - if (Minimize%BOBYQA(this, match_fde_zc, 2, 5, log_params,param_min, & - param_max, 0.8_dl,1e-4_dl,this%DebugLevel,2000)) then + ! param_min(1) = log(0.001_dl) + ! param_min(2) = log(1d-58) + ! param_max(1) = log(1e5_dl) + ! param_max(2) = log(1d-50) + ! if (Minimize%BOBYQA(this, match_fde_zc, 2, 5, log_params,param_min, & + ! param_max, 0.8_dl,1e-4_dl,this%DebugLevel,2000)) then + + if (Minimize%NEWUOA(this, match_fde_zc, 2, 5, log_params,& + 0.8_dl,1e-4_dl,this%DebugLevel,500)) then + if (Minimize%Last_bestfit > 1e-3) then global_error_flag = error_darkenergy global_error_message= 'TEarlyQuintessence ERROR converging solution for fde, zc' - print *, 'last-bestfit= ', Minimize%Last_bestfit + write(*,*) 'last-bestfit= ', Minimize%Last_bestfit return end if this%f = exp(log_params(1)) this%m = exp(log_params(2)) if (this%DebugLevel>0) then call this%calc_zc_fde(fzero, xzero) - print *, 'matched outputs Bobyqa zc, fde = ', fzero, xzero + write(*,*) 'matched outputs Bobyqa zc, fde = ', fzero, xzero end if else global_error_flag = error_darkenergy @@ -455,6 +460,7 @@ subroutine TEarlyQuintessence_Init(this, State) sampled_a(ix)=exp(aend) a2 = sampled_a(ix)**2 call dverk(this,NumEqs,EvolveBackgroundLog,afrom,y,aend,this%integrate_tol,ind,c,NumEqs,w) + if (.not. this%check_error(exp(afrom), exp(aend))) return call EvolveBackgroundLog(this,NumEqs,aend,y,w(:,1)) phi_a(ix)=y(1) phidot_a(ix)=y(2)/a2 @@ -505,6 +511,7 @@ subroutine TEarlyQuintessence_Init(this, State) a2 =aend**2 this%sampled_a(ix)=aend call dverk(this,NumEqs,EvolveBackground,afrom,y,aend,this%integrate_tol,ind,c,NumEqs,w) + if (.not. this%check_error(afrom, aend)) return call EvolveBackground(this,NumEqs,aend,y,w(:,1)) this%phi_a(ix)=y(1) this%phidot_a(ix)=y(2)/a2 @@ -542,6 +549,20 @@ subroutine TEarlyQuintessence_Init(this, State) end subroutine TEarlyQuintessence_Init + logical function check_error(this, afrom, aend) + class(TEarlyQuintessence) :: this + real(dl) afrom, aend + + if (global_error_flag/=0) then + write(*,*) 'TEarlyQuintessence error integrating', afrom, aend + write(*,*) this%n, this%f, this%m, this%theta_i + stop + check_error = .false. + return + end if + check_error= .true. + end function check_error + logical function fde_peak(this, peak, xlo, xhi, Flo, Fhi, ddFlo, ddFhi) class(TEarlyQuintessence) :: this real(dl), intent(out) :: peak @@ -605,10 +626,12 @@ function match_fde_zc(this, x) this%f = exp(x(1)) this%m = exp(x(2)) call this%calc_zc_fde(zc, fde_zc) - + match_fde_zc = (log(this%fde_zc)-log(fde_zc))**2 + (log(zc)-log(this%zc))**2 - !print *, 'f, m, zc, fde_zc', this%f, this%m, zc, fde_zc, match_fde_zc - + if (this%DebugLevel>1) then + write(*,*) 'search f, m, zc, fde_zc, chi2', this%f, this%m, zc, fde_zc, match_fde_zc + end if + end function match_fde_zc subroutine calc_zc_fde(this, z_c, fde_zc) @@ -649,6 +672,7 @@ subroutine calc_zc_fde(this, z_c, fde_zc) sampled_a(ix)=exp(aend) a2 = sampled_a(ix)**2 call dverk(this,NumEqs,EvolveBackgroundLog,afrom,y,aend,this%integrate_tol,ind,c,NumEqs,w) + if (.not. this%check_error(exp(afrom), exp(aend))) return call EvolveBackgroundLog(this,NumEqs,aend,y,w(:,1)) fde(ix) = 1/((this%state%grho_no_de(sampled_a(ix)) + this%frac_lambda0*this%State%grhov*a2**2) & /((0.5d0*y(2)**2/a2 + a2**2*this%Vofphi(y(1),0))) + 1) @@ -676,7 +700,7 @@ subroutine calc_zc_fde(this, z_c, fde_zc) b0 = 1 - a0 fde_zc=b0*fde(ix+1) + a0*(fde(ix)-b0*((a0+1)*ddfde(ix)+(2-a0)*ddfde(ix+1))*da**2/6._dl) else - print *, 'calc_zc_fde: NO PEAK' + write(*,*) 'calc_zc_fde: NO PEAK' z_c = -1 fde_zc = 0 end if diff --git a/fortran/Makefile_main b/fortran/Makefile_main index 9126f7d3..b886a584 100644 --- a/fortran/Makefile_main +++ b/fortran/Makefile_main @@ -5,7 +5,7 @@ POWERSPECTRUM_FILES ?= InitialPower REIONIZATION_FILES ?= reionization RECOMBINATION_FILES ?= recfast NONLINEAR_FILES ?= halofit SecondOrderPK -DARKENERGY_FILES ?= DarkEnergyFluid DarkEnergyPPF PowellConstrainedMinimize DarkEnergyQuintessence +DARKENERGY_FILES ?= DarkEnergyFluid DarkEnergyPPF PowellMinimize DarkEnergyQuintessence BISPECTRUM ?= SeparableBispectrum diff --git a/fortran/PowellConstrainedMinimize.f90 b/fortran/PowellMinimize.f90 similarity index 59% rename from fortran/PowellConstrainedMinimize.f90 rename to fortran/PowellMinimize.f90 index 684acf99..8b049720 100644 --- a/fortran/PowellConstrainedMinimize.f90 +++ b/fortran/PowellMinimize.f90 @@ -1,12 +1,32 @@ - !####### constrained minimization ###### - !F90 translated version of plato.asu.edu/ftp/other_software/bobyqa.zip + !####### constrained and unconstrained minimization ###### + !F90 translated version of plato.asu.edu/ftp/other_software/bobyqa.zip and newuoa !Powell 2009 Method http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf !AL Sept 2012/2020: translated to F90, changed to input "funkk" class function argument !to minimize rather than calfun subroutine as original + !Typical Usage (typically Npt= 2*dimension+1), to get best fit in params(:) starting from guess + + ! Type(TBOBYQA) :: Minimize !bounded minimization + + ! if (Minimize%BOBYQA(this_class, class_function, dimension, Npt, params,param_min, & + ! param_max, step_radius, tolerance, DebugLevel, max_eval)) ... + ! + + !Type(TNEWUOA) :: Minimize !unbounded minimization + + ! if (Minimize%NEWUOA(this_class, class_function, dimension, Npt, param_guess, step_radius, & + ! tolerance, DebugLevel, max_eval)) ... + ! + ! The class_function is of the form + ! + ! real(dp) function func(this, x) + ! class(TMyType) :: this + ! real(dp), intent(in) :: x(:) + ! func = ...f(x)... + ! end function func + MODULE Powell - use MathUtils implicit none private @@ -14,6 +34,7 @@ MODULE Powell INTEGER, PARAMETER :: func_dp = dp !SELECTED_REAL_KIND(12, 60) INTEGER, PARAMETER :: Powell_CO_prec = dp + ! the type of function minimized, a class procedure taking in an array of values interface FUNCTION obj_vec_function(obj, x) use precision @@ -23,21 +44,30 @@ FUNCTION obj_vec_function(obj, x) END FUNCTION obj_vec_function end interface - type TBOBYQA + type TMinimizer REAL(dp) :: Last_bestfit - REAL(dp) :: FVAL_Converge_difference = 0._dp procedure(obj_vec_function), pointer, nopass :: funkk class(*), pointer :: obj + end type + + type, extends(TMinimizer) :: TBOBYQA + REAL(dp) :: FVAL_Converge_difference = 0._dp contains procedure :: BOBYQA end type - PUBLIC :: TBOBYQA , Powell_CO_prec + type, extends(TMinimizer) :: TNEWUOA + contains + procedure :: NEWUOA + procedure, private :: newuob + end type + + PUBLIC :: TBOBYQA , TNEWUOA, Powell_CO_prec CONTAINS function BOBYQA (this, obj, funk, N,NPT,X,XL,XU,RHOBEG,RHOEND,IPRINT,MAXFUN) - !Main function + !Main function for bounded minimization use iso_c_binding class(TBOBYQA) this class(*), target :: obj @@ -2317,4 +2347,1627 @@ SUBROUTINE UPDATE (N,NPT,BMAT,ZMAT,NDIM,VLAG,BETA,DENOM,KNEW,W) END SUBROUTINE UPDATE + + !***************************************************************************************** + !> + ! This subroutine seeks the least value of a function of many variables, + ! by a trust region method that forms quadratic models by interpolation. + ! There can be some freedom in the interpolation conditions, which is + ! taken up by minimizing the Frobenius norm of the change to the second + ! derivative of the quadratic model, beginning with a zero matrix. + + logical function newuoa (this, obj, funk, n, npt, x, rhobeg, rhoend, iprint, maxfun) + use iso_c_binding + class(TNEWUOA) this + class(*), target :: obj + real(dp), external :: funk !a class function f(obj,x) where obj is any class instance + + integer,intent(in) :: n !! the number of variables. must be at least 2. + integer,intent(in) :: npt !! The number of interpolation conditions. + !! Its value must be in the interval `[N+2,(N+1)(N+2)/2]`. + real(dp),dimension(*),intent(inout) :: x !! Initial values of the variables must be set in X(1),X(2),...,X(N). They + !! will be changed to the values that give the least calculated F. + real(dp),intent(in) :: rhobeg !! RHOBEG and RHOEND must be set to the initial and final values of a trust + !! region radius, so both must be positive with RHOEND<=RHOBEG. Typically + !! RHOBEG should be about one tenth of the greatest expected change to a + !! variable, and RHOEND should indicate the accuracy that is required in + !! the final values of the variables. + real(dp),intent(in) :: rhoend !! RHOBEG and RHOEND must be set to the initial and final values of a trust + !! region radius, so both must be positive with RHOEND<=RHOBEG. Typically + !! RHOBEG should be about one tenth of the greatest expected change to a + !! variable, and RHOEND should indicate the accuracy that is required in + !! the final values of the variables. + integer,intent(in) :: iprint !! The value of IPRINT should be set to 0, 1, 2 or 3, which controls the + !! amount of printing. Specifically, there is no output if IPRINT=0 and + !! there is output only at the return if IPRINT=1. Otherwise, each new + !! value of RHO is printed, with the best vector of variables so far and + !! the corresponding value of the objective function. Further, each new + !! value of F with its variables are output if IPRINT=3. + integer,intent(in) :: maxfun !! an upper bound on the number of calls of CALFUN. + !! for the variables `X(1),X(2),...,X(N)`. + + real(dp),dimension(:),allocatable :: w + integer :: np,nptm,ndim,ixb,ixo,ixn,ixp,ifv,igq,ihq,ipq,ibmat,izmat,id,ivl,iw + + ! Partition the working space array, so that different parts of it can be + ! treated separately by the subroutine that performs the main calculation. + + call C_F_PROCPOINTER(c_funloc(funk), this%funkk) + this%obj => obj + this%Last_bestfit = 1d30 + + np = n + 1 + nptm = npt - np + if (npt < n+2 .or. npt > ((n+2)*np)/2) then + write(*,*) 'Return from NEWUOA because NPT is not in the required interval' + newuoa = .false. + return + end if + + ! The array W will be used for working space + allocate(w((NPT+13)*(NPT+N)+3*N*(N+3)/2)) + + ndim = npt + n + ixb = 1 + ixo = ixb + n + ixn = ixo + n + ixp = ixn + n + ifv = ixp + n * npt + igq = ifv + npt + ihq = igq + n + ipq = ihq + (n*np) / 2 + ibmat = ipq + npt + izmat = ibmat + ndim * n + id = izmat + npt * nptm + ivl = id + n + iw = ivl + ndim + + ! The above settings provide a partition of W for subroutine NEWUOB. + ! The partition requires the first NPT*(NPT+N)+5*N*(N+3)/2 elements of + ! W plus the space that is needed by the last array of NEWUOB. + + newuoa = this%newuob (n, npt, x, rhobeg, rhoend, iprint, maxfun, w(ixb), w(ixo), w(ixn), & + w(ixp), w(ifv), w(igq), w(ihq), w(ipq), w(ibmat), w(izmat), ndim, & + w(id), w(ivl), w(iw)) + + + deallocate(w) + + end function newuoa + !***************************************************************************************** + + logical function newuob (this, n, npt, x, rhobeg, rhoend, iprint, maxfun, xbase, xopt, & + xnew, xpt, fval, gq, hq, pq, bmat, zmat, ndim, d, vlag, w) + class(TNEWUOA) this + + real(dp) alpha , beta , bsum , crvmin , delta , detrat , & + & diff , diffa , diffb , diffc , distsq , dnorm , dsq , & + & dstep , dx , f , fbeg , fopt + real(dp) fsave , gisq , gqsq , half , hdiag , one , ratio , recip , reciq , & + rho , Rhobeg , Rhoend , rhosq , sum , suma , sumb + real(dp) sumz , temp , tempq , tenth , vquad , xjpt , xoptsq , zero, xipt + INTEGER i , idz , ih , ip , Iprint , ipt , itemp , itest , & + & j , jp , jpt , k , knew , kopt , ksave , ktemp , Maxfun , & + & N , Ndim + INTEGER nf , nfm , nfmm , nfsav , nftest , nh , np , Npt , nptm + real(dp) x (*), xbase (*), xopt (*), xnew (*), xpt (npt,*), fval (*), gq (*), hq(*), & + pq (*), bmat (ndim,*), zmat (npt,*), d (*), vlag (*), w (*) + ! + ! The arguments N, NPT, X, RHOBEG, RHOEND, IPRINT and MAXFUN are identical + ! to the corresponding arguments in SUBROUTINE NEWUOA. + ! XBASE will hold a shift of origin that should reduce the contributions + ! from rounding errors to values of the model and Lagrange functions. + ! XOPT will be set to the displacement from XBASE of the vector of + ! variables that provides the least calculated F so far. + ! XNEW will be set to the displacement from XBASE of the vector of + ! variables for the current calculation of F. + ! XPT will contain the interpolation point coordinates relative to XBASE. + ! FVAL will hold the values of F at the interpolation points. + ! GQ will hold the gradient of the quadratic model at XBASE. + ! HQ will hold the explicit second derivatives of the quadratic model. + ! PQ will contain the parameters of the implicit second derivatives of + ! the quadratic model. + ! BMAT will hold the last N columns of H. + ! ZMAT will hold the factorization of the leading NPT by NPT submatrix of + ! H, this factorization being ZMAT times Diag(DZ) times ZMAT^T, where + ! the elements of DZ are plus or minus one, as specified by IDZ. + ! NDIM is the first dimension of BMAT and has the value NPT+N. + ! D is reserved for trial steps from XOPT. + ! VLAG will contain the values of the Lagrange functions at a new point X. + ! They are part of a product that requires VLAG to be of length NDIM. + ! The array W will be used for working space. Its length must be at least + ! 10*NDIM = 10*(NPT+N). + ! + ! Set some constants. + ! + newuob = .true. + half = 0.5_dp + one = 1.0_dp + tenth = 0.1_dp + zero = 0.0_dp + np = n + 1 + nh = (n*np) / 2 + nptm = npt - np + nftest = max (maxfun, 1) + ! + ! Set the initial elements of XPT, BMAT, HQ, PQ and ZMAT to zero. + ! + do j = 1, n + xbase (j) = x (j) + do k = 1, npt + xpt (k, j) = zero + end do + do i = 1, ndim + bmat (i, j) = zero + end do + end do + do ih = 1, nh + hq (ih) = zero + end do + do k = 1, npt + pq (k) = zero + do j = 1, nptm + zmat (k, j) = zero + end do + end do + ! + ! Begin the initialization procedure. NF becomes one more than the number + ! of function values so far. The coordinates of the displacement of the + ! next initial interpolation point from XBASE are set in XPT(NF,.). + ! + rhosq = rhobeg * rhobeg + recip = one / rhosq + reciq = sqrt (half) / rhosq + nf = 0 +50 nfm = nf + nfmm = nf - n + nf = nf + 1 + if (nfm <= 2*n) then + if (nfm >= 1 .and. nfm <= n) then + xpt (nf, nfm) = rhobeg + else if (nfm > n) then + xpt (nf, nfmm) = - rhobeg + end if + else + itemp = (nfmm-1) / n + jpt = nfm - itemp * n - n + ipt = jpt + itemp + if (ipt > n) then + itemp = jpt + jpt = ipt - n + ipt = itemp + end if + xipt = rhobeg + if (fval(ipt+np) < fval(ipt+1)) xipt = - xipt + xjpt = rhobeg + if (fval(jpt+np) < fval(jpt+1)) xjpt = - xjpt + xpt (nf, ipt) = xipt + xpt (nf, jpt) = xjpt + end if + ! + ! Calculate the next value of F, label 70 being reached immediately + ! after this calculation. The least function value so far and its index + ! are required. + ! + do j = 1, n + x (j) = xpt (nf, j) + xbase (j) + end do + go to 310 +70 fval (nf) = f + if (nf == 1) then + fbeg = f + fopt = f + kopt = 1 + else if (f < fopt) then + fopt = f + kopt = nf + end if + ! + ! Set the nonzero initial elements of BMAT and the quadratic model in + ! the cases when NF is at most 2*N+1. + ! + if (nfm <= 2*n) then + if (nfm >= 1 .and. nfm <= n) then + gq (nfm) = (f-fbeg) / rhobeg + if (npt < nf+n) then + bmat (1, nfm) = - one / rhobeg + bmat (nf, nfm) = one / rhobeg + bmat (npt+nfm, nfm) = - half * rhosq + end if + else if (nfm > n) then + bmat (nf-n, nfmm) = half / rhobeg + bmat (nf, nfmm) = - half / rhobeg + zmat (1, nfmm) = - reciq - reciq + zmat (nf-n, nfmm) = reciq + zmat (nf, nfmm) = reciq + ih = (nfmm*(nfmm+1)) / 2 + temp = (fbeg-f) / rhobeg + hq (ih) = (gq(nfmm)-temp) / rhobeg + gq (nfmm) = half * (gq(nfmm)+temp) + end if + ! + ! Set the off-diagonal second derivatives of the Lagrange functions and + ! the initial quadratic model. + ! + else + ih = (ipt*(ipt-1)) / 2 + jpt + if (xipt < zero) ipt = ipt + n + if (xjpt < zero) jpt = jpt + n + zmat (1, nfmm) = recip + zmat (nf, nfmm) = recip + zmat (ipt+1, nfmm) = - recip + zmat (jpt+1, nfmm) = - recip + hq (ih) = (fbeg-fval(ipt+1)-fval(jpt+1)+f) / (xipt*xjpt) + end if + if (nf < npt) go to 50 + ! + ! Begin the iterative procedure, because the initial model is complete. + ! + rho = rhobeg + delta = rho + idz = 1 + diffa = zero + diffb = zero + itest = 0 + xoptsq = zero + do i = 1, n + xopt (i) = xpt (kopt, i) + xoptsq = xoptsq + xopt (i) ** 2 + end do +90 nfsav = nf + ! + ! Generate the next trust region step and test its length. Set KNEW + ! to -1 if the purpose of the next F will be to improve the model. + ! +100 knew = 0 + call trsapp (n, npt, xopt, xpt, gq, hq, pq, delta, d, w, w(np), w(np+n), & + w(np+2*n), crvmin) + dsq = zero + do i = 1, n + dsq = dsq + d (i) ** 2 + end do + dnorm = min (delta, sqrt(dsq)) + if (dnorm < half*rho) then + knew = - 1 + delta = tenth * delta + ratio = - 1.0_dp + if (delta <= 1.5_dp*rho) delta = rho + if (nf <= nfsav+2) go to 460 + temp = 0.125_dp * crvmin * rho * rho + if (temp <= max(diffa, diffb, diffc)) go to 460 + go to 490 + end if + ! + ! Shift XBASE if XOPT may be too far from XBASE. First make the changes + ! to BMAT that do not depend on ZMAT. + ! +120 if (dsq <= 1.0e-3_dp*xoptsq) then + tempq = 0.25_dp * xoptsq + do k = 1, npt + sum = zero + do i = 1, n + sum = sum + xpt (k, i) * xopt (i) + end do + temp = pq (k) * sum + sum = sum - half * xoptsq + w (npt+k) = sum + do i = 1, n + gq (i) = gq (i) + temp * xpt (k, i) + xpt (k, i) = xpt (k, i) - half * xopt (i) + vlag (i) = bmat (k, i) + w (i) = sum * xpt (k, i) + tempq * xopt (i) + ip = npt + i + do j = 1, i + bmat (ip, j) = bmat (ip, j) + vlag (i) * w (j) + w (i) * vlag (j) + end do + end do + end do + ! + ! Then the revisions of BMAT that depend on ZMAT are calculated. + ! + do k = 1, nptm + sumz = zero + do i = 1, npt + sumz = sumz + zmat (i, k) + w (i) = w (npt+i) * zmat (i, k) + end do + do j = 1, n + sum = tempq * sumz * xopt (j) + do i = 1, npt + sum = sum + w (i) * xpt (i, j) + end do + vlag (j) = sum + if (k < idz) sum = - sum + do i = 1, npt + bmat (i, j) = bmat (i, j) + sum * zmat (i, k) + end do + end do + do i = 1, n + ip = i + npt + temp = vlag (i) + if (k < idz) temp = - temp + do j = 1, i + bmat (ip, j) = bmat (ip, j) + temp * vlag (j) + end do + end do + end do + ! + ! The following instructions complete the shift of XBASE, including + ! the changes to the parameters of the quadratic model. + ! + ih = 0 + do j = 1, n + w (j) = zero + do k = 1, npt + w (j) = w (j) + pq (k) * xpt (k, j) + xpt (k, j) = xpt (k, j) - half * xopt (j) + end do + do i = 1, j + ih = ih + 1 + if (i < j) gq (j) = gq (j) + hq (ih) * xopt (i) + gq (i) = gq (i) + hq (ih) * xopt (j) + hq (ih) = hq (ih) + w (i) * xopt (j) + xopt (i) * w (j) + bmat (npt+i, j) = bmat (npt+j, i) + end do + end do + do j = 1, n + xbase (j) = xbase (j) + xopt (j) + xopt (j) = zero + end do + xoptsq = zero + end if + ! + ! Pick the model step if KNEW is positive. A different choice of D + ! may be made later, if the choice of D by BIGLAG causes substantial + ! cancellation in DENOM. + ! + if (knew > 0) then + call biglag (n, npt, xopt, xpt, bmat, zmat, idz, ndim, knew, dstep, d, alpha, & + vlag, vlag(npt+1), w, w(np), w(np+n)) + end if + ! + ! Calculate VLAG and BETA for the current choice of D. The first NPT + ! components of W_check will be held in W. + ! + do k = 1, npt + suma = zero + sumb = zero + sum = zero + do j = 1, n + suma = suma + xpt (k, j) * d (j) + sumb = sumb + xpt (k, j) * xopt (j) + sum = sum + bmat (k, j) * d (j) + end do + w (k) = suma * (half*suma+sumb) + vlag (k) = sum + end do + beta = zero + do k = 1, nptm + sum = zero + do i = 1, npt + sum = sum + zmat (i, k) * w (i) + end do + if (k < idz) then + beta = beta + sum * sum + sum = - sum + else + beta = beta - sum * sum + end if + do i = 1, npt + vlag (i) = vlag (i) + sum * zmat (i, k) + end do + end do + bsum = zero + dx = zero + do j = 1, n + sum = zero + do i = 1, npt + sum = sum + w (i) * bmat (i, j) + end do + bsum = bsum + sum * d (j) + jp = npt + j + do k = 1, n + sum = sum + bmat (jp, k) * d (k) + end do + vlag (jp) = sum + bsum = bsum + sum * d (j) + dx = dx + d (j) * xopt (j) + end do + beta = dx * dx + dsq * (xoptsq+dx+dx+half*dsq) + beta - bsum + vlag (kopt) = vlag (kopt) + one + ! + ! If KNEW is positive and if the cancellation in DENOM is unacceptable, + ! then BIGDEN calculates an alternative model step, XNEW being used for + ! working space. + ! + if (knew > 0) then + temp = one + alpha * beta / vlag (knew) ** 2 + if (abs(temp) <= 0.8_dp) then + call bigden (n, npt, xopt, xpt, bmat, zmat, idz, ndim, kopt, knew, d, w, & + vlag, beta, xnew, w(ndim+1), w(6*ndim+1)) + end if + end if + ! + ! Calculate the next value of the objective function. + ! +290 do i = 1, n + xnew (i) = xopt (i) + d (i) + x (i) = xbase (i) + xnew (i) + end do + nf = nf + 1 +310 if (nf > nftest) then + nf = nf - 1 + if (iprint > 0) print 320 +320 format (/ 4 x, 'Return from NEWUOA because CALFUN has been',& + ' called MAXFUN times.') + newuob = .false. + go to 530 + end if + f = this%funkk(this%obj, X(1:N)) + if (iprint == 3) then + print 330, nf, f, (x(i), i=1, n) +330 format (/ 4 x, 'Function number', i6, ' F =', 1 pd18.10,& + ' The corresponding X is:' / (2 x, 5d15.6)) + end if + if (nf <= npt) go to 70 + if (knew ==-1) go to 530 + ! + ! Use the quadratic model to predict the change in F due to the step D, + ! and set DIFF to the error of this prediction. + ! + vquad = zero + ih = 0 + do j = 1, n + vquad = vquad + d (j) * gq (j) + do i = 1, j + ih = ih + 1 + temp = d (i) * xnew (j) + d (j) * xopt (i) + if (i == j) temp = half * temp + vquad = vquad + temp * hq (ih) + end do + end do + do k = 1, npt + vquad = vquad + pq (k) * w (k) + end do + diff = f - fopt - vquad + diffc = diffb + diffb = diffa + diffa = abs (diff) + if (dnorm > rho) nfsav = nf + ! + ! Update FOPT and XOPT if the new F is the least value of the objective + ! function so far. The branch when KNEW is positive occurs if D is not + ! a trust region step. + ! + fsave = fopt + if (f < fopt) then + fopt = f + xoptsq = zero + do i = 1, n + xopt (i) = xnew (i) + xoptsq = xoptsq + xopt (i) ** 2 + end do + end if + ksave = knew + if (knew > 0) go to 410 + ! + ! Pick the next value of DELTA after a trust region step. + ! + if (vquad >= zero) then + if (iprint > 0) print 370 +370 format (/ 4 x, 'Return from NEWUOA because a trust',& + ' region step has failed to reduce Q.') + newuob = .false. + go to 530 + end if + ratio = (f-fsave) / vquad + if (ratio <= tenth) then + delta = half * dnorm + else if (ratio <= 0.7_dp) then + delta = max (half*delta, dnorm) + else + delta = max (half*delta, dnorm+dnorm) + end if + if (delta <= 1.5_dp*rho) delta = rho + ! + ! Set KNEW to the index of the next interpolation point to be deleted. + ! + rhosq = max (tenth*delta, rho) ** 2 + ktemp = 0 + detrat = zero + if (f >= fsave) then + ktemp = kopt + detrat = one + end if + do k = 1, npt + hdiag = zero + do j = 1, nptm + temp = one + if (j < idz) temp = - one + hdiag = hdiag + temp * zmat (k, j) ** 2 + end do + temp = abs (beta*hdiag+vlag(k)**2) + distsq = zero + do j = 1, n + distsq = distsq + (xpt(k, j)-xopt(j)) ** 2 + end do + if (distsq > rhosq) temp = temp * (distsq/rhosq) ** 3 + if (temp > detrat .and. k /= ktemp) then + detrat = temp + knew = k + end if + end do + if (knew == 0) go to 460 + ! + ! Update BMAT, ZMAT and IDZ, so that the KNEW-th interpolation point + ! can be moved. Begin the updating of the quadratic model, starting + ! with the explicit second derivative term. + ! +410 call update_newuoa (n, npt, bmat, zmat, idz, ndim, vlag, beta, knew, w) + fval (knew) = f + ih = 0 + do i = 1, n + temp = pq (knew) * xpt (knew, i) + do j = 1, i + ih = ih + 1 + hq (ih) = hq (ih) + temp * xpt (knew, j) + end do + end do + pq (knew) = zero + ! + ! Update the other second derivative parameters, and then the gradient + ! vector of the model. Also include the new interpolation point. + ! + do j = 1, nptm + temp = diff * zmat (knew, j) + if (j < idz) temp = - temp + do k = 1, npt + pq (k) = pq (k) + temp * zmat (k, j) + end do + end do + gqsq = zero + do i = 1, n + gq (i) = gq (i) + diff * bmat (knew, i) + gqsq = gqsq + gq (i) ** 2 + xpt (knew, i) = xnew (i) + end do + ! + ! If a trust region step makes a small change to the objective function, + ! then calculate the gradient of the least Frobenius norm interpolant at + ! XBASE, and store it in W, using VLAG for a vector of right hand sides. + ! + if (ksave == 0 .and. delta == rho) then + if (abs(ratio) > 1.0e-2_dp) then + itest = 0 + else + do k = 1, npt + vlag (k) = fval (k) - fval (kopt) + end do + gisq = zero + do i = 1, n + sum = zero + do k = 1, npt + sum = sum + bmat (k, i) * vlag (k) + end do + gisq = gisq + sum * sum + w (i) = sum + end do + ! + ! Test whether to replace the new quadratic model by the least Frobenius + ! norm interpolant, making the replacement if the test is satisfied. + ! + itest = itest + 1 + if (gqsq < 100.0_dp*gisq) itest = 0 + if (itest >= 3) then + do i = 1, n + gq (i) = w (i) + end do + do ih = 1, nh + hq (ih) = zero + end do + do j = 1, nptm + w (j) = zero + do k = 1, npt + w (j) = w (j) + vlag (k) * zmat (k, j) + end do + if (j < idz) w (j) = - w (j) + end do + do k = 1, npt + pq (k) = zero + do j = 1, nptm + pq (k) = pq (k) + zmat (k, j) * w (j) + end do + end do + itest = 0 + end if + end if + end if + if (f < fsave) kopt = knew + ! + ! If a trust region step has provided a sufficient decrease in F, then + ! branch for another trust region calculation. The case KSAVE>0 occurs + ! when the new function value was calculated by a model step. + ! + if (f <= fsave+tenth*vquad) go to 100 + if (ksave > 0) go to 100 + ! + ! Alternatively, find out if the interpolation points are close enough + ! to the best point so far. + ! + knew = 0 +460 distsq = 4.0_dp * delta * delta + do k = 1, npt + sum = zero + do j = 1, n + sum = sum + (xpt(k, j)-xopt(j)) ** 2 + end do + if (sum > distsq) then + knew = k + distsq = sum + end if + end do + ! + ! If KNEW is positive, then set DSTEP, and branch back for the next + ! iteration, which will generate a "model step". + ! + if (knew > 0) then + dstep = max (min(tenth*sqrt(distsq), half*delta), rho) + dsq = dstep * dstep + go to 120 + end if + if (ratio > zero) go to 100 + if (max(delta, dnorm) > rho) go to 100 + ! + ! The calculations with the current value of RHO are complete. Pick the + ! next values of RHO and DELTA. + ! +490 if (rho > rhoend) then + delta = half * rho + ratio = rho / rhoend + if (ratio <= 16.0_dp) then + rho = rhoend + else if (ratio <= 250.0_dp) then + rho = sqrt (ratio) * rhoend + else + rho = tenth * rho + end if + delta = max (delta, rho) + this%Last_bestfit = fopt + if (iprint >= 2) then + if (iprint >= 3) print 500 +500 format (5 x) + print 510, rho, nf +510 format (/ 4 x, 'New RHO =', 1 pd11.4, 5 x, 'Number of',& + ' function values =', i6) + print 520, fopt, (xbase(i)+xopt(i), i=1, n) +520 format (4 x, 'Least value of F =', 1 pd23.15, 9 x,& + 'The corresponding X is:'/(2 x, 5d15.6)) + end if + go to 90 + end if + ! + ! Return from the calculation, after another Newton-Raphson step, if + ! it is too short to have been tried before. + ! + if (knew ==-1) go to 290 +530 if (fopt <= f) then + do i = 1, n + x (i) = xbase (i) + xopt (i) + end do + f = fopt + end if + if (iprint >= 1) then + print 550, nf +550 format (/ 4 x, 'At the return from NEWUOA', 5 x,& + 'Number of function values =', i6) + print 520, f, (x(i), i=1, n) + end if + + end function newuob + + subroutine bigden (n, npt, xopt, xpt, bmat, zmat, idz, ndim, kopt, knew, d, w, vlag, & + beta, s, wvec, prod) + real(dp) alpha , angle , Beta , dd , denmax , denold , densav , diff , ds , dstemp , & + dtest , half , one , quart + real(dp) ss , ssden , sstemp , step , sum , sumold , tau , & + & temp , tempa , tempb , tempc , two , twopi , xoptd , xopts + real(dp) xoptsq , zero + INTEGER i , Idz , ip , isave , iterc , iu , j , jc , k , Knew , & + & Kopt , ksav , N , Ndim , Npt , nptm , nw + real(dp) xopt (*), xpt (npt,*), bmat (ndim,*), zmat (npt,*), d (*), w (*), vlag & + (*), s (*), wvec (ndim,*), prod (ndim,*) + real(dp) den (9), denex (9), par (9) + ! + ! N is the number of variables. + ! NPT is the number of interpolation equations. + ! XOPT is the best interpolation point so far. + ! XPT contains the coordinates of the current interpolation points. + ! BMAT provides the last N columns of H. + ! ZMAT and IDZ give a factorization of the first NPT by NPT submatrix of H. + ! NDIM is the first dimension of BMAT and has the value NPT+N. + ! KOPT is the index of the optimal interpolation point. + ! KNEW is the index of the interpolation point that is going to be moved. + ! D will be set to the step from XOPT to the new point, and on entry it + ! should be the D that was calculated by the last call of BIGLAG. The + ! length of the initial D provides a trust region bound on the final D. + ! W will be set to Wcheck for the final choice of D. + ! VLAG will be set to Theta*Wcheck+e_b for the final choice of D. + ! BETA will be set to the value that will occur in the updating formula + ! when the KNEW-th interpolation point is moved to its new position. + ! S, WVEC, PROD and the private arrays DEN, DENEX and PAR will be used + ! for working space. + ! + ! D is calculated in a way that should provide a denominator with a large + ! modulus in the updating formula when the KNEW-th interpolation point is + ! shifted to the new position XOPT+D. + ! + ! Set some constants. + ! + half = 0.5_dp + one = 1.0_dp + quart = 0.25_dp + two = 2.0_dp + zero = 0.0_dp + twopi = 8.0_dp * atan (one) + nptm = npt - n - 1 + ! + ! Store the first NPT elements of the KNEW-th column of H in W(N+1) + ! to W(N+NPT). + ! + do k = 1, npt + w (n+k) = zero + end do + do j = 1, nptm + temp = zmat (knew, j) + if (j < idz) temp = - temp + do k = 1, npt + w (n+k) = w (n+k) + temp * zmat (k, j) + end do + end do + alpha = w (n+knew) + ! + ! The initial search direction D is taken from the last call of BIGLAG, + ! and the initial S is set below, usually to the direction from X_OPT + ! to X_KNEW, but a different direction to an interpolation point may + ! be chosen, in order to prevent S from being nearly parallel to D. + ! + dd = zero + ds = zero + ss = zero + xoptsq = zero + do i = 1, n + dd = dd + d (i) ** 2 + s (i) = xpt (knew, i) - xopt (i) + ds = ds + d (i) * s (i) + ss = ss + s (i) ** 2 + xoptsq = xoptsq + xopt (i) ** 2 + end do + if (ds*ds > 0.99_dp*dd*ss) then + ksav = knew + dtest = ds * ds / ss + do k = 1, npt + if (k /= kopt) then + dstemp = zero + sstemp = zero + do i = 1, n + diff = xpt (k, i) - xopt (i) + dstemp = dstemp + d (i) * diff + sstemp = sstemp + diff * diff + end do + if (dstemp*dstemp/sstemp < dtest) then + ksav = k + dtest = dstemp * dstemp / sstemp + ds = dstemp + ss = sstemp + end if + end if + end do + do i = 1, n + s (i) = xpt (ksav, i) - xopt (i) + end do + end if + ssden = dd * ss - ds * ds + iterc = 0 + densav = zero + ! + ! Begin the iteration by overwriting S with a vector that has the + ! required length and direction. + ! +70 iterc = iterc + 1 + temp = one / sqrt (ssden) + xoptd = zero + xopts = zero + do i = 1, n + s (i) = temp * (dd*s(i)-ds*d(i)) + xoptd = xoptd + xopt (i) * d (i) + xopts = xopts + xopt (i) * s (i) + end do + ! + ! Set the coefficients of the first two terms of BETA. + ! + tempa = half * xoptd * xoptd + tempb = half * xopts * xopts + den (1) = dd * (xoptsq+half*dd) + tempa + tempb + den (2) = two * xoptd * dd + den (3) = two * xopts * dd + den (4) = tempa - tempb + den (5) = xoptd * xopts + do i = 6, 9 + den (i) = zero + end do + ! + ! Put the coefficients of Wcheck in WVEC. + ! + do k = 1, npt + tempa = zero + tempb = zero + tempc = zero + do i = 1, n + tempa = tempa + xpt (k, i) * d (i) + tempb = tempb + xpt (k, i) * s (i) + tempc = tempc + xpt (k, i) * xopt (i) + end do + wvec (k, 1) = quart * (tempa*tempa+tempb*tempb) + wvec (k, 2) = tempa * tempc + wvec (k, 3) = tempb * tempc + wvec (k, 4) = quart * (tempa*tempa-tempb*tempb) + wvec (k, 5) = half * tempa * tempb + end do + do i = 1, n + ip = i + npt + wvec (ip, 1) = zero + wvec (ip, 2) = d (i) + wvec (ip, 3) = s (i) + wvec (ip, 4) = zero + wvec (ip, 5) = zero + end do + ! + ! Put the coefficents of THETA*Wcheck in PROD. + ! + do jc = 1, 5 + nw = npt + if (jc == 2 .or. jc == 3) nw = ndim + do k = 1, npt + prod (k, jc) = zero + end do + do j = 1, nptm + sum = zero + do k = 1, npt + sum = sum + zmat (k, j) * wvec (k, jc) + end do + if (j < idz) sum = - sum + do k = 1, npt + prod (k, jc) = prod (k, jc) + sum * zmat (k, j) + end do + end do + if (nw == ndim) then + do k = 1, npt + sum = zero + do j = 1, n + sum = sum + bmat (k, j) * wvec (npt+j, jc) + end do + prod (k, jc) = prod (k, jc) + sum + end do + end if + do j = 1, n + sum = zero + do i = 1, nw + sum = sum + bmat (i, j) * wvec (i, jc) + end do + prod (npt+j, jc) = sum + end do + end do + ! + ! Include in DEN the part of BETA that depends on THETA. + ! + do k = 1, ndim + sum = zero + do i = 1, 5 + par (i) = half * prod (k, i) * wvec (k, i) + sum = sum + par (i) + end do + den (1) = den (1) - par (1) - sum + tempa = prod (k, 1) * wvec (k, 2) + prod (k, 2) * wvec (k, 1) + tempb = prod (k, 2) * wvec (k, 4) + prod (k, 4) * wvec (k, 2) + tempc = prod (k, 3) * wvec (k, 5) + prod (k, 5) * wvec (k, 3) + den (2) = den (2) - tempa - half * (tempb+tempc) + den (6) = den (6) - half * (tempb-tempc) + tempa = prod (k, 1) * wvec (k, 3) + prod (k, 3) * wvec (k, 1) + tempb = prod (k, 2) * wvec (k, 5) + prod (k, 5) * wvec (k, 2) + tempc = prod (k, 3) * wvec (k, 4) + prod (k, 4) * wvec (k, 3) + den (3) = den (3) - tempa - half * (tempb-tempc) + den (7) = den (7) - half * (tempb+tempc) + tempa = prod (k, 1) * wvec (k, 4) + prod (k, 4) * wvec (k, 1) + den (4) = den (4) - tempa - par (2) + par (3) + tempa = prod (k, 1) * wvec (k, 5) + prod (k, 5) * wvec (k, 1) + tempb = prod (k, 2) * wvec (k, 3) + prod (k, 3) * wvec (k, 2) + den (5) = den (5) - tempa - half * tempb + den (8) = den (8) - par (4) + par (5) + tempa = prod (k, 4) * wvec (k, 5) + prod (k, 5) * wvec (k, 4) + den (9) = den (9) - half * tempa + end do + ! + ! Extend DEN so that it holds all the coefficients of DENOM. + ! + sum = zero + do i = 1, 5 + par (i) = half * prod (knew, i) ** 2 + sum = sum + par (i) + end do + denex (1) = alpha * den (1) + par (1) + sum + tempa = two * prod (knew, 1) * prod (knew, 2) + tempb = prod (knew, 2) * prod (knew, 4) + tempc = prod (knew, 3) * prod (knew, 5) + denex (2) = alpha * den (2) + tempa + tempb + tempc + denex (6) = alpha * den (6) + tempb - tempc + tempa = two * prod (knew, 1) * prod (knew, 3) + tempb = prod (knew, 2) * prod (knew, 5) + tempc = prod (knew, 3) * prod (knew, 4) + denex (3) = alpha * den (3) + tempa + tempb - tempc + denex (7) = alpha * den (7) + tempb + tempc + tempa = two * prod (knew, 1) * prod (knew, 4) + denex (4) = alpha * den (4) + tempa + par (2) - par (3) + tempa = two * prod (knew, 1) * prod (knew, 5) + denex (5) = alpha * den (5) + tempa + prod (knew, 2) * prod (knew, 3) + denex (8) = alpha * den (8) + par (4) - par (5) + denex (9) = alpha * den (9) + prod (knew, 4) * prod (knew, 5) + ! + ! Seek the value of the angle that maximizes the modulus of DENOM. + ! + sum = denex (1) + denex (2) + denex (4) + denex (6) + denex (8) + denold = sum + denmax = sum + isave = 0 + iu = 49 + temp = twopi / real (iu+1, dp) + par (1) = one + do i = 1, iu + angle = real (i, dp) * temp + par (2) = cos (angle) + par (3) = sin (angle) + do j = 4, 8, 2 + par (j) = par (2) * par (j-2) - par (3) * par (j-1) + par (j+1) = par (2) * par (j-1) + par (3) * par (j-2) + end do + sumold = sum + sum = zero + do j = 1, 9 + sum = sum + denex (j) * par (j) + end do + if (abs(sum) > abs(denmax)) then + denmax = sum + isave = i + tempa = sumold + else if (i == isave+1) then + tempb = sum + end if + end do + if (isave == 0) tempa = sum + if (isave == iu) tempb = denold + step = zero + if (tempa /= tempb) then + tempa = tempa - denmax + tempb = tempb - denmax + step = half * (tempa-tempb) / (tempa+tempb) + end if + angle = temp * (real(isave, dp)+step) + ! + ! Calculate the new parameters of the denominator, the new VLAG vector + ! and the new D. Then test for convergence. + ! + par (2) = cos (angle) + par (3) = sin (angle) + do j = 4, 8, 2 + par (j) = par (2) * par (j-2) - par (3) * par (j-1) + par (j+1) = par (2) * par (j-1) + par (3) * par (j-2) + end do + beta = zero + denmax = zero + do j = 1, 9 + beta = beta + den (j) * par (j) + denmax = denmax + denex (j) * par (j) + end do + do k = 1, ndim + vlag (k) = zero + do j = 1, 5 + vlag (k) = vlag (k) + prod (k, j) * par (j) + end do + end do + tau = vlag (knew) + dd = zero + tempa = zero + tempb = zero + do i = 1, n + d (i) = par (2) * d (i) + par (3) * s (i) + w (i) = xopt (i) + d (i) + dd = dd + d (i) ** 2 + tempa = tempa + d (i) * w (i) + tempb = tempb + w (i) * w (i) + end do + if (iterc >= n) go to 340 + if (iterc > 1) densav = max (densav, denold) + if (abs(denmax) <= 1.1_dp*abs(densav)) go to 340 + densav = denmax + ! + ! Set S to half the gradient of the denominator with respect to D. + ! Then branch for the next iteration. + ! + do i = 1, n + temp = tempa * xopt (i) + tempb * d (i) - vlag (npt+i) + s (i) = tau * bmat (knew, i) + alpha * temp + end do + do k = 1, npt + sum = zero + do j = 1, n + sum = sum + xpt (k, j) * w (j) + end do + temp = (tau*w(n+k)-alpha*vlag(k)) * sum + do i = 1, n + s (i) = s (i) + temp * xpt (k, i) + end do + end do + ss = zero + ds = zero + do i = 1, n + ss = ss + s (i) ** 2 + ds = ds + d (i) * s (i) + end do + ssden = dd * ss - ds * ds + if (ssden >= 1.0e-8_dp*dd*ss) go to 70 + ! + ! Set the vector W before the RETURN from the subroutine. + ! +340 do k = 1, ndim + w (k) = zero + do j = 1, 5 + w (k) = w (k) + wvec (k, j) * par (j) + end do + end do + vlag (kopt) = vlag (kopt) + one + + end subroutine bigden + + subroutine biglag (n, npt, xopt, xpt, bmat, zmat, idz, ndim, knew, delta, d, alpha, & + hcol, gc, gd, s, w) + + real(dp) Alpha , angle , cf1 , cf2 , cf3 , cf4 , cf5 , cth , & + & dd , delsq , Delta , denom , dhd , gg , half + INTEGER i , Idz , isave , iterc , iu , j , k , Knew , N , Ndim , & + & Npt , nptm + real(dp) one , scale , sp , ss , step , sth , sum , tau , & + & taubeg , taumax , tauold , temp , tempa , tempb , twopi , zero + real(dp) xopt (*), xpt (npt,*), bmat (ndim,*), zmat (npt,*), d (*), hcol (*), & + gc(*), gd (*), s (*), w (*) + ! + ! N is the number of variables. + ! NPT is the number of interpolation equations. + ! XOPT is the best interpolation point so far. + ! XPT contains the coordinates of the current interpolation points. + ! BMAT provides the last N columns of H. + ! ZMAT and IDZ give a factorization of the first NPT by NPT submatrix of H. + ! NDIM is the first dimension of BMAT and has the value NPT+N. + ! KNEW is the index of the interpolation point that is going to be moved. + ! DELTA is the current trust region bound. + ! D will be set to the step from XOPT to the new point. + ! ALPHA will be set to the KNEW-th diagonal element of the H matrix. + ! HCOL, GC, GD, S and W will be used for working space. + ! + ! The step D is calculated in a way that attempts to maximize the modulus + ! of LFUNC(XOPT+D), subject to the bound ||D|| .LE. DELTA, where LFUNC is + ! the KNEW-th Lagrange function. + ! + ! Set some constants. + ! + half = 0.5_dp + one = 1.0_dp + zero = 0.0_dp + twopi = 8.0_dp * atan (one) + delsq = delta * delta + nptm = npt - n - 1 + ! + ! Set the first NPT components of HCOL to the leading elements of the + ! KNEW-th column of H. + ! + iterc = 0 + do k = 1, npt + hcol (k) = zero + end do + do j = 1, nptm + temp = zmat (knew, j) + if (j < idz) temp = - temp + do k = 1, npt + hcol (k) = hcol (k) + temp * zmat (k, j) + end do + end do + alpha = hcol (knew) + ! + ! Set the unscaled initial direction D. Form the gradient of LFUNC at + ! XOPT, and multiply D by the second derivative matrix of LFUNC. + ! + dd = zero + do i = 1, n + d (i) = xpt (knew, i) - xopt (i) + gc (i) = bmat (knew, i) + gd (i) = zero + dd = dd + d (i) ** 2 + end do + do k = 1, npt + temp = zero + sum = zero + do j = 1, n + temp = temp + xpt (k, j) * xopt (j) + sum = sum + xpt (k, j) * d (j) + end do + temp = hcol (k) * temp + sum = hcol (k) * sum + do i = 1, n + gc (i) = gc (i) + temp * xpt (k, i) + gd (i) = gd (i) + sum * xpt (k, i) + end do + end do + ! + ! Scale D and GD, with a sign change if required. Set S to another + ! vector in the initial two dimensional subspace. + ! + gg = zero + sp = zero + dhd = zero + do i = 1, n + gg = gg + gc (i) ** 2 + sp = sp + d (i) * gc (i) + dhd = dhd + d (i) * gd (i) + end do + scale = delta / sqrt (dd) + if (sp*dhd < zero) scale = - scale + temp = zero + if (sp*sp > 0.99_dp*dd*gg) temp = one + tau = scale * (abs(sp)+half*scale*abs(dhd)) + if (gg*delsq < 0.01_dp*tau*tau) temp = one + do i = 1, n + d (i) = scale * d (i) + gd (i) = scale * gd (i) + s (i) = gc (i) + temp * gd (i) + end do + ! + ! Begin the iteration by overwriting S with a vector that has the + ! required length and direction, except that termination occurs if + ! the given D and S are nearly parallel. + ! +80 iterc = iterc + 1 + dd = zero + sp = zero + ss = zero + do i = 1, n + dd = dd + d (i) ** 2 + sp = sp + d (i) * s (i) + ss = ss + s (i) ** 2 + end do + temp = dd * ss - sp * sp + if (temp <= 1.0e-8_dp*dd*ss) go to 160 + denom = sqrt (temp) + do i = 1, n + s (i) = (dd*s(i)-sp*d(i)) / denom + w (i) = zero + end do + ! + ! Calculate the coefficients of the objective function on the circle, + ! beginning with the multiplication of S by the second derivative matrix. + ! + do k = 1, npt + sum = zero + do j = 1, n + sum = sum + xpt (k, j) * s (j) + end do + sum = hcol (k) * sum + do i = 1, n + w (i) = w (i) + sum * xpt (k, i) + end do + end do + cf1 = zero + cf2 = zero + cf3 = zero + cf4 = zero + cf5 = zero + do i = 1, n + cf1 = cf1 + s (i) * w (i) + cf2 = cf2 + d (i) * gc (i) + cf3 = cf3 + s (i) * gc (i) + cf4 = cf4 + d (i) * gd (i) + cf5 = cf5 + s (i) * gd (i) + end do + cf1 = half * cf1 + cf4 = half * cf4 - cf1 + ! + ! Seek the value of the angle that maximizes the modulus of TAU. + ! + taubeg = cf1 + cf2 + cf4 + taumax = taubeg + tauold = taubeg + isave = 0 + iu = 49 + temp = twopi / real (iu+1, dp) + do i = 1, iu + angle = real (i, dp) * temp + cth = cos (angle) + sth = sin (angle) + tau = cf1 + (cf2+cf4*cth) * cth + (cf3+cf5*cth) * sth + if (abs(tau) > abs(taumax)) then + taumax = tau + isave = i + tempa = tauold + else if (i == isave+1) then + tempb = tau + end if + tauold = tau + end do + if (isave == 0) tempa = tau + if (isave == iu) tempb = taubeg + step = zero + if (tempa /= tempb) then + tempa = tempa - taumax + tempb = tempb - taumax + step = half * (tempa-tempb) / (tempa+tempb) + end if + angle = temp * (real(isave, dp)+step) + ! + ! Calculate the new D and GD. Then test for convergence. + ! + cth = cos (angle) + sth = sin (angle) + tau = cf1 + (cf2+cf4*cth) * cth + (cf3+cf5*cth) * sth + do i = 1, n + d (i) = cth * d (i) + sth * s (i) + gd (i) = cth * gd (i) + sth * w (i) + s (i) = gc (i) + gd (i) + end do + if (abs(tau) <= 1.1_dp*abs(taubeg)) go to 160 + if (iterc < n) go to 80 +160 return + end subroutine biglag + + subroutine trsapp (n, npt, xopt, xpt, gq, hq, pq, delta, step, d, g, hd, hs, crvmin) + real(dp) alpha , angle , angtest , bstep , cf , Crvmin , cth , & + & dd , delsq , Delta , dg , dhd , dhs , ds , gg , ggbeg ,& + & ggsav + real(dp) half , qadd , qbeg , qmin , qnew , qred , qsav , ratio , reduc , & + sg , sgk , shs , ss , sth , temp, tempa , tempb , twopi , zero + INTEGER i , ih , isave , iterc , itermax , itersw , iu , j , k , & + & N , Npt + real(dp) xopt (*), xpt (npt,*), gq (*), hq (*), pq (*), step (*), d (*), g (*), & + hd (*), hs (*) + ! + ! N is the number of variables of a quadratic objective function, Q say. + ! The arguments NPT, XOPT, XPT, GQ, HQ and PQ have their usual meanings, + ! in order to define the current quadratic model Q. + ! DELTA is the trust region radius, and has to be positive. + ! STEP will be set to the calculated trial step. + ! The arrays D, G, HD and HS will be used for working space. + ! CRVMIN will be set to the least curvature of H along the conjugate + ! directions that occur, except that it is set to zero if STEP goes + ! all the way to the trust region boundary. + ! + ! The calculation of STEP begins with the truncated conjugate gradient + ! method. If the boundary of the trust region is reached, then further + ! changes to STEP may be made, each one being in the 2D space spanned + ! by the current STEP and the corresponding gradient of Q. Thus STEP + ! should provide a substantial reduction to Q within the trust region. + ! + ! Initialization, which includes setting HD to H times XOPT. + ! + half = 0.5_dp + zero = 0.0_dp + twopi = 8.0_dp * atan (1.0_dp) + delsq = delta * delta + iterc = 0 + itermax = n + itersw = itermax + do i = 1, n + d (i) = xopt (i) + end do + go to 170 + ! + ! Prepare for the first line search. + ! +20 qred = zero + dd = zero + do i = 1, n + step (i) = zero + hs (i) = zero + g (i) = gq (i) + hd (i) + d (i) = - g (i) + dd = dd + d (i) ** 2 + end do + crvmin = zero + if (dd == zero) go to 160 + ds = zero + ss = zero + gg = dd + ggbeg = gg + ! + ! Calculate the step to the trust region boundary and the product HD. + ! +40 iterc = iterc + 1 + temp = delsq - ss + bstep = temp / (ds+sqrt(ds*ds+dd*temp)) + go to 170 +50 dhd = zero + do j = 1, n + dhd = dhd + d (j) * hd (j) + end do + ! + ! Update CRVMIN and set the step-length ALPHA. + ! + alpha = bstep + if (dhd > zero) then + temp = dhd / dd + if (iterc == 1) crvmin = temp + crvmin = min (crvmin, temp) + alpha = min (alpha, gg/dhd) + end if + qadd = alpha * (gg-half*alpha*dhd) + qred = qred + qadd + ! + ! Update STEP and HS. + ! + ggsav = gg + gg = zero + do i = 1, n + step (i) = step (i) + alpha * d (i) + hs (i) = hs (i) + alpha * hd (i) + gg = gg + (g(i)+hs(i)) ** 2 + end do + ! + ! Begin another conjugate direction iteration if required. + ! + if (alpha < bstep) then + if (qadd <= 0.01_dp*qred) go to 160 + if (gg <= 1.0e-4_dp*ggbeg) go to 160 + if (iterc == itermax) go to 160 + temp = gg / ggsav + dd = zero + ds = zero + ss = zero + do i = 1, n + d (i) = temp * d (i) - g (i) - hs (i) + dd = dd + d (i) ** 2 + ds = ds + d (i) * step (i) + ss = ss + step (i) ** 2 + end do + if (ds <= zero) go to 160 + if (ss < delsq) go to 40 + end if + crvmin = zero + itersw = iterc + ! + ! Test whether an alternative iteration is required. + ! +90 if (gg <= 1.0e-4_dp*ggbeg) go to 160 + sg = zero + shs = zero + do i = 1, n + sg = sg + step (i) * g (i) + shs = shs + step (i) * hs (i) + end do + sgk = sg + shs + angtest = sgk / sqrt (gg*delsq) + if (angtest <=-0.99_dp) go to 160 + ! + ! Begin the alternative iteration by calculating D and HD and some + ! scalar products. + ! + iterc = iterc + 1 + temp = sqrt (delsq*gg-sgk*sgk) + tempa = delsq / temp + tempb = sgk / temp + do i = 1, n + d (i) = tempa * (g(i)+hs(i)) - tempb * step (i) + end do + go to 170 +120 dg = zero + dhd = zero + dhs = zero + do i = 1, n + dg = dg + d (i) * g (i) + dhd = dhd + hd (i) * d (i) + dhs = dhs + hd (i) * step (i) + end do + ! + ! Seek the value of the angle that minimizes Q. + ! + cf = half * (shs-dhd) + qbeg = sg + cf + qsav = qbeg + qmin = qbeg + isave = 0 + iu = 49 + temp = twopi / real (iu+1, dp) + do i = 1, iu + angle = real (i, dp) * temp + cth = cos (angle) + sth = sin (angle) + qnew = (sg+cf*cth) * cth + (dg+dhs*cth) * sth + if (qnew < qmin) then + qmin = qnew + isave = i + tempa = qsav + else if (i == isave+1) then + tempb = qnew + end if + qsav = qnew + end do + if (isave == zero) tempa = qnew + if (isave == iu) tempb = qbeg + angle = zero + if (tempa /= tempb) then + tempa = tempa - qmin + tempb = tempb - qmin + angle = half * (tempa-tempb) / (tempa+tempb) + end if + angle = temp * (real(isave, dp)+angle) + ! + ! Calculate the new STEP and HS. Then test for convergence. + ! + cth = cos (angle) + sth = sin (angle) + reduc = qbeg - (sg+cf*cth) * cth - (dg+dhs*cth) * sth + gg = zero + do i = 1, n + step (i) = cth * step (i) + sth * d (i) + hs (i) = cth * hs (i) + sth * hd (i) + gg = gg + (g(i)+hs(i)) ** 2 + end do + qred = qred + reduc + ratio = reduc / qred + if (iterc < itermax .and. ratio > 0.01_dp) go to 90 +160 return + ! + ! The following instructions act as a subroutine for setting the vector + ! HD to the vector D multiplied by the second derivative matrix of Q. + ! They are called from three different places, which are distinguished + ! by the value of ITERC. + ! +170 do i = 1, n + hd (i) = zero + end do + do k = 1, npt + temp = zero + do j = 1, n + temp = temp + xpt (k, j) * d (j) + end do + temp = temp * pq (k) + do i = 1, n + hd (i) = hd (i) + temp * xpt (k, i) + end do + end do + ih = 0 + do j = 1, n + do i = 1, j + ih = ih + 1 + if (i < j) hd (j) = hd (j) + hq (ih) * d (i) + hd (i) = hd (i) + hq (ih) * d (j) + end do + end do + if (iterc == 0) go to 20 + if (iterc <= itersw) go to 50 + go to 120 + end subroutine trsapp + + subroutine update_newuoa (n, npt, bmat, zmat, idz, ndim, vlag, beta, knew, w) + real(dp) alpha , Beta , denom , one , scala , scalb , tau , & + & tausq , temp , tempa , tempb , zero + INTEGER i , Idz , iflag , j , ja , jb , jl , jp , Knew , N , & + & Ndim , Npt , nptm + real(dp) bmat (ndim,*), zmat (npt,*), vlag (*), w (*) + ! + ! The arrays BMAT and ZMAT with IDZ are updated, in order to shift the + ! interpolation point that has index KNEW. On entry, VLAG contains the + ! components of the vector Theta*Wcheck+e_b of the updating formula + ! (6.11), and BETA holds the value of the parameter that has this name. + ! The vector W is used for working space. + ! + ! Set some constants. + ! + one = 1.0_dp + zero = 0.0_dp + nptm = npt - n - 1 + ! + ! Apply the rotations that put zeros in the KNEW-th row of ZMAT. + ! + jl = 1 + do j = 2, nptm + if (j == idz) then + jl = idz + else if (zmat(knew, j) /= zero) then + temp = sqrt (zmat(knew, jl)**2+zmat(knew, j)**2) + tempa = zmat (knew, jl) / temp + tempb = zmat (knew, j) / temp + do i = 1, npt + temp = tempa * zmat (i, jl) + tempb * zmat (i, j) + zmat (i, j) = tempa * zmat (i, j) - tempb * zmat (i, jl) + zmat (i, jl) = temp + end do + zmat (knew, j) = zero + end if + end do + ! + ! Put the first NPT components of the KNEW-th column of HLAG into W, + ! and calculate the parameters of the updating formula. + ! + tempa = zmat (knew, 1) + if (idz >= 2) tempa = - tempa + if (jl > 1) tempb = zmat (knew, jl) + do i = 1, npt + w (i) = tempa * zmat (i, 1) + if (jl > 1) w (i) = w (i) + tempb * zmat (i, jl) + end do + alpha = w (knew) + tau = vlag (knew) + tausq = tau * tau + denom = alpha * beta + tausq + vlag (knew) = vlag (knew) - one + ! + ! Complete the updating of ZMAT when there is only one nonzero element + ! in the KNEW-th row of the new matrix ZMAT, but, if IFLAG is set to one, + ! then the first column of ZMAT will be exchanged with another one later. + ! + iflag = 0 + if (jl == 1) then + temp = sqrt (abs(denom)) + tempb = tempa / temp + tempa = tau / temp + do i = 1, npt + zmat (i, 1) = tempa * zmat (i, 1) - tempb * vlag (i) + end do + if (idz == 1 .and. temp < zero) idz = 2 + if (idz >= 2 .and. temp >= zero) iflag = 1 + else + ! + ! Complete the updating of ZMAT in the alternative case. + ! + ja = 1 + if (beta >= zero) ja = jl + jb = jl + 1 - ja + temp = zmat (knew, jb) / denom + tempa = temp * beta + tempb = temp * tau + temp = zmat (knew, ja) + scala = one / sqrt (abs(beta)*temp*temp+tausq) + scalb = scala * sqrt (abs(denom)) + do i = 1, npt + zmat (i, ja) = scala * (tau*zmat(i, ja)-temp*vlag(i)) + zmat (i, jb) = scalb * (zmat(i, jb)-tempa*w(i)-tempb*vlag(i)) + end do + if (denom <= zero) then + if (beta < zero) idz = idz + 1 + if (beta >= zero) iflag = 1 + end if + end if + ! + ! IDZ is reduced in the following case, and usually the first column + ! of ZMAT is exchanged with a later one. + ! + if (iflag == 1) then + idz = idz - 1 + do i = 1, npt + temp = zmat (i, 1) + zmat (i, 1) = zmat (i, idz) + zmat (i, idz) = temp + end do + end if + ! + ! Finally, update the matrix BMAT. + ! + do j = 1, n + jp = npt + j + w (jp) = bmat (knew, j) + tempa = (alpha*vlag(jp)-tau*w(jp)) / denom + tempb = (-beta*w(jp)-tau*vlag(jp)) / denom + do i = 1, jp + bmat (i, j) = bmat (i, j) + tempa * vlag (i) + tempb * w (i) + if (i > npt) bmat (jp, i-npt) = bmat (i, j) + end do + end do + + end subroutine update_newuoa + END MODULE Powell diff --git a/fortran/VisualStudio/CAMBLib.vfproj b/fortran/VisualStudio/CAMBLib.vfproj index 1fafe9ec..6e4fab92 100644 --- a/fortran/VisualStudio/CAMBLib.vfproj +++ b/fortran/VisualStudio/CAMBLib.vfproj @@ -43,7 +43,7 @@ - +