// Solve solves non-linear problem f(x) == 0
func (o *NlSolver) Solve(x []float64, silent bool) (err error) {
// compute scaling vector
la.VecScaleAbs(o.scal, o.atol, o.rtol, x) // scal := Atol + Rtol*abs(x)
// evaluate function @ x
err = o.Ffcn(o.fx, x) // fx := f(x)
o.NFeval, o.NJeval = 1, 0
if err != nil {
return
}
// show message
if !silent {
o.msg("", 0, 0, 0, true, false)
}
// iterations
var Ldx, Ldx_prev, Θ float64 // RMS norm of delta x, convergence rate
var fx_max float64
var nfv int
for o.It = 0; o.It < o.MaxIt; o.It++ {
// check convergence on f(x)
fx_max = la.VecLargest(o.fx, 1.0) // den = 1.0
if fx_max < o.ftol {
if !silent {
o.msg("fx_max(ini)", o.It, Ldx, fx_max, false, true)
}
break
}
// show message
if !silent {
o.msg("", o.It, Ldx, fx_max, false, false)
}
// output
if o.Out != nil {
o.Out(x)
}
// evaluate Jacobian @ x
if o.It == 0 || !o.CteJac {
if o.useDn {
err = o.JfcnDn(o.J, x)
} else {
if o.numJ {
err = Jacobian(&o.Jtri, o.Ffcn, x, o.fx, o.w, false)
o.NFeval += o.neq
} else {
err = o.JfcnSp(&o.Jtri, x)
}
}
o.NJeval += 1
if err != nil {
return
}
}
// dense solution
if o.useDn {
// invert matrix
err = la.MatInvG(o.Ji, o.J, 1e-10)
if err != nil {
return chk.Err(_nls_err1, err.Error())
}
// solve linear system (compute mdx) and compute lin-search data
o.φ = 0.0
for i := 0; i < o.neq; i++ {
o.mdx[i], o.dφdx[i] = 0.0, 0.0
for j := 0; j < o.neq; j++ {
o.mdx[i] += o.Ji[i][j] * o.fx[j] // mdx = inv(J) * fx
o.dφdx[i] += o.J[j][i] * o.fx[j] // dφdx = tra(J) * fx
}
o.φ += o.fx[i] * o.fx[i]
}
o.φ *= 0.5
// sparse solution
} else {
// init sparse solver
if o.It == 0 {
symmetric, verbose, timing := false, false, false
err := o.lis.InitR(&o.Jtri, symmetric, verbose, timing)
if err != nil {
return chk.Err(_nls_err9, err.Error())
}
}
// factorisation (must be done for all iterations)
o.lis.Fact()
// solve linear system => compute mdx
o.lis.SolveR(o.mdx, o.fx, false) // mdx = inv(J) * fx false => !sumToRoot
// compute lin-search data
if o.Lsearch {
o.φ = 0.5 * la.VecDot(o.fx, o.fx)
la.SpTriMatTrVecMul(o.dφdx, &o.Jtri, o.fx) // dφdx := transpose(J) * fx
}
}
//io.Pforan("φ = %v\n", o.φ)
//io.Pforan("dφdx = %v\n", o.dφdx)
// update x
Ldx = 0.0
for i := 0; i < o.neq; i++ {
o.x0[i] = x[i]
x[i] -= o.mdx[i]
Ldx += (o.mdx[i] / o.scal[i]) * (o.mdx[i] / o.scal[i])
}
Ldx = math.Sqrt(Ldx / float64(o.neq))
// calculate fx := f(x) @ update x
err = o.Ffcn(o.fx, x)
o.NFeval += 1
if err != nil {
return
}
// check convergence on f(x) => avoid line-search if converged already
fx_max = la.VecLargest(o.fx, 1.0) // den = 1.0
if fx_max < o.ftol {
if !silent {
o.msg("fx_max", o.It, Ldx, fx_max, false, true)
}
break
}
// check convergence on Ldx
if Ldx < o.fnewt {
if !silent {
o.msg("Ldx", o.It, Ldx, fx_max, false, true)
}
break
}
// call line-search => update x and fx
if o.Lsearch {
nfv, err = LineSearch(x, o.fx, o.Ffcn, o.mdx, o.x0, o.dφdx, o.φ, o.LsMaxIt, true)
o.NFeval += nfv
if err != nil {
return chk.Err(_nls_err2, err.Error())
}
Ldx = 0.0
for i := 0; i < o.neq; i++ {
Ldx += ((x[i] - o.x0[i]) / o.scal[i]) * ((x[i] - o.x0[i]) / o.scal[i])
}
Ldx = math.Sqrt(Ldx / float64(o.neq))
fx_max = la.VecLargest(o.fx, 1.0) // den = 1.0
if Ldx < o.fnewt {
if !silent {
o.msg("Ldx(linsrch)", o.It, Ldx, fx_max, false, true)
}
break
}
}
// check convergence rate
if o.It > 0 && o.ChkConv {
Θ = Ldx / Ldx_prev
if Θ > 0.99 {
return chk.Err(_nls_err3, Θ, Ldx, Ldx_prev)
}
}
Ldx_prev = Ldx
}
// output
if o.Out != nil {
o.Out(x)
}
// check convergence
if o.It == o.MaxIt {
err = chk.Err(_nls_err4, o.It)
}
return
}
// Solve solves from (xa,ya) to (xb,yb) => find yb (stored in y)
func (o *ODE) Solve(y []float64, x, xb, Δx float64, fixstp bool, args ...interface{}) (err error) {
// check
if xb < x {
err = chk.Err(_ode_err3, xb, x)
return
}
// derived variables
o.fnewt = max(10.0*o.ϵ/o.Rtol, min(0.03, math.Sqrt(o.Rtol)))
// initial step size
Δx = min(Δx, xb-x)
if fixstp {
o.h = Δx
} else {
o.h = min(Δx, o.IniH)
}
o.hprev = o.h
// output initial state
if o.out != nil {
o.out(true, o.h, x, y, args...)
}
// stat variables
o.nfeval = 0
o.njeval = 0
o.nsteps = 0
o.naccepted = 0
o.nrejected = 0
o.ndecomp = 0
o.nlinsol = 0
o.nitmax = 0
// control variables
o.doinit = true
o.first = true
o.last = false
o.reject = false
o.diverg = false
o.dvfac = 0
o.η = 1.0
o.jacIsOK = false
o.reuseJdec = false
o.reuseJ = false
o.nit = 0
o.hopt = o.h
o.θ = o.θmax
// local error indicator
var rerr float64
// linear solver
lsname := "umfpack"
if o.Distr {
lsname = "mumps"
}
o.lsolR = la.GetSolver(lsname)
o.lsolC = la.GetSolver(lsname)
// clean up and show stat before leaving
defer func() {
o.lsolR.Clean()
o.lsolC.Clean()
if !o.silent {
o.Stat()
}
}()
// first scaling variable
la.VecScaleAbs(o.scal, o.Atol, o.Rtol, y) // o.scal := o.Atol + o.Rtol * abs(y)
// fixed steps
if fixstp {
la.VecCopy(o.w[0], 1, y) // copy initial values to worksapce
if o.Verbose {
io.Pfgreen("x = %v\n", x)
}
for x < xb {
//if x + o.h > xb { o.h = xb - x }
if o.jac == nil { // numerical Jacobian
if o.method == "Radau5" {
o.nfeval += 1
o.fcn(o.f0, x, y, args...)
}
}
o.reuseJdec = false
o.reuseJ = false
o.jacIsOK = false
o.step(o, y, x, args...)
o.nsteps += 1
o.doinit = false
o.first = false
o.hprev = o.h
x += o.h
o.accept(o, y)
if o.out != nil {
o.out(false, o.h, x, y, args...)
}
if o.Verbose {
io.Pfgreen("x = %v\n", x)
}
}
return
}
// first function evaluation
o.nfeval += 1
o.fcn(o.f0, x, y, args...) // o.f0 := f(x,y)
// time loop
var dxmax, xstep, fac, div, dxnew, facgus, old_h, old_rerr float64
var dxratio float64
var failed bool
for x < xb {
dxmax, xstep = Δx, x+Δx
failed = false
for iss := 0; iss < o.NmaxSS+1; iss++ {
// total number of substeps
o.nsteps += 1
// error: did not converge
if iss == o.NmaxSS {
failed = true
break
}
// converged?
if x-xstep >= 0.0 {
break
}
// step update
rerr, err = o.step(o, y, x, args...)
// initialise only once
o.doinit = false
// iterations diverging ?
if o.diverg {
o.diverg = false
o.reject = true
o.last = false
o.h = o.dvfac * o.h
continue
}
// step size change
fac = min(o.Mfac, o.Mfac*float64(1+2*o.NmaxIt)/float64(o.nit+2*o.NmaxIt))
div = max(o.Mmin, min(o.Mmax, math.Pow(rerr, 0.25)/fac))
dxnew = o.h / div
// accepted
if rerr < 1.0 {
// set flags
o.naccepted += 1
o.first = false
o.jacIsOK = false
// update x and y
o.hprev = o.h
x += o.h
o.accept(o, y)
// output
if o.out != nil {
o.out(false, o.h, x, y, args...)
}
// converged ?
if o.last {
o.hopt = o.h // optimal h
break
}
// predictive controller of Gustafsson
if o.PredCtrl {
if o.naccepted > 1 {
facgus = (old_h / o.h) * math.Pow(math.Pow(rerr, 2.0)/old_rerr, 0.25) / o.Mfac
facgus = max(o.Mmin, min(o.Mmax, facgus))
div = max(div, facgus)
dxnew = o.h / div
}
old_h = o.h
old_rerr = max(1.0e-2, rerr)
}
// calc new scal and f0
la.VecScaleAbs(o.scal, o.Atol, o.Rtol, y) // o.scal := o.Atol + o.Rtol * abs(y)
o.nfeval += 1
o.fcn(o.f0, x, y, args...) // o.f0 := f(x,y)
// new step size
dxnew = min(dxnew, dxmax)
if o.reject { // do not alow o.h to grow if previous was a reject
dxnew = min(o.h, dxnew)
}
o.reject = false
// do not reuse current Jacobian and decomposition by default
o.reuseJdec = false
// last step ?
if x+dxnew-xstep >= 0.0 {
o.last = true
o.h = xstep - x
} else {
dxratio = dxnew / o.h
o.reuseJdec = (o.θ <= o.θmax && dxratio >= o.C1h && dxratio <= o.C2h)
if !o.reuseJdec {
o.h = dxnew
}
}
// check θ to decide if at least the Jacobian can be reused
if !o.reuseJdec {
o.reuseJ = (o.θ <= o.θmax)
}
// rejected
} else {
// set flags
if o.naccepted > 0 {
o.nrejected += 1
}
o.reject = true
o.last = false
// new step size
if o.first {
o.h = 0.1 * o.h
} else {
o.h = dxnew
}
// last step
if x+o.h > xstep {
o.h = xstep - x
}
}
}
// sub-stepping failed
if failed {
err = chk.Err(_ode_err2, o.NmaxSS)
break
}
}
return
}