func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
if mpi.Rank() == 0 {
chk.PrintTitle("Test ODE 02b")
io.Pfcyan("Hairer-Wanner VII-p5 Eq.(1.5) Van der Pol's Equation (MPI)\n")
}
if mpi.Size() != 2 {
chk.Panic(">> error: this test requires 2 MPI processors\n")
return
}
eps := 1.0e-6
w := make([]float64, 2) // workspace
fcn := func(f []float64, x float64, y []float64, args ...interface{}) error {
f[0], f[1] = 0, 0
switch mpi.Rank() {
case 0:
f[0] = y[1]
case 1:
f[1] = ((1.0-y[0]*y[0])*y[1] - y[0]) / eps
}
// join all f
mpi.AllReduceSum(f, w)
return nil
}
jac := func(dfdy *la.Triplet, x float64, y []float64, args ...interface{}) error {
if dfdy.Max() == 0 {
dfdy.Init(2, 2, 4)
}
dfdy.Start()
if false { // per column
switch mpi.Rank() {
case 0:
dfdy.Put(0, 0, 0.0)
dfdy.Put(1, 0, (-2.0*y[0]*y[1]-1.0)/eps)
case 1:
dfdy.Put(0, 1, 1.0)
dfdy.Put(1, 1, (1.0-y[0]*y[0])/eps)
}
} else { // per row
switch mpi.Rank() {
case 0:
dfdy.Put(0, 0, 0.0)
dfdy.Put(0, 1, 1.0)
case 1:
dfdy.Put(1, 0, (-2.0*y[0]*y[1]-1.0)/eps)
dfdy.Put(1, 1, (1.0-y[0]*y[0])/eps)
}
}
return nil
}
// data
silent := false
fixstp := false
//method := "Dopri5"
method := "Radau5"
xa, xb := 0.0, 2.0
ya := []float64{2.0, -0.6}
ndim := len(ya)
// output
var b bytes.Buffer
out := func(first bool, dx, x float64, y []float64, args ...interface{}) error {
if mpi.Rank() == 0 {
if first {
fmt.Fprintf(&b, "%23s %23s %23s %23s\n", "dx", "x", "y0", "y1")
}
fmt.Fprintf(&b, "%23.15E %23.15E %23.15E %23.15E\n", dx, x, y[0], y[1])
}
return nil
}
defer func() {
if mpi.Rank() == 0 {
extra := "d2 = Read('data/vdpol_radau5_for.dat')\n" +
"subplot(3,1,1)\n" +
"plot(d2['x'],d2['y0'],'k+',label='res',ms=10)\n" +
"subplot(3,1,2)\n" +
"plot(d2['x'],d2['y1'],'k+',label='res',ms=10)\n"
ode.Plot("/tmp/gosl", "vdpolB", method, &b, []int{0, 1}, ndim, nil, xa, xb, true, false, extra)
}
}()
// one run
var o ode.ODE
o.Distr = true
//numjac := true
numjac := false
if numjac {
o.Init(method, ndim, fcn, nil, nil, out, silent)
} else {
o.Init(method, ndim, fcn, jac, nil, out, silent)
}
// tolerances and initial step size
rtol := 1e-4
atol := rtol
o.SetTol(atol, rtol)
o.IniH = 1.0e-4
//o.NmaxSS = 2
y := make([]float64, ndim)
copy(y, ya)
t0 := time.Now()
if fixstp {
o.Solve(y, xa, xb, 0.05, fixstp)
} else {
o.Solve(y, xa, xb, xb-xa, fixstp)
}
if mpi.Rank() == 0 {
io.Pfmag("elapsed time = %v\n", time.Now().Sub(t0))
}
}
// run_iterations solves the nonlinear problem
func run_iterations(t, Δt float64, d *Domain, sum *Summary) (diverging, ok bool) {
// zero accumulated increments
la.VecFill(d.Sol.ΔY, 0)
// calculate global starred vectors and interpolate starred variables from nodes to integration points
if LogErr(d.star_vars(Δt), "cannot compute starred variables") {
return
}
// auxiliary variables
var it int
var largFb, largFb0, Lδu float64
var prevFb, prevLδu float64
// message
if Global.Sim.Data.ShowR {
io.Pf("\n%13s%4s%23s%23s\n", "t", "it", "largFb", "Lδu")
defer func() {
io.Pf("%13.6e%4d%23.15e%23.15e\n", t, it, largFb, Lδu)
}()
}
// iterations
for it = 0; it < Global.Sim.Solver.NmaxIt; it++ {
// assemble right-hand side vector (fb) with negative of residuals
la.VecFill(d.Fb, 0)
for _, e := range d.Elems {
if !e.AddToRhs(d.Fb, d.Sol) {
break
}
}
if Stop() {
return
}
// join all fb
if Global.Distr {
mpi.AllReduceSum(d.Fb, d.Wb) // this must be done here because there might be nodes sharing boundary conditions
}
// point natural boundary conditions; e.g. concentrated loads
d.PtNatBcs.AddToRhs(d.Fb, t)
// essential boundary conditioins; e.g. constraints
d.EssenBcs.AddToRhs(d.Fb, d.Sol)
// debug
if Global.Debug {
//la.PrintVec("fb", d.Fb[:d.Ny], "%13.10f ", false)
//panic("stop")
}
// find largest absolute component of fb
largFb = la.VecLargest(d.Fb, 1)
// save residual
if Global.Stat {
sum.Resids.Append(it == 0, largFb)
}
// check largFb value
if it == 0 {
// store largest absolute component of fb
largFb0 = largFb
} else {
// check convergence on Lf0
if largFb < Global.Sim.Solver.FbTol*largFb0 { // converged on fb
break
}
// check convergence on fb_min
if largFb < Global.Sim.Solver.FbMin { // converged with smallest value of fb
break
}
}
// check divergence on fb
if it > 1 && Global.Sim.Solver.DvgCtrl {
if largFb > prevFb {
diverging = true
break
}
}
prevFb = largFb
// assemble Jacobian matrix
do_asm_fact := (it == 0 || !Global.Sim.Data.CteTg)
if do_asm_fact {
// assemble element matrices
d.Kb.Start()
for _, e := range d.Elems {
if !e.AddToKb(d.Kb, d.Sol, it == 0) {
break
}
}
if Stop() {
return
}
// debug
if Global.DebugKb != nil {
Global.DebugKb(d, it)
}
// join A and tr(A) matrices into Kb
if Global.Root {
d.Kb.PutMatAndMatT(&d.EssenBcs.A)
}
// initialise linear solver
if d.InitLSol {
if LogErr(d.LinSol.InitR(d.Kb, Global.Sim.LinSol.Symmetric, Global.Sim.LinSol.Verbose, Global.Sim.LinSol.Timing), "cannot initialise linear solver") {
return
}
d.InitLSol = false
}
// perform factorisation
LogErr(d.LinSol.Fact(), "factorisation")
if Stop() {
return
}
}
// debug
//KK := d.Kb.ToMatrix(nil).ToDense()
//la.PrintMat("KK", KK, "%20.10f", false)
//panic("stop")
// solve for wb := δyb
LogErr(d.LinSol.SolveR(d.Wb, d.Fb, false), "solve")
if Stop() {
return
}
// debug
if Global.Debug {
//la.PrintVec("wb", d.Wb[:d.Ny], "%13.10f ", false)
}
// update primary variables (y)
for i := 0; i < d.Ny; i++ {
d.Sol.Y[i] += d.Wb[i] // y += δy
d.Sol.ΔY[i] += d.Wb[i] // ΔY += δy
}
if !Global.Sim.Data.Steady {
for _, I := range d.T1eqs {
d.Sol.Dydt[I] = Global.DynCoefs.β1*d.Sol.Y[I] - d.Sol.Psi[I]
}
for _, I := range d.T2eqs {
d.Sol.Dydt[I] = Global.DynCoefs.α4*d.Sol.Y[I] - d.Sol.Chi[I]
d.Sol.D2ydt2[I] = Global.DynCoefs.α1*d.Sol.Y[I] - d.Sol.Zet[I]
}
}
// update Lagrange multipliers (λ)
for i := 0; i < d.Nlam; i++ {
d.Sol.L[i] += d.Wb[d.Ny+i] // λ += δλ
}
// backup / restore
if it == 0 {
// create backup copy of all secondary variables
for _, e := range d.ElemIntvars {
e.BackupIvs(false)
}
} else {
// recover last converged state from backup copy
for _, e := range d.ElemIntvars {
e.RestoreIvs(false)
}
}
// update secondary variables
for _, e := range d.Elems {
if !e.Update(d.Sol) {
break
}
}
if Stop() {
return
}
// compute RMS norm of δu and check convegence on δu
Lδu = la.VecRmsErr(d.Wb[:d.Ny], Global.Sim.Solver.Atol, Global.Sim.Solver.Rtol, d.Sol.Y[:d.Ny])
// message
if Global.Sim.Data.ShowR {
io.Pf("%13.6e%4d%23.15e%23.15e\n", t, it, largFb, Lδu)
}
// stop if converged on δu
if Lδu < Global.Sim.Solver.Itol {
break
}
// check divergence on Lδu
if it > 1 && Global.Sim.Solver.DvgCtrl {
if Lδu > prevLδu {
diverging = true
break
}
}
prevLδu = Lδu
}
// check if iterations diverged
if it == Global.Sim.Solver.NmaxIt {
io.PfMag("max number of iterations reached: it = %d\n", it)
return
}
// success
ok = true
return
}
// run_iterations solves the nonlinear problem
func run_iterations(t, Δt float64, d *Domain, dc *DynCoefs, sum *Summary, dbgKb DebugKb_t) (diverging bool, err error) {
// zero accumulated increments
la.VecFill(d.Sol.ΔY, 0)
// calculate global starred vectors and interpolate starred variables from nodes to integration points
if !d.Sim.Data.Steady {
// compute starred vectors
for _, I := range d.T1eqs {
d.Sol.Psi[I] = dc.β1*d.Sol.Y[I] + dc.β2*d.Sol.Dydt[I]
}
for _, I := range d.T2eqs {
d.Sol.Zet[I] = dc.α1*d.Sol.Y[I] + dc.α2*d.Sol.Dydt[I] + dc.α3*d.Sol.D2ydt2[I]
d.Sol.Chi[I] = dc.α4*d.Sol.Y[I] + dc.α5*d.Sol.Dydt[I] + dc.α6*d.Sol.D2ydt2[I]
}
// set internal starred variables
for _, e := range d.Elems {
err = e.InterpStarVars(d.Sol)
if err != nil {
err = chk.Err("cannot compute starred variables:\n%v", err)
return
}
}
}
// auxiliary variables
var it int
var largFb, largFb0, Lδu float64
var prevFb, prevLδu float64
dat := d.Sim.Solver
// message
if dat.ShowR {
io.Pf("\n%13s%4s%23s%23s\n", "t", "it", "largFb", "Lδu")
defer func() {
io.Pf("%13.6e%4d%23.15e%23.15e\n", t, it, largFb, Lδu)
}()
}
// iterations
for it = 0; it < dat.NmaxIt; it++ {
// assemble right-hand side vector (fb) with negative of residuals
la.VecFill(d.Fb, 0)
for _, e := range d.Elems {
err = e.AddToRhs(d.Fb, d.Sol)
if err != nil {
return
}
}
// join all fb
if d.Distr {
mpi.AllReduceSum(d.Fb, d.Wb) // this must be done here because there might be nodes sharing boundary conditions
}
// point natural boundary conditions; e.g. concentrated loads
d.PtNatBcs.AddToRhs(d.Fb, t)
// essential boundary conditioins; e.g. constraints
d.EssenBcs.AddToRhs(d.Fb, d.Sol)
// find largest absolute component of fb
largFb = la.VecLargest(d.Fb, 1)
// save residual
if d.Sim.Data.Stat {
if sum != nil {
sum.Resids.Append(it == 0, largFb)
}
}
// check largFb value
if it == 0 {
// store largest absolute component of fb
largFb0 = largFb
} else {
// check convergence on Lf0
if largFb < dat.FbTol*largFb0 { // converged on fb
break
}
// check convergence on fb_min
if largFb < dat.FbMin { // converged with smallest value of fb
break
}
}
// check divergence on fb
if it > 1 && dat.DvgCtrl {
if largFb > prevFb {
diverging = true
break
}
}
prevFb = largFb
// assemble Jacobian matrix
do_asm_fact := (it == 0 || !dat.CteTg)
if do_asm_fact {
// assemble element matrices
d.Kb.Start()
for _, e := range d.Elems {
err = e.AddToKb(d.Kb, d.Sol, it == 0)
if err != nil {
return
}
}
// debug
if dbgKb != nil {
dbgKb(d, it)
}
// join A and tr(A) matrices into Kb
if d.Proc == 0 {
d.Kb.PutMatAndMatT(&d.EssenBcs.A)
}
// initialise linear solver
if d.InitLSol {
err = d.LinSol.InitR(d.Kb, d.Sim.LinSol.Symmetric, d.Sim.LinSol.Verbose, d.Sim.LinSol.Timing)
if err != nil {
err = chk.Err("cannot initialise linear solver:\n%v", err)
return
}
d.InitLSol = false
}
// perform factorisation
err = d.LinSol.Fact()
if err != nil {
err = chk.Err("factorisation failed:\n%v", err)
return
}
}
// solve for wb := δyb
err = d.LinSol.SolveR(d.Wb, d.Fb, false)
if err != nil {
err = chk.Err("solve failed:%v\n", err)
return
}
// update primary variables (y)
for i := 0; i < d.Ny; i++ {
d.Sol.Y[i] += d.Wb[i] // y += δy
d.Sol.ΔY[i] += d.Wb[i] // ΔY += δy
}
if !d.Sim.Data.Steady {
for _, I := range d.T1eqs {
d.Sol.Dydt[I] = dc.β1*d.Sol.Y[I] - d.Sol.Psi[I]
}
for _, I := range d.T2eqs {
d.Sol.Dydt[I] = dc.α4*d.Sol.Y[I] - d.Sol.Chi[I]
d.Sol.D2ydt2[I] = dc.α1*d.Sol.Y[I] - d.Sol.Zet[I]
}
}
// update Lagrange multipliers (λ)
for i := 0; i < d.Nlam; i++ {
d.Sol.L[i] += d.Wb[d.Ny+i] // λ += δλ
}
// backup / restore
if it == 0 {
// create backup copy of all secondary variables
for _, e := range d.ElemIntvars {
e.BackupIvs(false)
}
} else {
// recover last converged state from backup copy
for _, e := range d.ElemIntvars {
e.RestoreIvs(false)
}
}
// update secondary variables
for _, e := range d.Elems {
err = e.Update(d.Sol)
if err != nil {
break
}
}
// compute RMS norm of δu and check convegence on δu
Lδu = la.VecRmsErr(d.Wb[:d.Ny], dat.Atol, dat.Rtol, d.Sol.Y[:d.Ny])
// message
if dat.ShowR {
io.Pf("%13.6e%4d%23.15e%23.15e\n", t, it, largFb, Lδu)
}
// stop if converged on δu
if Lδu < dat.Itol {
break
}
// check divergence on Lδu
if it > 1 && dat.DvgCtrl {
if Lδu > prevLδu {
diverging = true
break
}
}
prevLδu = Lδu
}
// check if iterations diverged
if it == dat.NmaxIt {
err = chk.Err("max number of iterations reached: it = %d\n", it)
}
return
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
if mpi.Rank() == 0 {
chk.PrintTitle("Test ODE 04b (MPI)")
io.Pfcyan("Hairer-Wanner VII-p376 Transistor Amplifier (MPI)\n")
io.Pfcyan("(from E Hairer's website, not the system in the book)\n")
}
if mpi.Size() != 3 {
chk.Panic(">> error: this test requires 3 MPI processors\n")
return
}
// RIGHT-HAND SIDE OF THE AMPLIFIER PROBLEM
w := make([]float64, 8) // workspace
fcn := func(f []float64, x float64, y []float64, args ...interface{}) error {
d := args[0].(*HWtransData)
UET := d.UE * math.Sin(d.W*x)
FAC1 := d.BETA * (math.Exp((y[3]-y[2])/d.UF) - 1.0)
FAC2 := d.BETA * (math.Exp((y[6]-y[5])/d.UF) - 1.0)
la.VecFill(f, 0)
switch mpi.Rank() {
case 0:
f[0] = y[0] / d.R9
case 1:
f[1] = (y[1]-d.UB)/d.R8 + d.ALPHA*FAC1
f[2] = y[2]/d.R7 - FAC1
case 2:
f[3] = y[3]/d.R5 + (y[3]-d.UB)/d.R6 + (1.0-d.ALPHA)*FAC1
f[4] = (y[4]-d.UB)/d.R4 + d.ALPHA*FAC2
f[5] = y[5]/d.R3 - FAC2
f[6] = y[6]/d.R1 + (y[6]-d.UB)/d.R2 + (1.0-d.ALPHA)*FAC2
f[7] = (y[7] - UET) / d.R0
}
mpi.AllReduceSum(f, w)
return nil
}
// JACOBIAN OF THE AMPLIFIER PROBLEM
jac := func(dfdy *la.Triplet, x float64, y []float64, args ...interface{}) error {
d := args[0].(*HWtransData)
FAC14 := d.BETA * math.Exp((y[3]-y[2])/d.UF) / d.UF
FAC27 := d.BETA * math.Exp((y[6]-y[5])/d.UF) / d.UF
if dfdy.Max() == 0 {
dfdy.Init(8, 8, 16)
}
NU := 2
dfdy.Start()
switch mpi.Rank() {
case 0:
dfdy.Put(2+0-NU, 0, 1.0/d.R9)
dfdy.Put(2+1-NU, 1, 1.0/d.R8)
dfdy.Put(1+2-NU, 2, -d.ALPHA*FAC14)
dfdy.Put(0+3-NU, 3, d.ALPHA*FAC14)
dfdy.Put(2+2-NU, 2, 1.0/d.R7+FAC14)
case 1:
dfdy.Put(1+3-NU, 3, -FAC14)
dfdy.Put(2+3-NU, 3, 1.0/d.R5+1.0/d.R6+(1.0-d.ALPHA)*FAC14)
dfdy.Put(3+2-NU, 2, -(1.0-d.ALPHA)*FAC14)
dfdy.Put(2+4-NU, 4, 1.0/d.R4)
dfdy.Put(1+5-NU, 5, -d.ALPHA*FAC27)
case 2:
dfdy.Put(0+6-NU, 6, d.ALPHA*FAC27)
dfdy.Put(2+5-NU, 5, 1.0/d.R3+FAC27)
dfdy.Put(1+6-NU, 6, -FAC27)
dfdy.Put(2+6-NU, 6, 1.0/d.R1+1.0/d.R2+(1.0-d.ALPHA)*FAC27)
dfdy.Put(3+5-NU, 5, -(1.0-d.ALPHA)*FAC27)
dfdy.Put(2+7-NU, 7, 1.0/d.R0)
}
return nil
}
// MATRIX "M"
c1, c2, c3, c4, c5 := 1.0e-6, 2.0e-6, 3.0e-6, 4.0e-6, 5.0e-6
var M la.Triplet
M.Init(8, 8, 14)
M.Start()
NU := 1
switch mpi.Rank() {
case 0:
M.Put(1+0-NU, 0, -c5)
M.Put(0+1-NU, 1, c5)
M.Put(2+0-NU, 0, c5)
M.Put(1+1-NU, 1, -c5)
M.Put(1+2-NU, 2, -c4)
M.Put(1+3-NU, 3, -c3)
case 1:
M.Put(0+4-NU, 4, c3)
M.Put(2+3-NU, 3, c3)
M.Put(1+4-NU, 4, -c3)
case 2:
M.Put(1+5-NU, 5, -c2)
M.Put(1+6-NU, 6, -c1)
M.Put(0+7-NU, 7, c1)
M.Put(2+6-NU, 6, c1)
M.Put(1+7-NU, 7, -c1)
}
// WRITE FILE FUNCTION
idxstp := 1
var b bytes.Buffer
out := func(first bool, dx, x float64, y []float64, args ...interface{}) error {
if mpi.Rank() == 0 {
if first {
fmt.Fprintf(&b, "%6s%23s%23s%23s%23s%23s%23s%23s%23s%23s\n", "ns", "x", "y0", "y1", "y2", "y3", "y4", "y5", "y6", "y7")
}
fmt.Fprintf(&b, "%6d%23.15E", idxstp, x)
for j := 0; j < len(y); j++ {
fmt.Fprintf(&b, "%23.15E", y[j])
}
fmt.Fprintf(&b, "\n")
idxstp += 1
}
return nil
}
defer func() {
if mpi.Rank() == 0 {
io.WriteFileD("/tmp/gosl", "hwamplifierB.res", &b)
}
}()
// INITIAL DATA
D, xa, xb, ya := HWtransIni()
// SET ODE SOLVER
silent := false
fixstp := false
//method := "Dopri5"
method := "Radau5"
ndim := len(ya)
//numjac := true
numjac := false
var osol ode.ODE
osol.Pll = true
if numjac {
osol.Init(method, ndim, fcn, nil, &M, out, silent)
} else {
osol.Init(method, ndim, fcn, jac, &M, out, silent)
}
osol.IniH = 1.0e-6 // initial step size
// SET TOLERANCES
atol, rtol := 1e-11, 1e-5
osol.SetTol(atol, rtol)
// RUN
t0 := time.Now()
if fixstp {
osol.Solve(ya, xa, xb, 0.01, fixstp, &D)
} else {
osol.Solve(ya, xa, xb, xb-xa, fixstp, &D)
}
if mpi.Rank() == 0 {
io.Pfmag("elapsed time = %v\n", time.Now().Sub(t0))
}
}
// Radau5 step function
func radau5_step_mpi(o *Solver, y0 []float64, x0 float64, args ...interface{}) (rerr float64, err error) {
// factors
α := r5.α_ / o.h
β := r5.β_ / o.h
γ := r5.γ_ / o.h
// Jacobian and decomposition
if o.reuseJdec {
o.reuseJdec = false
} else {
// calculate only first Jacobian for all iterations (simple/modified Newton's method)
if o.reuseJ {
o.reuseJ = false
} else if !o.jacIsOK {
// Jacobian triplet
if o.jac == nil { // numerical
//if x0 == 0.0 { io.Pfgrey(" > > > > > > > > . . . numerical Jacobian . . . < < < < < < < < <\n") }
err = num.JacobianMpi(&o.dfdyT, func(fy, y []float64) (e error) {
e = o.fcn(fy, o.h, x0, y, args...)
return
}, y0, o.f0, o.w[0], o.Distr) // w works here as workspace variable
} else { // analytical
//if x0 == 0.0 { io.Pfgrey(" > > > > > > > > . . . analytical Jacobian . . . < < < < < < < < <\n") }
err = o.jac(&o.dfdyT, o.h, x0, y0, args...)
}
if err != nil {
return
}
// create M matrix
if o.doinit && !o.hasM {
o.mTri = new(la.Triplet)
if o.Distr {
id, sz := mpi.Rank(), mpi.Size()
start, endp1 := (id*o.ndim)/sz, ((id+1)*o.ndim)/sz
o.mTri.Init(o.ndim, o.ndim, endp1-start)
for i := start; i < endp1; i++ {
o.mTri.Put(i, i, 1.0)
}
} else {
o.mTri.Init(o.ndim, o.ndim, o.ndim)
for i := 0; i < o.ndim; i++ {
o.mTri.Put(i, i, 1.0)
}
}
}
o.njeval += 1
o.jacIsOK = true
}
// initialise triplets
if o.doinit {
o.rctriR = new(la.Triplet)
o.rctriC = new(la.TripletC)
o.rctriR.Init(o.ndim, o.ndim, o.mTri.Len()+o.dfdyT.Len())
xzmono := o.Distr
o.rctriC.Init(o.ndim, o.ndim, o.mTri.Len()+o.dfdyT.Len(), xzmono)
}
// update triplets
la.SpTriAdd(o.rctriR, γ, o.mTri, -1, &o.dfdyT) // rctriR := γ*M - dfdy
la.SpTriAddR2C(o.rctriC, α, β, o.mTri, -1, &o.dfdyT) // rctriC := (α+βi)*M - dfdy
// initialise solver
if o.doinit {
err = o.lsolR.InitR(o.rctriR, false, false, false)
if err != nil {
return
}
err = o.lsolC.InitC(o.rctriC, false, false, false)
if err != nil {
return
}
}
// perform factorisation
o.lsolR.Fact()
o.lsolC.Fact()
o.ndecomp += 1
}
// updated u[i]
o.u[0] = x0 + r5.c[0]*o.h
o.u[1] = x0 + r5.c[1]*o.h
o.u[2] = x0 + r5.c[2]*o.h
// (trial/initial) updated z[i] and w[i]
if o.first || o.ZeroTrial {
for m := 0; m < o.ndim; m++ {
o.z[0][m], o.w[0][m] = 0.0, 0.0
o.z[1][m], o.w[1][m] = 0.0, 0.0
o.z[2][m], o.w[2][m] = 0.0, 0.0
}
} else {
c3q := o.h / o.hprev
c1q := r5.μ1 * c3q
c2q := r5.μ2 * c3q
for m := 0; m < o.ndim; m++ {
o.z[0][m] = c1q * (o.ycol[0][m] + (c1q-r5.μ4)*(o.ycol[1][m]+(c1q-r5.μ3)*o.ycol[2][m]))
o.z[1][m] = c2q * (o.ycol[0][m] + (c2q-r5.μ4)*(o.ycol[1][m]+(c2q-r5.μ3)*o.ycol[2][m]))
o.z[2][m] = c3q * (o.ycol[0][m] + (c3q-r5.μ4)*(o.ycol[1][m]+(c3q-r5.μ3)*o.ycol[2][m]))
o.w[0][m] = r5.Ti[0][0]*o.z[0][m] + r5.Ti[0][1]*o.z[1][m] + r5.Ti[0][2]*o.z[2][m]
o.w[1][m] = r5.Ti[1][0]*o.z[0][m] + r5.Ti[1][1]*o.z[1][m] + r5.Ti[1][2]*o.z[2][m]
o.w[2][m] = r5.Ti[2][0]*o.z[0][m] + r5.Ti[2][1]*o.z[1][m] + r5.Ti[2][2]*o.z[2][m]
}
}
// iterations
o.nit = 0
o.η = math.Pow(max(o.η, o.ϵ), 0.8)
o.θ = o.θmax
o.diverg = false
var Lδw, oLδw, thq, othq, iterr, itRerr, qnewt float64
var it int
for it = 0; it < o.NmaxIt; it++ {
// max iterations ?
o.nit = it + 1
if o.nit > o.nitmax {
o.nitmax = o.nit
}
// evaluate f(x,y) at (u[i],v[i]=y0+z[i])
for i := 0; i < 3; i++ {
for m := 0; m < o.ndim; m++ {
o.v[i][m] = y0[m] + o.z[i][m]
}
o.nfeval += 1
err = o.fcn(o.f[i], o.h, o.u[i], o.v[i], args...)
if err != nil {
return
}
}
// calc rhs
if o.hasM {
// using δw as workspace here
la.SpMatVecMul(o.δw[0], 1, o.mMat, o.w[0]) // δw0 := M * w0
la.SpMatVecMul(o.δw[1], 1, o.mMat, o.w[1]) // δw1 := M * w1
la.SpMatVecMul(o.δw[2], 1, o.mMat, o.w[2]) // δw2 := M * w2
if o.Distr {
mpi.AllReduceSum(o.δw[0], o.v[0]) // v is used as workspace here
mpi.AllReduceSum(o.δw[1], o.v[1]) // v is used as workspace here
mpi.AllReduceSum(o.δw[2], o.v[2]) // v is used as workspace here
}
for m := 0; m < o.ndim; m++ {
o.v[0][m] = r5.Ti[0][0]*o.f[0][m] + r5.Ti[0][1]*o.f[1][m] + r5.Ti[0][2]*o.f[2][m] - γ*o.δw[0][m]
o.v[1][m] = r5.Ti[1][0]*o.f[0][m] + r5.Ti[1][1]*o.f[1][m] + r5.Ti[1][2]*o.f[2][m] - α*o.δw[1][m] + β*o.δw[2][m]
o.v[2][m] = r5.Ti[2][0]*o.f[0][m] + r5.Ti[2][1]*o.f[1][m] + r5.Ti[2][2]*o.f[2][m] - β*o.δw[1][m] - α*o.δw[2][m]
}
} else {
for m := 0; m < o.ndim; m++ {
o.v[0][m] = r5.Ti[0][0]*o.f[0][m] + r5.Ti[0][1]*o.f[1][m] + r5.Ti[0][2]*o.f[2][m] - γ*o.w[0][m]
o.v[1][m] = r5.Ti[1][0]*o.f[0][m] + r5.Ti[1][1]*o.f[1][m] + r5.Ti[1][2]*o.f[2][m] - α*o.w[1][m] + β*o.w[2][m]
o.v[2][m] = r5.Ti[2][0]*o.f[0][m] + r5.Ti[2][1]*o.f[1][m] + r5.Ti[2][2]*o.f[2][m] - β*o.w[1][m] - α*o.w[2][m]
}
}
// solve linear system
o.nlinsol += 1
var errR, errC error
if !o.Distr && o.Pll {
wg := new(sync.WaitGroup)
wg.Add(2)
go func() {
errR = o.lsolR.SolveR(o.δw[0], o.v[0], false)
wg.Done()
}()
go func() {
errC = o.lsolC.SolveC(o.δw[1], o.δw[2], o.v[1], o.v[2], false)
wg.Done()
}()
wg.Wait()
} else {
errR = o.lsolR.SolveR(o.δw[0], o.v[0], false)
errC = o.lsolC.SolveC(o.δw[1], o.δw[2], o.v[1], o.v[2], false)
}
// check for errors from linear solution
if errR != nil || errC != nil {
var errmsg string
if errR != nil {
errmsg += errR.Error()
}
if errC != nil {
if errR != nil {
errmsg += "\n"
}
errmsg += errC.Error()
}
err = errors.New(errmsg)
return
}
// update w and z
for m := 0; m < o.ndim; m++ {
o.w[0][m] += o.δw[0][m]
o.w[1][m] += o.δw[1][m]
o.w[2][m] += o.δw[2][m]
o.z[0][m] = r5.T[0][0]*o.w[0][m] + r5.T[0][1]*o.w[1][m] + r5.T[0][2]*o.w[2][m]
o.z[1][m] = r5.T[1][0]*o.w[0][m] + r5.T[1][1]*o.w[1][m] + r5.T[1][2]*o.w[2][m]
o.z[2][m] = r5.T[2][0]*o.w[0][m] + r5.T[2][1]*o.w[1][m] + r5.T[2][2]*o.w[2][m]
}
// rms norm of δw
Lδw = 0.0
for m := 0; m < o.ndim; m++ {
Lδw += math.Pow(o.δw[0][m]/o.scal[m], 2.0) + math.Pow(o.δw[1][m]/o.scal[m], 2.0) + math.Pow(o.δw[2][m]/o.scal[m], 2.0)
}
Lδw = math.Sqrt(Lδw / float64(3*o.ndim))
// check convergence
if it > 0 {
thq = Lδw / oLδw
if it == 1 {
o.θ = thq
} else {
o.θ = math.Sqrt(thq * othq)
}
othq = thq
if o.θ < 0.99 {
o.η = o.θ / (1.0 - o.θ)
iterr = Lδw * math.Pow(o.θ, float64(o.NmaxIt-o.nit)) / (1.0 - o.θ)
itRerr = iterr / o.fnewt
if itRerr >= 1.0 { // diverging
qnewt = max(1.0e-4, min(20.0, itRerr))
o.dvfac = 0.8 * math.Pow(qnewt, -1.0/(4.0+float64(o.NmaxIt)-1.0-float64(o.nit)))
o.diverg = true
break
}
} else { // diverging badly (unexpected step-rejection)
o.dvfac = 0.5
o.diverg = true
break
}
}
// save old norm
oLδw = Lδw
// converged
if o.η*Lδw < o.fnewt {
break
}
}
// did not converge
if it == o.NmaxIt-1 {
chk.Panic("radau5_step failed with it=%d", it)
}
// diverging => stop
if o.diverg {
rerr = 2.0 // must leave state intact, any rerr is OK
return
}
// error estimate
if o.LerrStrat == 1 {
// simple strategy => HW-VII p123 Eq.(8.17) (not good for stiff problems)
for m := 0; m < o.ndim; m++ {
o.ez[m] = r5.e0*o.z[0][m] + r5.e1*o.z[1][m] + r5.e2*o.z[2][m]
o.lerr[m] = r5.γ0*o.h*o.f0[m] + o.ez[m]
rerr += math.Pow(o.lerr[m]/o.scal[m], 2.0)
}
rerr = max(math.Sqrt(rerr/float64(o.ndim)), 1.0e-10)
} else {
// common
if o.hasM {
for m := 0; m < o.ndim; m++ {
o.ez[m] = r5.e0*o.z[0][m] + r5.e1*o.z[1][m] + r5.e2*o.z[2][m]
o.rhs[m] = o.f0[m]
}
if o.Distr {
la.SpMatVecMul(o.δw[0], γ, o.mMat, o.ez) // δw[0] = γ * M * ez (δw[0] is workspace)
mpi.AllReduceSumAdd(o.rhs, o.δw[0], o.δw[1]) // rhs += join_with_sum(δw[0]) (δw[1] is workspace)
} else {
la.SpMatVecMulAdd(o.rhs, γ, o.mMat, o.ez) // rhs += γ * M * ez
}
} else {
for m := 0; m < o.ndim; m++ {
o.ez[m] = r5.e0*o.z[0][m] + r5.e1*o.z[1][m] + r5.e2*o.z[2][m]
o.rhs[m] = o.f0[m] + γ*o.ez[m]
}
}
// HW-VII p123 Eq.(8.19)
if o.LerrStrat == 2 {
o.lsolR.SolveR(o.lerr, o.rhs, false)
rerr = o.rms_norm(o.lerr)
// HW-VII p123 Eq.(8.20)
} else {
o.lsolR.SolveR(o.lerr, o.rhs, false)
rerr = o.rms_norm(o.lerr)
if !(rerr < 1.0) {
if o.first || o.reject {
for m := 0; m < o.ndim; m++ {
o.v[0][m] = y0[m] + o.lerr[m] // y0perr
}
o.nfeval += 1
err = o.fcn(o.f[0], o.h, x0, o.v[0], args...) // f0perr
if err != nil {
return
}
if o.hasM {
la.VecCopy(o.rhs, 1, o.f[0]) // rhs := f0perr
if o.Distr {
la.SpMatVecMul(o.δw[0], γ, o.mMat, o.ez) // δw[0] = γ * M * ez (δw[0] is workspace)
mpi.AllReduceSumAdd(o.rhs, o.δw[0], o.δw[1]) // rhs += join_with_sum(δw[0]) (δw[1] is workspace)
} else {
la.SpMatVecMulAdd(o.rhs, γ, o.mMat, o.ez) // rhs += γ * M * ez
}
} else {
la.VecAdd2(o.rhs, 1, o.f[0], γ, o.ez) // rhs = f0perr + γ * ez
}
o.lsolR.SolveR(o.lerr, o.rhs, false)
rerr = o.rms_norm(o.lerr)
}
}
}
}
return
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
if mpi.Rank() == 0 {
io.PfYel("\nTest MPI 01\n")
}
if mpi.Size() != 3 {
chk.Panic("this test needs 3 processors")
}
n := 11
x := make([]float64, n)
id, sz := mpi.Rank(), mpi.Size()
start, endp1 := (id*n)/sz, ((id+1)*n)/sz
for i := start; i < endp1; i++ {
x[i] = float64(i)
}
// Barrier
mpi.Barrier()
io.Pfgrey("x @ proc # %d = %v\n", id, x)
// SumToRoot
r := make([]float64, n)
mpi.SumToRoot(r, x)
var tst testing.T
if id == 0 {
chk.Vector(&tst, fmt.Sprintf("SumToRoot: r @ proc # %d", id), 1e-17, r, []float64{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10})
} else {
chk.Vector(&tst, fmt.Sprintf("SumToRoot: r @ proc # %d", id), 1e-17, r, make([]float64, n))
}
// BcastFromRoot
r[0] = 666
mpi.BcastFromRoot(r)
chk.Vector(&tst, fmt.Sprintf("BcastFromRoot: r @ proc # %d", id), 1e-17, r, []float64{666, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10})
// AllReduceSum
setslice(x)
w := make([]float64, n)
mpi.AllReduceSum(x, w)
chk.Vector(&tst, fmt.Sprintf("AllReduceSum: w @ proc # %d", id), 1e-17, w, []float64{110, 110, 110, 1021, 1021, 1021, 2032, 2032, 2032, 3043, 3043})
// AllReduceSumAdd
setslice(x)
y := []float64{-1000, -1000, -1000, -1000, -1000, -1000, -1000, -1000, -1000, -1000, -1000}
mpi.AllReduceSumAdd(y, x, w)
chk.Vector(&tst, fmt.Sprintf("AllReduceSumAdd: y @ proc # %d", id), 1e-17, y, []float64{-890, -890, -890, 21, 21, 21, 1032, 1032, 1032, 2043, 2043})
// AllReduceMin
setslice(x)
mpi.AllReduceMin(x, w)
chk.Vector(&tst, fmt.Sprintf("AllReduceMin: x @ proc # %d", id), 1e-17, x, []float64{0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3})
// AllReduceMax
setslice(x)
mpi.AllReduceMax(x, w)
chk.Vector(&tst, fmt.Sprintf("AllReduceMax: x @ proc # %d", id), 1e-17, x, []float64{100, 100, 100, 1000, 1000, 1000, 2000, 2000, 2000, 3000, 3000})
}
// solve_linear_problem solves the linear problem
func solve_linear_problem(t float64, d *Domain, dc *DynCoefs, sum *Summary, first bool) (err error) {
// assemble right-hand side vector (fb) with **negative** of residuals
la.VecFill(d.Fb, 0)
for _, e := range d.Elems {
err = e.AddToRhs(d.Fb, d.Sol)
if err != nil {
return
}
}
// join all fb
if d.Distr {
mpi.AllReduceSum(d.Fb, d.Wb) // this must be done here because there might be nodes sharing boundary conditions
}
// point natural boundary conditions; e.g. concentrated loads
d.PtNatBcs.AddToRhs(d.Fb, t)
// essential boundary conditioins; e.g. constraints
d.EssenBcs.AddToRhs(d.Fb, d.Sol)
// assemble and factorise Jacobian matrix just once
if first {
// assemble element matrices
d.Kb.Start()
for _, e := range d.Elems {
err = e.AddToKb(d.Kb, d.Sol, true)
if err != nil {
return
}
}
// join A and tr(A) matrices into Kb
if d.Proc == 0 {
d.Kb.PutMatAndMatT(&d.EssenBcs.A)
}
// initialise linear solver (just once)
if d.InitLSol {
err = d.LinSol.InitR(d.Kb, d.Sim.LinSol.Symmetric, d.Sim.LinSol.Verbose, d.Sim.LinSol.Timing)
if err != nil {
err = chk.Err("cannot initialise linear solver:\n%v", err)
return
}
d.InitLSol = false
}
// perform factorisation (always if not CteTg)
err = d.LinSol.Fact()
if err != nil {
err = chk.Err("factorisation failed:\n%v", err)
return
}
}
// solve for wb
err = d.LinSol.SolveR(d.Wb, d.Fb, false)
if err != nil {
err = chk.Err("solve failed:%v\n", err)
return
}
// update primary variables (y)
for i := 0; i < d.Ny; i++ {
d.Sol.Y[i] += d.Wb[i] // y += δy
d.Sol.ΔY[i] = d.Wb[i] // ΔY = δy
}
// update Lagrange multipliers (λ)
for i := 0; i < d.Nlam; i++ {
d.Sol.L[i] += d.Wb[d.Ny+i] // λ += δλ
}
// update secondary variables
for _, e := range d.Elems {
err = e.Update(d.Sol)
if err != nil {
break
}
}
return
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
if mpi.Rank() == 0 {
chk.PrintTitle("ode04: Hairer-Wanner VII-p376 Transistor Amplifier\n")
}
if mpi.Size() != 3 {
chk.Panic(">> error: this test requires 3 MPI processors\n")
return
}
// data
UE, UB, UF, ALPHA, BETA := 0.1, 6.0, 0.026, 0.99, 1.0e-6
R0, R1, R2, R3, R4, R5 := 1000.0, 9000.0, 9000.0, 9000.0, 9000.0, 9000.0
R6, R7, R8, R9 := 9000.0, 9000.0, 9000.0, 9000.0
W := 2.0 * 3.141592654 * 100.0
// initial values
xa := 0.0
ya := []float64{0.0,
UB,
UB / (R6/R5 + 1.0),
UB / (R6/R5 + 1.0),
UB,
UB / (R2/R1 + 1.0),
UB / (R2/R1 + 1.0),
0.0}
// endpoint of integration
xb := 0.05
//xb = 0.0123 // OK
//xb = 0.01235 // !OK
// right-hand side of the amplifier problem
w := make([]float64, 8) // workspace
fcn := func(f []float64, dx, x float64, y []float64, args ...interface{}) error {
UET := UE * math.Sin(W*x)
FAC1 := BETA * (math.Exp((y[3]-y[2])/UF) - 1.0)
FAC2 := BETA * (math.Exp((y[6]-y[5])/UF) - 1.0)
la.VecFill(f, 0)
switch mpi.Rank() {
case 0:
f[0] = y[0] / R9
case 1:
f[1] = (y[1]-UB)/R8 + ALPHA*FAC1
f[2] = y[2]/R7 - FAC1
case 2:
f[3] = y[3]/R5 + (y[3]-UB)/R6 + (1.0-ALPHA)*FAC1
f[4] = (y[4]-UB)/R4 + ALPHA*FAC2
f[5] = y[5]/R3 - FAC2
f[6] = y[6]/R1 + (y[6]-UB)/R2 + (1.0-ALPHA)*FAC2
f[7] = (y[7] - UET) / R0
}
mpi.AllReduceSum(f, w)
return nil
}
// Jacobian of the amplifier problem
jac := func(dfdy *la.Triplet, dx, x float64, y []float64, args ...interface{}) error {
FAC14 := BETA * math.Exp((y[3]-y[2])/UF) / UF
FAC27 := BETA * math.Exp((y[6]-y[5])/UF) / UF
if dfdy.Max() == 0 {
dfdy.Init(8, 8, 16)
}
NU := 2
dfdy.Start()
switch mpi.Rank() {
case 0:
dfdy.Put(2+0-NU, 0, 1.0/R9)
dfdy.Put(2+1-NU, 1, 1.0/R8)
dfdy.Put(1+2-NU, 2, -ALPHA*FAC14)
dfdy.Put(0+3-NU, 3, ALPHA*FAC14)
dfdy.Put(2+2-NU, 2, 1.0/R7+FAC14)
case 1:
dfdy.Put(1+3-NU, 3, -FAC14)
dfdy.Put(2+3-NU, 3, 1.0/R5+1.0/R6+(1.0-ALPHA)*FAC14)
dfdy.Put(3+2-NU, 2, -(1.0-ALPHA)*FAC14)
dfdy.Put(2+4-NU, 4, 1.0/R4)
dfdy.Put(1+5-NU, 5, -ALPHA*FAC27)
case 2:
dfdy.Put(0+6-NU, 6, ALPHA*FAC27)
dfdy.Put(2+5-NU, 5, 1.0/R3+FAC27)
dfdy.Put(1+6-NU, 6, -FAC27)
dfdy.Put(2+6-NU, 6, 1.0/R1+1.0/R2+(1.0-ALPHA)*FAC27)
dfdy.Put(3+5-NU, 5, -(1.0-ALPHA)*FAC27)
dfdy.Put(2+7-NU, 7, 1.0/R0)
}
return nil
}
// matrix "M"
c1, c2, c3, c4, c5 := 1.0e-6, 2.0e-6, 3.0e-6, 4.0e-6, 5.0e-6
var M la.Triplet
M.Init(8, 8, 14)
M.Start()
NU := 1
switch mpi.Rank() {
case 0:
M.Put(1+0-NU, 0, -c5)
M.Put(0+1-NU, 1, c5)
M.Put(2+0-NU, 0, c5)
M.Put(1+1-NU, 1, -c5)
M.Put(1+2-NU, 2, -c4)
M.Put(1+3-NU, 3, -c3)
case 1:
M.Put(0+4-NU, 4, c3)
M.Put(2+3-NU, 3, c3)
M.Put(1+4-NU, 4, -c3)
case 2:
M.Put(1+5-NU, 5, -c2)
M.Put(1+6-NU, 6, -c1)
M.Put(0+7-NU, 7, c1)
M.Put(2+6-NU, 6, c1)
M.Put(1+7-NU, 7, -c1)
}
// flags
silent := false
fixstp := false
//method := "Dopri5"
method := "Radau5"
ndim := len(ya)
numjac := false
// structure to hold numerical results
res := ode.Results{Method: method}
// ODE solver
var osol ode.Solver
osol.Pll = true
// solve problem
if numjac {
osol.Init(method, ndim, fcn, nil, &M, ode.SimpleOutput, silent)
} else {
osol.Init(method, ndim, fcn, jac, &M, ode.SimpleOutput, silent)
}
osol.IniH = 1.0e-6 // initial step size
// set tolerances
atol, rtol := 1e-11, 1e-5
osol.SetTol(atol, rtol)
// run
t0 := time.Now()
if fixstp {
osol.Solve(ya, xa, xb, 0.01, fixstp, &res)
} else {
osol.Solve(ya, xa, xb, xb-xa, fixstp, &res)
}
// plot
if mpi.Rank() == 0 {
io.Pfmag("elapsed time = %v\n", time.Now().Sub(t0))
plt.SetForEps(2.0, 400)
args := "'b-', marker='.', lw=1, clip_on=0"
ode.Plot("/tmp/gosl/ode", "hwamplifier_mpi.eps", &res, nil, xa, xb, "", args, func() {
_, T, err := io.ReadTable("data/radau5_hwamplifier.dat")
if err != nil {
chk.Panic("%v", err)
}
for j := 0; j < ndim; j++ {
plt.Subplot(ndim+1, 1, j+1)
plt.Plot(T["x"], T[io.Sf("y%d", j)], "'k+',label='reference',ms=10")
}
})
}
}
func main() {
mpi.Start(false)
defer func() {
mpi.Stop(false)
}()
if mpi.Rank() == 0 {
chk.PrintTitle("ode02: Hairer-Wanner VII-p5 Eq.(1.5) Van der Pol's Equation")
}
if mpi.Size() != 2 {
chk.Panic(">> error: this test requires 2 MPI processors\n")
return
}
eps := 1.0e-6
w := make([]float64, 2) // workspace
fcn := func(f []float64, dx, x float64, y []float64, args ...interface{}) error {
f[0], f[1] = 0, 0
switch mpi.Rank() {
case 0:
f[0] = y[1]
case 1:
f[1] = ((1.0-y[0]*y[0])*y[1] - y[0]) / eps
}
// join all f
mpi.AllReduceSum(f, w)
return nil
}
jac := func(dfdy *la.Triplet, dx, x float64, y []float64, args ...interface{}) error {
if dfdy.Max() == 0 {
dfdy.Init(2, 2, 4)
}
dfdy.Start()
if false { // per column
switch mpi.Rank() {
case 0:
dfdy.Put(0, 0, 0.0)
dfdy.Put(1, 0, (-2.0*y[0]*y[1]-1.0)/eps)
case 1:
dfdy.Put(0, 1, 1.0)
dfdy.Put(1, 1, (1.0-y[0]*y[0])/eps)
}
} else { // per row
switch mpi.Rank() {
case 0:
dfdy.Put(0, 0, 0.0)
dfdy.Put(0, 1, 1.0)
case 1:
dfdy.Put(1, 0, (-2.0*y[0]*y[1]-1.0)/eps)
dfdy.Put(1, 1, (1.0-y[0]*y[0])/eps)
}
}
return nil
}
// method and flags
silent := false
fixstp := false
//method := "Dopri5"
method := "Radau5"
numjac := false
xa, xb := 0.0, 2.0
ya := []float64{2.0, -0.6}
ndim := len(ya)
// structure to hold numerical results
res := ode.Results{Method: method}
// allocate ODE object
var o ode.Solver
o.Distr = true
if numjac {
o.Init(method, ndim, fcn, nil, nil, ode.SimpleOutput, silent)
} else {
o.Init(method, ndim, fcn, jac, nil, ode.SimpleOutput, silent)
}
// tolerances and initial step size
rtol := 1e-4
atol := rtol
o.IniH = 1.0e-4
o.SetTol(atol, rtol)
//o.NmaxSS = 2
// solve problem
y := make([]float64, ndim)
copy(y, ya)
t0 := time.Now()
if fixstp {
o.Solve(y, xa, xb, 0.05, fixstp, &res)
} else {
o.Solve(y, xa, xb, xb-xa, fixstp, &res)
}
// plot
if mpi.Rank() == 0 {
io.Pfmag("elapsed time = %v\n", time.Now().Sub(t0))
plt.SetForEps(1.5, 400)
args := "'b-', marker='.', lw=1, ms=4, clip_on=0"
ode.Plot("/tmp/gosl/ode", "vdpolA_mpi.eps", &res, nil, xa, xb, "", args, func() {
_, T, err := io.ReadTable("data/vdpol_radau5_for.dat")
if err != nil {
chk.Panic("%v", err)
}
plt.Subplot(3, 1, 1)
plt.Plot(T["x"], T["y0"], "'k+',label='reference',ms=7")
plt.Subplot(3, 1, 2)
plt.Plot(T["x"], T["y1"], "'k+',label='reference',ms=7")
})
}
}