func parabolaTestPoints() []vec.Vector {
sk := 0.1 // small value of k
ssk := sk / math.Sqrt(2)
// unique point
zero := vec.ZeroVector(3)
// basis vectors
xb := []float64{sk, 0.0, 0.0}
yb := []float64{0.0, sk, 0.0}
zb := []float64{0.0, 0.0, sk}
xyb := []float64{ssk, ssk, 0.0}
xzb := []float64{ssk, 0.0, ssk}
yzb := []float64{0.0, ssk, ssk}
basis := []vec.Vector{xb, yb, zb, xyb, xzb, yzb}
// create points from basis
numRadialPoints := 3
points := []vec.Vector{zero}
for _, v := range basis {
for i := 1; i <= numRadialPoints; i++ {
pt := vec.ZeroVector(len(v))
copy(pt, v)
pt.Mul(float64(i), &pt)
points = append(points, pt)
}
}
return points
}
// Produce a a slice of vectors whose values cover each first Brillouin zone
// point once.
func bzPoints(L, d int) []vec.Vector {
cachedL, okL := pointsCache[L]
if okL {
cachedD, okD := cachedL[d]
if okD {
return cachedD
}
} else {
pointsCache[L] = make(map[int][]vec.Vector)
}
points := make([]vec.Vector, pow(L, d))
// start is the minumum value of any component of a point
start := -math.Pi
// (finish - step) is the maximum value of any component of a point
finish := -start
// step is the separation between point components
step := (finish - start) / float64(L)
k := vec.ZeroVector(d)
kIndex := make([]int, d)
// set initial value for k
for i := 0; i < d; i++ {
k[i] = start
kIndex[i] = 0
}
// iterate over Brillouin zone
done := false
for i := 0; !done; i++ {
points[i] = vec.ZeroVector(len(k))
copy(points[i], k)
done = bzAdvance(k, kIndex, start, step, L, d)
}
pointsCache[L][d] = points
return points
}
func VectorSum(pointsPerSide, gridDim, fnDim int, fn BzVectorFunc) vec.Vector {
c := vec.ZeroVector(fnDim)
add := func(next vec.Vector, total *vec.Vector) {
for i := 0; i < fnDim; i++ {
x := (*total)[i]
y := next[i] - c[i]
t := x + y
c[i] = (t - x) - y
(*total)[i] = t
}
}
start := vec.ZeroVector(fnDim)
return bzVectorReduce(add, start, pointsPerSide, gridDim, fnDim, fn)
}
// Convert v back to Go format.
func VecFromGSL(v *C.gsl_vector) vec.Vector {
dim := int(v.size)
u := vec.ZeroVector(dim)
for i := 0; i < dim; i++ {
u[i] = float64(C.gsl_vector_get(v, C.size_t(i)))
}
return u
}
// Combine fns into one function, suitable for passing to MultiDim.
// All funcs passed in must have the same dimension.
func Combine(fns []Diffable) DiffSystem {
NumFuncs := len(fns)
Dimension := fns[0].Dimension
// F(v) = \sum_i fns[i].F(v) e_i
// (e_i is unit vector in i'th direction)
F := func(v vec.Vector) (vec.Vector, error) {
var err error
ret := vec.ZeroVector(NumFuncs)
for i := 0; i < NumFuncs; i++ {
ret[i], err = fns[i].F(v)
if err != nil {
return ret, err
}
}
return ret, nil
}
// Df(v) = \sum_i fns[i].Df(v)
Df := func(v vec.Vector) ([]vec.Vector, error) {
var err error
ret := make([]vec.Vector, NumFuncs)
for i := 0; i < NumFuncs; i++ {
ret[i], err = fns[i].Df(v)
if err != nil {
return ret, err
}
}
return ret, nil
}
Fdf := func(v vec.Vector) (vec.Vector, []vec.Vector, error) {
var err error
ret_f := vec.ZeroVector(NumFuncs)
ret_df := make([]vec.Vector, NumFuncs)
for i := 0; i < NumFuncs; i++ {
ret_f[i], err = fns[i].F(v)
if err != nil {
return ret_f, ret_df, err
}
ret_df[i], err = fns[i].Df(v)
if err != nil {
return ret_f, ret_df, err
}
}
return ret_f, ret_df, nil
}
return DiffSystem{F, Df, Fdf, NumFuncs, Dimension}
}
func VectorAvg(pointsPerSide, gridDim, fnDim int, fn BzVectorFunc) vec.Vector {
N := math.Pow(float64(pointsPerSide), float64(gridDim))
sum := VectorSum(pointsPerSide, gridDim, fnDim, fn)
avg := vec.ZeroVector(fnDim)
for i := 0; i < fnDim; i++ {
avg[i] = sum[i] / N
}
return avg
}
func bzVectorReduce(combine bzVectorConsumer, start vec.Vector, L, d, fnDim int, fn BzVectorFunc) vec.Vector {
points := bzPoints(L, d)
total := start
out := vec.ZeroVector(fnDim)
for i := 0; i < len(points); i++ {
k := points[i]
fn(k, &out)
combine(out, &total)
}
return total
}
// Gradient of fn at v within tolerance epsabs. h is the initial step size.
func Gradient(fn vec.FnDim0, v vec.Vector, h, epsabs float64) (vec.Vector, error) {
grad := vec.ZeroVector(len(v))
// calculate derivative of fn w.r.t. each component of v
for i := 0; i < len(v); i++ {
deriv, err := Derivative(fn, v, i, h, epsabs)
if err != nil {
return grad, err
}
grad[i] = deriv
}
return grad, nil
}
// Calculate Mu_b - (-Omega_+(0))
func AbsErrorMu_b(env *tempAll.Environment, variables []string) solve.Diffable {
F := func(v vec.Vector) (float64, error) {
if v.ContainsNaN() {
fmt.Printf("got NaN in AbsErrorMu_b (v=%v)\n", v)
return 0.0, errors.New("NaN in input")
}
env.Set(v, variables)
zv := vec.ZeroVector(3)
omega0, err := tempCrit.OmegaPlus(env, zv)
if err != nil {
return 0.0, err
}
lhs := env.Mu_b
rhs := -omega0
return lhs - rhs, nil
}
h := 1e-5
epsabs := 1e-4
return solve.SimpleDiffable(F, len(variables), h, epsabs)
}
// Return a list of all k points surveyed by OmegaCoeffs().
func OmegaCoeffsPoints(numRadial int, sk float64) []vec.Vector {
ssk := sk / math.Sqrt(2)
// basis vectors
xb := []float64{sk, 0.0, 0.0}
yb := []float64{0.0, sk, 0.0}
zb := []float64{0.0, 0.0, sk}
xyb := []float64{ssk, ssk, 0.0}
xzb := []float64{ssk, 0.0, ssk}
yzb := []float64{0.0, ssk, ssk}
basis := []vec.Vector{xb, yb, zb, xyb, xzb, yzb}
// create points from basis
points := []vec.Vector{}
for _, v := range basis {
for i := 1; i <= numRadial; i++ {
pt := vec.ZeroVector(len(v))
copy(pt, v)
pt.Mul(float64(i), &pt)
points = append(points, pt)
}
}
return points
}
// Return a vector with the fit parameters [a_x, a_y, b, mu_pair] to the
// given functions.
func omegaFitHelper(env *tempAll.Environment, fn OmegaFunc, points []vec.Vector) (vec.Vector, error) {
// evaluate omega_+/-(k) at each point
omegas := []float64{}
Xs := []vec.Vector{}
for _, q := range points {
omega, err := fn(env, q)
if err != nil {
continue
}
X := vec.ZeroVector(4)
X[0] = q[0] * q[0]
X[1] = q[1] * q[1]
X[2] = q[2] * q[2]
X[3] = -1
Xs = append(Xs, X)
omegas = append(omegas, omega)
}
if len(omegas) < 3 {
return nil, fmt.Errorf("not enough omega_+/- values can be found")
}
return fit.Linear(omegas, Xs), nil
}
func TestFitParabolaLinear(t *testing.T) {
epsAbs := 1e-9
ax, ay, b, mu_b := 1.01, 0.99, 0.1, -0.01
omegaGenerator := func(q vec.Vector) float64 {
return ax*q[0]*q[0] + ay*q[1]*q[1] + b*q[2]*q[2] - mu_b
}
points := parabolaTestPoints()
y := make([]float64, len(points))
X := make([]vec.Vector, len(points))
for i, q := range points {
y[i] = omegaGenerator(q)
X[i] = vec.ZeroVector(4)
X[i][0] = q[0] * q[0]
X[i][1] = q[1] * q[1]
X[i][2] = q[2] * q[2]
X[i][3] = -1
}
coeffs := Linear(y, X)
if math.Abs(coeffs[0]-ax) > epsAbs || math.Abs(coeffs[1]-ay) > epsAbs || math.Abs(coeffs[2]-b) > epsAbs || math.Abs(coeffs[3]-mu_b) > epsAbs {
t.Fatalf("unexpected coefficients; got %s, expected %s", coeffs, []float64{ax, ay, b, mu_b})
}
}
// Solve the (D1, Mu_h, Mu_b) system with Beta and x fixed.
func SolveD1Mu_hMu_b(env *tempAll.Environment, epsAbs, epsRel float64) (vec.Vector, error) {
/*
// fix pair coefficients
if env.A == 0.0 && env.B == 0.0 && env.FixedPairCoeffs {
D1, Mu_h, Mu_b, Beta, Be_field := env.D1, env.Mu_h, env.Mu_b, env.Beta, env.Be_field
env.Mu_b = 0.0 // Mu_b is 0 at T_c
env.Be_field = 0.0
_, err := tempCrit.CritTempSolve(env, epsAbs, epsRel)
if err != nil {
return nil, err
}
omegaFit, err := tempCrit.OmegaFit(env, tempCrit.OmegaPlus)
if err != nil {
return nil, err
}
env.A, env.B = omegaFit[0], omegaFit[2]
env.PairCoeffsReady = true
// uncache env
env.D1, env.Mu_h, env.Mu_b, env.Beta, env.Be_field = D1, Mu_h, Mu_b, Beta, Be_field
}
*/
maxIters := 1000
oldMu_b := env.Mu_b
for i := 0; i < maxIters; i++ {
// iterate D1/Mu_h
solution, err := SolveD1Mu_h(env, epsAbs, epsRel)
if err != nil {
return nil, err
}
// iterate Mu_b
Be_field := env.Be_field
env.Be_field = 0.0
zv := vec.ZeroVector(3)
omega0, err := tempCrit.OmegaPlus(env, zv)
//omegaFit, err := tempCrit.OmegaFit(env, tempCrit.OmegaPlus)
if err != nil {
return nil, err
}
env.Mu_b = -omega0
env.Be_field = Be_field
//A, Mub_eff := omegaFit[0], omegaFit[3]
//env.Mu_b = -omega0 + 2.0 * env.Be_field * env.A
//Mub_eff := omegaFit[3]
//env.Mu_b = Mub_eff
//fmt.Printf("iterating Mu_b: now %f, before %f\n", env.Mu_b, oldMu_b)
// check if done
if math.Abs(env.Mu_b-oldMu_b) < epsAbs || !env.IterateD1Mu_hMu_b {
return []float64{solution[0], solution[1], env.Mu_b}, nil
}
oldMu_b = env.Mu_b
}
return []float64{0.0, 0.0, 0.0}, fmt.Errorf("failed to find D1/Mu_h/Mu_b solution for env=%s\n", env.String())
/*
system, start := D1Mu_hMu_bSystem(env)
solution, err := solve.MultiDim(system, start, epsAbs, epsRel)
if err != nil {
return nil, err
}
return solution, nil
*/
}