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Lagrange Barycentric Interpolation

See Also:
Ilya Yaroshenko
import mir.algorithm.iteration: all;
import mir.math;
import mir.ndslice;

auto f(double x) { return (x-2) * (x-5) * (x-9); }
auto fd(double x) { return (x-5) * (x-9) + (x-2) * (x-5) + (x-2) * (x-9); }
auto fd2(double x) { return (x-5) + (x-9) + (x-2) + (x-5) + (x-2) + (x-9); }
double[2] fWithDerivative(double x) { return [f(x), fd(x)]; }
double[3] fWithTwoDerivatives(double x) { return [f(x), fd(x), fd2(x)]; }

auto x = [0.0, 3, 5, 10];
auto y =!f.slice.field;
// `!2` adds first two derivatives: f, f', and f''
// default parameter is 0
auto l = x.lagrange!2(y);

foreach(test; x ~ [2.0, 5, 9] ~ [double(PI), double(E), 10, 100, 1e3])
foreach(scale; [-1, +1, 1 + double.epsilon, 1 + double.epsilon.sqrt])
foreach(shift; [0, double.min_normal/2, double.min_normal*2, double.epsilon, double.epsilon.sqrt])
    auto testX = test * scale + shift;

    auto lx = l(testX);
    auto l_ldx = l.withDerivative(testX);
    auto l_ld_ld2x = l.withTwoDerivatives(testX);

Lagrange!(T, maxDerivative) lagrange(uint maxDerivative = 0, T, X)(scope const T[] x, scope const X[] y)
if (maxDerivative < 16);

@trusted Lagrange!(Unqual!(Slice!(YIterator, 1, ykind).DeepElement), maxDerivative) lagrange(uint maxDerivative = 0, XIterator, SliceKind xkind, YIterator, SliceKind ykind)(scope Slice!(XIterator, 1, xkind) x, scope Slice!(YIterator, 1, ykind) y)
if (maxDerivative < 16);
Constructs barycentric lagrange interpolant.
struct Lagrange(T, uint maxAdditionalFunctions = 0) if (isFloatingPoint!T && (maxAdditionalFunctions < 16));
RCArray!(immutable(T)) _grid;
for internal use only.
RCArray!(immutable(T)) _inversedBarycentricWeights;
for internal use only.
RCArray!T[maxAdditionalFunctions + 1] _normalizedValues;
for internal use only.
T[maxAdditionalFunctions + 1] _asums;
for internal use only.
this(RCArray!(immutable(T)) grid, RCArray!T values, RCArray!(immutable(T)) inversedBarycentricWeights);

Complexity O(N)

this(RCArray!(immutable(T)) grid, RCArray!T values);

Complexity O(N^^2)

ref const(RCArray!(immutable(T))) grid();
ref const(RCArray!(immutable(T))) inversedBarycentricWeights();
ref const(RCArray!T)[maxAdditionalFunctions + 1] normalizedValues();
ref const(T)[maxAdditionalFunctions + 1] asums();
size_t intervalCount(size_t dimension = 0)();
alias withDerivative = opCall!1;
alias withTwoDerivatives = opCall!2;
@nogc RCArray!(immutable(T)) inversedBarycentricWeights(T)(scope Slice!(const(T)*) x)
if (isFloatingPoint!T);
@nogc Slice!(T*) polynomialDerivativeValues(T)(return Slice!(T*) d, Slice!(const(T)*) x, Slice!(const(T)*) y, Slice!(const(T)*) w)
if (isFloatingPoint!T);
Computes derivative values in the same points
Slice!(T*) d derivative values (output)
Slice!(const(T)*) y values
Slice!(const(T)*) x grid
Slice!(const(T)*) w inversed barycentric weights
Derivative values in the same points
@nogc Slice!(T*) polynomialDerivativeValues(T)(return Slice!(T*) d, Slice!(const(T)*) x, Slice!(const(T)*) y)
if (isFloatingPoint!T);