Cubic Spline Interpolation

void riscv_spline_f32(riscv_spline_instance_f32 *S, const float32_t *xq, float32_t *pDst, uint32_t blockSize)

void riscv_spline_init_f32(riscv_spline_instance_f32 *S, riscv_spline_type type, const float32_t *x, const float32_t *y, uint32_t n, float32_t *coeffs, float32_t *tempBuffer)
 group SplineInterpolate
Spline interpolation is a method of interpolation where the interpolant is a piecewisedefined polynomial called “spline”.
Given a function f defined on the interval [a,b], a set of n nodes x(i) where a=x(1)<x(2)<…<x(n)=b and a set of n values y(i) = f(x(i)), a cubic spline interpolant S(x) is defined as:
 Introduction
where
Having defined h(i) = x(i+1)  x(i)
 Algorithm
It is possible to write the previous conditions in matrix form (Ax=B). In order to solve the system two boundary conidtions are needed.
Natural spline: S1’’(x1)=2*c(1)=0 ; Sn’’(xn)=2*c(n)=0 In matrix form:
Parabolic runout spline: S1’’(x1)=2*c(1)=S2’’(x2)=2*c(2) ; Sn1’’(xn1)=2*c(n1)=Sn’’(xn)=2*c(n) In matrix form:
A is a tridiagonal matrix (a band matrix of bandwidth 3) of size N=n+1. The factorization algorithms (A=LU) can be simplified considerably because a large number of zeros appear in regular patterns. The Crout method has been used: 1) Solve LZ=B
2) Solve UX=Z
c(i) for i=1, …, n1 are needed to compute the n1 polynomials. b(i) and d(i) are computed as:
b(i) = [y(i+1)y(i)]/h(i)h(i)*[c(i+1)+2*c(i)]/3
d(i) = [c(i+1)c(i)]/[3*h(i)] Moreover, a(i)=y(i).
It is possible to compute the interpolated vector for x values outside the input range (xq<x(1); xq>x(n)). The coefficients used to compute the y values for xq<x(1) are going to be the ones used for the first interval, while for xq>x(n) the coefficients used for the last interval.
 Behaviour outside the given intervals
The initialization function takes as input two arrays that the user has to allocate:
coeffs
will contain the b, c, and d coefficients for the (n1) intervals (n is the number of known points), hence its size must be 3*(n1);tempBuffer
is temporally used for internal computations and its size is n+n1. Initialization function
The x input array must be strictly sorted in ascending order and it must not contain twice the same value (x(i)<x(i+1)).
Functions

void riscv_spline_f32(riscv_spline_instance_f32 *S, const float32_t *xq, float32_t *pDst, uint32_t blockSize)
Processing function for the floatingpoint cubic spline interpolation.
 Parameters
S – [in] points to an instance of the floatingpoint spline structure.
xq – [in] points to the x values of the interpolated data points.
pDst – [out] points to the block of output data.
blockSize – [in] number of samples of output data.

void riscv_spline_init_f32(riscv_spline_instance_f32 *S, riscv_spline_type type, const float32_t *x, const float32_t *y, uint32_t n, float32_t *coeffs, float32_t *tempBuffer)
Initialization function for the floatingpoint cubic spline interpolation.
 Parameters
S – [inout] points to an instance of the floatingpoint spline structure.
type – [in] type of cubic spline interpolation (boundary conditions)
x – [in] points to the x values of the known data points.
y – [in] points to the y values of the known data points.
n – [in] number of known data points.
coeffs – [in] coefficients array for b, c, and d
tempBuffer – [in] buffer array for internal computations