三次样条插值(Cubic Spline Interpolation)及代码实现(C语言)

#define S_FUNCTION_NAME  cubic
#define S_FUNCTION_LEVEL 2
#include "simstruc.h"
#include "malloc.h"  //方便使用变量定义数组大小

static void mdlInitializeSizes(SimStruct *S)
{
/*参数只有一个，是n乘2的定点数组[xi, yi]:
* [ x1，y1;
*   x2, y2;
*   ..., ...;
*   xn, yn;
*/
ssSetNumSFcnParams(S, 1);
if (ssGetNumSFcnParams(S) != ssGetSFcnParamsCount(S)) return;

ssSetNumContStates(S, 0);
ssSetNumDiscStates(S, 0);

if (!ssSetNumInputPorts(S, 1)) return;  //输入是x
ssSetInputPortWidth(S, 0, 1);
ssSetInputPortRequiredContiguous(S, 0, true);
ssSetInputPortDirectFeedThrough(S, 0, 1);

if (!ssSetNumOutputPorts(S, 1)) return;  //输出是S(x)
ssSetOutputPortWidth(S, 0, 1);

ssSetNumSampleTimes(S, 1);
ssSetNumRWork(S, 0);
ssSetNumIWork(S, 0);
ssSetNumPWork(S, 0);
ssSetNumModes(S, 0);
ssSetNumNonsampledZCs(S, 0);

ssSetSimStateCompliance(S, USE_DEFAULT_SIM_STATE);

ssSetOptions(S, 0);
}

static void mdlInitializeSampleTimes(SimStruct *S)
{
ssSetSampleTime(S, 0, CONTINUOUS_SAMPLE_TIME);
ssSetOffsetTime(S, 0, 0.0);
}

#define MDL_INITIALIZE_CONDITIONS
#if defined(MDL_INITIALIZE_CONDITIONS)
static void mdlInitializeConditions(SimStruct *S)
{
}
#endif

#define MDL_START
#if defined(MDL_START)
static void mdlStart(SimStruct *S)
{
}
#endif /*  MDL_START */

static void mdlOutputs(SimStruct *S, int_T tid)
{
const real_T *map = mxGetPr(ssGetSFcnParam(S,0));  //获取定点数据
const int_T *mapSize = mxGetDimensions(ssGetSFcnParam(S,0));  //定点数组维数
const real_T *x = (const real_T*) ssGetInputPortSignal(S,0);  //输入x
real_T       *y = ssGetOutputPortSignal(S,0); //输出y
int_T step = 0;  //输入x在定点数中的位置
int_T i;
real_T yval;

for (i = 0; i < mapSize[0]; i++)
{
if (x[0] >= map[i] && x[0] < map[i + 1])
{
step = i;
break;
}
}

cubic_getval(&yval, mapSize, map, x[0], step);
y[0] = yval;

}

//自然边界的三次样条曲线函数
void cubic_getval(real_T* y, const int_T* size, const real_T* map, const real_T x, const int_T step)
{
int_T n = size[0];

//曲线系数
real_T* ai = (real_T*)malloc(sizeof(real_T) * (n-1));
real_T* bi = (real_T*)malloc(sizeof(real_T) * (n-1));
real_T* ci = (real_T*)malloc(sizeof(real_T) * (n-1));
real_T* di = (real_T*)malloc(sizeof(real_T) * (n-1));

real_T* h = (real_T*)malloc(sizeof(real_T) * (n-1));  //x的??

/* M矩阵的系数
*[B0, C0, ...
*[A1, B1, C1, ...
*[0,  A2, B2, C2, ...
*[0, ...             An-1, Bn-1]
*/
real_T* A = (real_T*)malloc(sizeof(real_T) * (n-2));
real_T* B = (real_T*)malloc(sizeof(real_T) * (n-2));
real_T* C = (real_T*)malloc(sizeof(real_T) * (n-2));
real_T* D = (real_T*)malloc(sizeof(real_T) * (n-2)); //等号右边的常数矩阵
real_T* E = (real_T*)malloc(sizeof(real_T) * (n-2)); //M矩阵

real_T* M = (real_T*)malloc(sizeof(real_T) * (n));  //包含端点的M矩阵

int_T i;

//计算x的步长
for ( i = 0; i < n -1; i++)
{
h[i] = map[i + 1] - map[i];
}

//指定系数
for( i = 0; i< n - 3; i++)
{
A[i] = h[i]; //忽略A[0]
B[i] = 2 * (h[i] + h[i+1]);
C[i] = h[i+1]; //忽略C(n-1)
}

//指定常数D
for (i = 0; i<n - 3; i++)
{
D[i] = 6 * ((map[n + i + 2] - map[n + i + 1]) / h[i + 1] - (map[n + i + 1] - map[n + i]) / h[i]);
}

//求解三对角矩阵，结果赋值给E
TDMA(E, n-3, A, B, C, D);

M[0] = 0; //自然边界的首端M为0
M[n-1] = 0;  //自然边界的末端M为0
for(i=1; i<n-1; i++)
{
M[i] = E[i-1]; //其它的M值
}

//?算三次?条曲?的系数
for( i = 0; i < n-1; i++)
{
ai[i] = map[n + i];
bi[i] = (map[n + i + 1] - map[n + i]) / h[i] - (2 * h[i] * M[i] + h[i] * M[i + 1]) / 6;
ci[i] = M[i] / 2;
di[i] = (M[i + 1] - M[i]) / (6 * h[i]);
}

*y = ai[step] + bi[step]*(x - map[step]) + ci[step] * (x - map[step]) * (x - map[step]) + di[step] * (x - map[step]) * (x - map[step]) * (x - map[step]);

free(h);
free(A);
free(B);
free(C);
free(D);
free(E);
free(M);
free(ai);
free(bi);
free(ci);
free(di);

}

void TDMA(real_T* X, const int_T n, real_T* A, real_T* B, real_T* C, real_T* D)
{
int_T i;
real_T tmp;

//上三角矩阵
C[0] = C[0] / B[0];
D[0] = D[0] / B[0];

for(i = 1; i<n; i++)
{
tmp = (B[i] - A[i] * C[i-1]);
C[i] = C[i] / tmp;
D[i] = (D[i] - A[i] * D[i-1]) / tmp;
}

//直接求出X的最后一个值
X[n-1] = D[n-1];

//逆向迭代， 求出X
for(i = n-2; i>=0; i--)
{
X[i] = D[i] - C[i] * X[i+1];
}
}

#define MDL_UPDATE
#if defined(MDL_UPDATE)
static void mdlUpdate(SimStruct *S, int_T tid)
{
}
#endif

#define MDL_DERIVATIVES
#if defined(MDL_DERIVATIVES)
static void mdlDerivatives(SimStruct *S)
{
}
#endif

static void mdlTerminate(SimStruct *S)
{
}

#ifdef  MATLAB_MEX_FILE
#include "simulink.c"
#else
#include "cg_sfun.h"
#endif