Rational interpolation is suitable for interpolating logarithmic or exponential relations of the data points
e.g. NTC thermistor characteristics than Lagrange interpolation, with its better accuracy and less computational complexity.

Example C program

list.1

/* riali : An interpolation program. Rev.1.11 : Oct. 3, 2011 * Copyright (c) 2011, Takayuki HOSODA. All rights reserved. * * SYNOPSIS * #include <math.h> * * double * riali(double x[3], double y[3], double xi, int mode) * * RETURN VALUE * The riali() function return NaN on errors. * * DESCRIPTION * The riali() calculate univariate interpolation for xi from a set of three data points. * The argument mode specifies interpolation method by one of the followings, * AIP_RATIONAL : 1st/1st order rational interpolation * AIP_LAGRANGE : 2nd order Lagrange interpolation * AIP_AUTO : use rational interpolation when the set of data points is * monotonic, otherwise use Lagrange interpolation instead. * LINK * http://www.finetune.jp/~lyuka/technote/riali/ */ #include <math.h> #define AIP_AUTO 0 #define AIP_RATIONAL 1 #define AIP_LAGRANGE 2 double riali(x, y, xi, mode) double x[3]; double y[3]; double xi; int mode; { register double t, u; if (y[0] == y[1] && y[1] == y[2]) return y[0]; if (mode == AIP_AUTO) { if (x[0] > x[1]) {t = x[0]; x[0] = x[1]; x[1] = t; t = y[0]; y[0] = y[1]; y[1] = t;} if (x[1] > x[2]) {t = x[1]; x[1] = x[2]; x[2] = t; t = y[1]; y[1] = y[2]; y[2] = t;} if (x[0] > x[1]) {t = x[0]; x[0] = x[1]; x[1] = t; t = y[0]; y[0] = y[1]; y[1] = t;} mode = ((y[0] < y[1] && y[1] < y[2]) || (y[0] > y[1] && y[1] > y[2])) ? AIP_RATIONAL : AIP_LAGRANGE; } if (mode == AIP_RATIONAL) { t = (xi - x[1]) * (x[2] - x[0]) * (y[2] - y[1]); u = (xi - x[0]) * (x[2] - x[1]) * (y[2] - y[0]); if ((t == 0.) && (u == 0)) return NAN; if ( t == 0.) return y[1]; if ( u == 0.) return y[0]; if (fabs(x[1] - xi) < fabs(x[0] - xi)) { t /= u; return (y[0] * t - y[1]) / (t - 1.0); } u /= t; return (y[1] * u - y[0]) / (u - 1.0); } else { if (x[0] == x[1] || x[1] == x[2] || x[2] == x[0]) return NAN; t = y[0] * (xi - x[1]) * (xi - x[2]) / ((x[0] - x[1]) * (x[0] - x[2])); t += y[1] * (xi - x[0]) * (xi - x[2]) / ((x[1] - x[0]) * (x[1] - x[2])); t += y[2] * (xi - x[0]) * (xi - x[1]) / ((x[2] - x[0]) * (x[2] - x[1])); return t; } } #ifdef DEBUG #include <stdio.h> int main() { int i; double xi, yi; /* Resistance V.S. Temperature table of an NTC thermistor */ double x[6] = { 19.847, 12.478, 8.068, 5.353, 3.635 }; double y[6] = { 10.000, 20.000, 30.000, 40.00, 50.000 }; printf("Res.\tTemp.\tRelative error\n"); /*Extrapolation*/ yi = ri(&x[1], &y[1], xi = 15.679); printf("%g\t%g\t%g\n", xi, yi, yi/15. -1.); /*Interpolation*/ yi = ri(&x[1], &y[1], xi = 12.478); printf("%g\t%g\t%g\n", xi, yi, yi/20. -1.); yi = ri(&x[1], &y[1], xi = 10.000); printf("%g\t%g\t%g\n", xi, yi, yi/25. -1.); yi = ri(&x[1], &y[1], xi = 8.068); printf("%g\t%g\t%g\n", xi, yi, yi/30. -1.); yi = ri(&x[1], &y[1], xi = 6.552); printf("%g\t%g\t%g\n", xi, yi, yi/35. -1.); yi = ri(&x[1], &y[1], xi = 5.353); printf("%g\t%g\t%g\n", xi, yi, yi/40. -1.); /*Extrapolation*/ yi = ri(&x[1], &y[1], xi = 4.399); printf("%g\t%g\t%g\n", xi, yi, yi/45. -1.); return 0; } #endif

Execution results of list.1 compiled with -DDEBUG option.

Execution results	Temperature V.S. Resistance characteristic of an NTC thermistor
Res. Temp. Relative error 15.679 15.2828 0.0188535 12.478 20 0 10 24.9431 -0.00227736 8.068 30 0 6.552 35.0529 0.00151246 5.353 40 0 4.399 44.7465 -0.00563339

Online interpolation program

Sample points
X1 :	Y1 :
X2 :	Y2 :
X3 :	Y3 :
Interpolation
Xi :	Yi :

Download

riali-1.11.js (3600 byte, JavaScript program and input form used in this page.)

Formulas used

Rational interpolation

Lagrange interpolation

Appendix

Interpolation and its error comparison of various transcendental functions

sin(x)	tan(x)
log(x + 1)	atan(x)
exp(-x)	exp(-x2)

SEE ALSO

Ostrowski's method

HP 15C - Rational interpolation

HP 42S - Rational interpolation