插值

Matlab 实现分段线性插值不需要编制函数程序，Matlab 中有现成的一维插值函
数 interp1。
y=interp1(x0,y0,x,‘method’)
method 指定插值的方法，默认为线性插值。其值可为：
‘nearest’ 最近项插值
‘linear’ 线性插值
‘spline’ 逐段 3 次样条插值
‘cubic’ 保凹凸性 3 次插值。
所有的插值方法要求 x0 是单调的。
当 x0 为等距时可以用快速插值法，使用快速插值法的格式为’*nearest’、’*linear’、
‘*spline’、’*cubic’。

function y=lagrange(x0,y0,x); 
n=length(x0);m=length(x); 
for i=1:m
    z=x(i); 
    s=0.0; 
    for k=1:n 
        p=1.0; 
    for j=1:n 
        if j~=k 
            p=p*(z-x0(j))/(x0(k)-x0(j)); 
        end 
    end 
    s=p*y0(k)+s; 
    end 
    y(i)=s; 
end

clc,clear 
x0=[0 3 5 7 9 11 12 13 14 15];
y0=[0 1.2 1.7 2.0 2.1 2.0 1.8 1.2 1.0 1.6]; 
x=0:0.1:15; 
y1=lagrange(x0,y0,x); %调用前面编写的Lagrange插值函数
y2=interp1(x0,y0,x); 
y3=interp1(x0,y0,x,'spline'); 
pp1=csape(x0,y0); y4=ppval(pp1,x); 
pp2=csape(x0,y0,'second'); y5=ppval(pp2,x); 
fprintf('比较一下不同插值方法和边界条件的结果:\n') 
fprintf('x y1 y2 y3 y4 y5\n') 
xianshi=[x',y1',y2',y3',y4',y5']; 
fprintf('%f\t%f\t%f\t%f\t%f\t%f\n',xianshi') 
subplot(2,2,1), plot(x0,y0,'+',x,y1), title('Lagrange') 
subplot(2,2,2), plot(x0,y0,'+',x,y2), title('Piecewise linear') 
subplot(2,2,3), plot(x0,y0,'+',x,y3), title('Spline1') 
subplot(2,2,4), plot(x0,y0,'+',x,y4), title('Spline2') 
dyx0=ppval(fnder(pp1),x0(1)) %求x=0处的导数
ytemp=y3(131:151); 
index=find(ytemp==min(ytemp)); 
xymin=[x(130+index),ytemp(index)]

result
比较一下不同插值方法和边界条件的结果:
x y1 y2 y3 y4 y5
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
0.100000 -4.943720 0.040000 0.049861 0.049716 0.044073
0.200000 -8.819979 0.080000 0.098997 0.098723 0.088118
0.300000 -11.772560 0.120000 0.147411 0.147027 0.132109
0.400000 -13.931018 0.160000 0.195109 0.194630 0.176019
0.500000 -15.411743 0.200000 0.242097 0.241537 0.219819
0.600000 -16.318971 0.240000 0.288378 0.287753 0.263484
0.700000 -16.745729 0.280000 0.333957 0.333280 0.306986
0.800000 -16.774737 0.320000 0.378840 0.378124 0.350298
0.900000 -16.479241 0.360000 0.423032 0.422288 0.393392
1.000000 -15.923810 0.400000 0.466537 0.465776 0.436241
1.100000 -15.165076 0.440000 0.509361 0.508593 0.478819
1.200000 -14.252433 0.480000 0.551507 0.550743 0.521098
1.300000 -13.228686 0.520000 0.592982 0.592230 0.563050
1.400000 -12.130662 0.560000 0.633790 0.633057 0.604649
1.500000 -10.989778 0.600000 0.673935 0.673230 0.645868
1.600000 -9.832573 0.640000 0.713423 0.712751 0.686679
1.700000 -8.681200 0.680000 0.752260 0.751626 0.727055
1.800000 -7.553886 0.720000 0.790448 0.789858 0.766969
1.900000 -6.465357 0.760000 0.827994 0.827451 0.806393
2.000000 -5.427232 0.800000 0.864903 0.864410 0.845301
2.100000 -4.448386 0.840000 0.901179 0.900738 0.883666
2.200000 -3.535284 0.880000 0.936827 0.936440 0.921460
2.300000 -2.692293 0.920000 0.971852 0.971520 0.958656
2.400000 -1.921962 0.960000 1.006259 1.005981 0.995226
2.500000 -1.225278 1.000000 1.040054 1.039829 1.031145
2.600000 -0.601908 1.040000 1.073240 1.073067 1.066384
2.700000 -0.050409 1.080000 1.105822 1.105699 1.100916
2.800000 0.431580 1.120000 1.137807 1.137729 1.134714
2.900000 0.847155 1.160000 1.169198 1.169161 1.167751
3.000000 1.200000 1.200000 1.200000 1.200000 1.200000
3.100000 1.494237 1.225000 1.230219 1.230250 1.231441
3.200000 1.734308 1.250000 1.259859 1.259915 1.262087
3.300000 1.924859 1.275000 1.288925 1.289002 1.291956
3.400000 2.070648 1.300000 1.317422 1.317515 1.321069
3.500000 2.176456 1.325000 1.345355 1.345459 1.349445
3.600000 2.247018 1.350000 1.372729 1.372841 1.377103
3.700000 2.286959 1.375000 1.399549 1.399664 1.404064
3.800000 2.300741 1.400000 1.425819 1.425935 1.430347
3.900000 2.292622 1.425000 1.451545 1.451659 1.455972
4.000000 2.266621 1.450000 1.476731 1.476840 1.480957
4.100000 2.226489 1.475000 1.501383 1.501485 1.505324
4.200000 2.175692 1.500000 1.525505 1.525598 1.529091
4.300000 2.117392 1.525000 1.549102 1.549185 1.552278
4.400000 2.054444 1.550000 1.572180 1.572251 1.574905
4.500000 1.989390 1.575000 1.594742 1.594801 1.596991
4.600000 1.924459 1.600000 1.616794 1.616841 1.618556
4.700000 1.861574 1.625000 1.638341 1.638375 1.639620
4.800000 1.802357 1.650000 1.659388 1.659410 1.660202
4.900000 1.748145 1.675000 1.679939 1.679950 1.680322
5.000000 1.700000 1.700000 1.700000 1.700000 1.700000
5.100000 1.658724 1.715000 1.719574 1.719565 1.719251
5.200000 1.624880 1.730000 1.738664 1.738647 1.738075
5.300000 1.598807 1.745000 1.757269 1.757246 1.756470
5.400000 1.580641 1.760000 1.775390 1.775361 1.774431
5.500000 1.570336 1.775000 1.793028 1.792995 1.791955
5.600000 1.567683 1.790000 1.810183 1.810147 1.809039
5.700000 1.572332 1.805000 1.826856 1.826818 1.825679
5.800000 1.583812 1.820000 1.843048 1.843009 1.841871
5.900000 1.601550 1.835000 1.858758 1.858720 1.857612
6.000000 1.624894 1.850000 1.873989 1.873950 1.872899
6.100000 1.653128 1.865000 1.888739 1.888702 1.887728
6.200000 1.685493 1.880000 1.903011 1.902976 1.902096
6.300000 1.721203 1.895000 1.916803 1.916771 1.915998
6.400000 1.759462 1.910000 1.930118 1.930089 1.929432
6.500000 1.799475 1.925000 1.942955 1.942931 1.942394
6.600000 1.840469 1.940000 1.955316 1.955295 1.954880
6.700000 1.881697 1.955000 1.967200 1.967184 1.966888
6.800000 1.922455 1.970000 1.978608 1.978597 1.978412
6.900000 1.962088 1.985000 1.989541 1.989536 1.989451
7.000000 2.000000 2.000000 2.000000 2.000000 2.000000
7.100000 2.035659 2.005000 2.009983 2.009989 2.010056
7.200000 2.068604 2.010000 2.019486 2.019497 2.019612
7.300000 2.098447 2.015000 2.028500 2.028516 2.028665
7.400000 2.124877 2.020000 2.037018 2.037040 2.037208
7.500000 2.147659 2.025000 2.045033 2.045060 2.045235
7.600000 2.166637 2.030000 2.052539 2.052570 2.052742
7.700000 2.181730 2.035000 2.059526 2.059562 2.059721
7.800000 2.192931 2.040000 2.065990 2.066029 2.066169
7.900000 2.200303 2.045000 2.071921 2.071963 2.072079
8.000000 2.203975 2.050000 2.077314 2.077358 2.077445
8.100000 2.204136 2.055000 2.082160 2.082205 2.082263
8.200000 2.201030 2.060000 2.086453 2.086498 2.086526
8.300000 2.194947 2.065000 2.090185 2.090229 2.090229
8.400000 2.186220 2.070000 2.093349 2.093391 2.093366
8.500000 2.175213 2.075000 2.095937 2.095976 2.095933
8.600000 2.162316 2.080000 2.097944 2.097978 2.097922
8.700000 2.147933 2.085000 2.099360 2.099389 2.099329
8.800000 2.132479 2.090000 2.100180 2.100201 2.100148
8.900000 2.116366 2.095000 2.100396 2.100407 2.100374
9.000000 2.100000 2.100000 2.100000 2.100000 2.100000
9.100000 2.083767 2.095000 2.098992 2.098979 2.099027
9.200000 2.068032 2.090000 2.097393 2.097366 2.097475
9.300000 2.053126 2.085000 2.095232 2.095190 2.095370
9.400000 2.039341 2.080000 2.092538 2.092480 2.092738
9.500000 2.026927 2.075000 2.089339 2.089265 2.089604
9.600000 2.016081 2.070000 2.085663 2.085573 2.085995
9.700000 2.006948 2.065000 2.081538 2.081434 2.081936
9.800000 1.999614 2.060000 2.076993 2.076876 2.077453
9.900000 1.994101 2.055000 2.072056 2.071928 2.072573
10.000000 1.990373 2.050000 2.066756 2.066619 2.067320
10.100000 1.988327 2.045000 2.061121 2.060978 2.061720
10.200000 1.987799 2.040000 2.055179 2.055032 2.055800
10.300000 1.988562 2.035000 2.048958 2.048813 2.049585
10.400000 1.990331 2.030000 2.042488 2.042347 2.043102
10.500000 1.992767 2.025000 2.035795 2.035664 2.036375
10.600000 1.995482 2.020000 2.028910 2.028792 2.029431
10.700000 1.998043 2.015000 2.021859 2.021762 2.022296
10.800000 1.999983 2.010000 2.014672 2.014600 2.014995
10.900000 2.000806 2.005000 2.007376 2.007337 2.007555
11.000000 2.000000 2.000000 2.000000 2.000000 2.000000
11.100000 1.997046 1.980000 1.992444 1.992490 1.992234
11.200000 1.991427 1.960000 1.984097 1.984191 1.983662
11.300000 1.982644 1.940000 1.974218 1.974358 1.973567
11.400000 1.970227 1.920000 1.962067 1.962246 1.961232
11.500000 1.953746 1.900000 1.946903 1.947110 1.945939
11.600000 1.932829 1.880000 1.927988 1.928205 1.926971
11.700000 1.907170 1.860000 1.904580 1.904787 1.903610
11.800000 1.876546 1.840000 1.875939 1.876110 1.875140
11.900000 1.840830 1.820000 1.841326 1.841429 1.840842
12.000000 1.800000 1.800000 1.800000 1.800000 1.800000
12.100000 1.754153 1.740000 1.751566 1.751427 1.752222
12.200000 1.703516 1.680000 1.697012 1.696712 1.698421
12.300000 1.648451 1.620000 1.637667 1.637206 1.639838
12.400000 1.589463 1.560000 1.574864 1.574259 1.577713
12.500000 1.527208 1.500000 1.509935 1.509224 1.513285
12.600000 1.462486 1.440000 1.444212 1.443451 1.447794
12.700000 1.396248 1.380000 1.379024 1.378291 1.382480
12.800000 1.329586 1.320000 1.315706 1.315095 1.318583
12.900000 1.263723 1.260000 1.255587 1.255214 1.257343
13.000000 1.200000 1.200000 1.200000 1.200000 1.200000
13.100000 1.139856 1.180000 1.150090 1.150601 1.147680
13.200000 1.084801 1.160000 1.106257 1.107361 1.101054
13.300000 1.036388 1.140000 1.068714 1.070419 1.060680
13.400000 0.996167 1.120000 1.037675 1.039916 1.027117
13.500000 0.965644 1.100000 1.013355 1.015994 1.000922
13.600000 0.946220 1.080000 0.995966 0.998791 0.982654
13.700000 0.939129 1.060000 0.985722 0.988450 0.972869
13.800000 0.945361 1.040000 0.982838 0.985111 0.972127
13.900000 0.965575 1.020000 0.987526 0.988914 0.980984
14.000000 1.000000 1.000000 1.000000 1.000000 1.000000
14.100000 1.048324 1.060000 1.020474 1.018567 1.029460
14.200000 1.109568 1.120000 1.049162 1.045044 1.068564
14.300000 1.181945 1.180000 1.086278 1.079918 1.116241
14.400000 1.262704 1.240000 1.132034 1.123675 1.171419
14.500000 1.347955 1.300000 1.186645 1.176801 1.233026
14.600000 1.432481 1.360000 1.250325 1.239783 1.299991
14.700000 1.509522 1.420000 1.323286 1.313108 1.371243
14.800000 1.570545 1.480000 1.405743 1.397261 1.445709
14.900000 1.604991 1.540000 1.497910 1.492730 1.522319
15.000000 1.600000 1.600000 1.600000 1.600000 1.600000

dyx0 =

0.5007

xymin =

13.8000 0.9828