我有一个拉格朗日插值算法，经过许多时间步后开始发散，我似乎无法弄清楚为什么。快速回顾一下，如果我有两个阵列
int x[11] = {0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100}
int y[11] = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
并且我在算法中输入了15的x值，输出（即插值的y值）应为3.以下算法使插值正确，但随着循环增量输入最终输出开始发散。我不确定是什么导致了分歧。代码创建两个从-100到+100的整数数组，并根据递增的x输入插值。值开始匹配，但大约55左右，插值的y值开始发散。代码如下。任何见解将不胜感激。
#include <stdio.h>
#define SIZE 201
int main()
{
double x[SIZE], y[SIZE], value, sum, factor[SIZE];
for (int i = 0; i < SIZE; i++)
{
x[i] = -100 + i;
}
for (int i = 0; i < SIZE; i++)
{
y[i] = -100 + i;
}
value = 0.0;
while (1)
{
sum = 0.0;
printf("Input is: %lf\n", value);
for(int i = 0; i < SIZE; i++)
{
factor[i] = 1.0;
for(int j = 0; j < SIZE; j++)
{
if(i != j)
{
factor[i] = factor[i] * (value - x[j])/(x[i] - x[j]);
}
}
sum = sum + factor[i] * y[i];
}
printf("Output is: %lf\n", sum);
// if ((value - sum) > 0.01) break;
if (value < 100) value += 0.001;
else break;
}
return 0;
}
答案 0 :(得分：2)
给定N个样本，拉格朗日多项式的度数为N，在您的情况下为200.这是一个相当大的度数，对于value足够大的中间数结果（即factor）开始表现得非常不稳定。我在外循环的每次迭代之后打印factor，这就是我在我的机器上的代码（代码在value处分开62.1）：
Input is: 62.100000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000002
-0.000010
0.000047
-0.000212
0.000916
-0.003834
0.015558
-0.061214
0.233667
-0.865800
3.115490
-10.892598
37.019453
-122.351630
393.413839
-1231.176112
3751.320027
-11132.603776
32188.920849
-90709.929863
249216.884601
-667734.556134
1745251.075381
-4451006.398268
11079451.201015
-26924437.939740
63892139.532124
-148088119.614110
335320733.480250
-741923910.135582
1604370277.576735
-3391407192.476863
7009154161.161494
-14165714298.372702
28000907070.092331
-54142322716.578796
102423636122.796417
-189594810732.400879
343460854643.929565
-608991796878.604858
1057024952593.679688
-1796190002106.606201
2988571833231.810547
-4869319260810.958008
7769844431032.450195
-12143392562153.164062
18590594785582.515625
-27881222667878.292969
40966915113033.500000
-58978430272092.585938
83200365623915.609375
-115016713573880.578125
155822462931701.031250
-206899948714767.906250
269263745978734.968750
-343484172924072.000000
429506076055356.812500
-526485323131795.875000
632668952077638.500000
-745344926690223.250000
860883054405210.375000
-974879663742953.625000
1082405867199472.750000
-1178344322529343.250000
1257784740974879.500000
-1316436663535827.500000
1351011674779421.000000
-1359527880018181.000000
1341497535715305.750000
-1297973187760748.750000
1231446261994579.000000
-1145611653890577.250000
1045029162299372.250000
-934724762872858.125000
819779917571950.250000
-704954924193421.250000
594383648437451.875000
-491363855983359.125000
398252366806871.062500
-316460003024013.937500
246529925760965.156250
-188275766805651.375000
140953330916004.437500
-103441104978326.546875
74409294797142.390625
-52463275136077.218750
36253867741005.359375
-24552698830046.890625
16295359517452.095703
-10597930155882.712891
6753701701870.307617
-4216931244837.877441
2579609371842.192871
-1545902253614.920410
907502140862.699341
-521815325006.697205
293869684950.453308
-162078786010.080200
87537719445.039368
-46294138480.248749
23970808358.990040
-12151499571.568396
6030219319.167710
-2929269302.567177
1392763710.430415
-648125926.103312
295175902.544337
-131559694.388017
57381635.324659
-24492187.039684
10230449.673919
-4182126.956840
1673313.368037
-655394.261464
251347.530884
-94414.677824
34753.964878
-12544.686326
4444.407825
-1547.546444
530.612164
-179.651152
60.317931
-20.218974
6.844738
-2.391284
0.904560
-0.429040
1.136162
0.029430
-0.003145
0.000450
-0.000070
0.000011
-0.000002
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
0.000000
-0.000000
Output is: 67.164298
很抱歉粘贴量。
您可以看到代码以非常大的factor s（例如155822462931701.031250）运行，而sum保持在1左右。这会导致由于规范化而导致精度下降。随着value的增长，精度的损失会放大。
底线是，天真的拉格朗日插值在数值上不稳定。查看Notes。