插值与拟合总复习

第二章 插值与拟合 .23 . xi yi -1.00 0.2200 -0.50 0.8000 0 2.0000 0.75 2.5000 1.00 3.7500 试对数据做出线性拟合并计算均方误差。

解 设拟合直线p(x)?a0?a1x，写出方程AX?Y：

?1?1.00??0.2200??1?0.50??0.8000????a???0?10?????2.0000? ???a1???10.752.5000???????11.00???3.7500??法方程ATAX?ATY：

?1???1.001?0.5010?1?1.00???1?0.5??a?11??0??10???0.751.00????a1??10.75???11.00???0.2200???0.8000?11????2.0000?0.751.00???2.5000????3.7500???5??0.25

?1????1.001?0.5010化简为

0.25??a0??9.45???a????

2.8125??1??5.005?经计算得：a0?1.80906,a1?1.61875。所以有

p(x)?1.80906?1.61875x R??(p(x)?y)iii?152?0.42

例2.10 对f(x)?xsin(x?2)?x2/5在区间[3,9]按下列数据构造二次拟合函数，

并计算拟合函数的均方误差。

. 24 . 实用数值分析解题指导

解 做二次拟合，设p1(x)?a0?a1x?a2x，则有

2?a0?a1x1?a2x12?y1?2?a0?a1x2?a2x2?y2 ????a?ax?ax2?y15255?0法方程组为：

?1?x?12??x11x22x2????1x11???1x2x5?????2x5?????1x5x???a0??1x????a1?x1?????2???ax?21???2x5??52i21221x22x2????y1?1??y2?????? x5y3??2?x5??????y5????5?5?xi??i?1?5??xi2?i?1??x?x?xi?1i?15i?155i2i3i??5?xy????i?i?1??a0??5i?1?5??3?? ?xi??a1????xiyi??i?1i?1a2?5???5???42??xiyi??xi????i?1??i?1?31.2207.5??a0??42.7766??5?31.2207.51456.66??a1???292.303? ????????207.51456.6610681.731????a2????2083.24??R???i?152i??(pi?152(xi)?f(xi))2?0.34288

例2.11 试对下列数据做出形如a?bx的拟合曲线。

2xi yi 2 1 23 6 5 22 7 46 8 61 解 写出?(x)?a?bx的矛盾方程组：

第二章 插值与拟合 .25 .

?a?bx12?y1?a?bx2?y?22 ???2??a?bx5?y5法方程为：

?1?2?x112x2???1x12??2?1??1x2??a???1???22?x5??????b??x1??21x?5????14???19??a??111????125?????64????b??49?149???164??52i12x2???y1???1?y2?? 2?x5???????y5??1???6??1???22? 64????46???61???11??49125149125149??5?5?x2?i??i?1?5??151?5??yx????a??i?1i?i?1? ?????55??4??b?2xxy?i???ii?i?1??i?1?151??a??136??????? 7219??b??6766?解得：a??3.00，b?1.00。所以有

?(x)??3.00?1.00x2

例2.12 试给出下列数据

xi yi 0.3 1.37731 0.5 1.48766 0.6 1.53879 0.7 1.58653 0.9 1.67 的形如?(x)?a?bsinx的拟合函数。

解 矛盾方程为：

. 26 . 实用数值分析解题指导

?a?bsinx1?y1??a?bsinx2?y2 ?????a?bsinx5?y5法方程组为：

?1??sinx11sinx2?1sinx1????1?1sinx2?a??1???????sinx?sinx5?????b???1???1sinx5??5??5?sinx?i??i?1?5??2.7671351sinx2?y1????1?y2?????

?sinx5?????y5??5??ysinx?i??i??a??i?1i?1? ?????55?2??b??sinx(sinx)y?iii????i?1??i?1?2.76713??a??7.66028???????

1.66462??b??11.7839?解方程组得：a??1.2，b?0.6。所以有

?(x)??1.2?0.6sinx

例2.13 给出下列数据 xi yi 1.25 20.8666 bx1.37 24.4819 1.45 28.667 1.69 39.6726 1.77 46.22 试对数据做出形如y?ae的拟合函数。

解 对函数y?ae两边取自然对数：

bxlna?bx?lny

得到数据表：

xi lnyi 得矛盾方程组：

1.25 3.03815 1.37 3.19793 1.45 3.35575 1.69 3.68.66 1.77 3.83341