误差理论与数据处理（费业泰）最全课后答案

得

d11?0.0819 d22?0.0819

得

?x??d11?0.03820.0819?0.0109

?y??d22?0.03820.0819?0.0109

方法二（按矩阵形式计算）：由误差方程V?L?AX v1?2.9?(3x?2y)?? v2?0.9?(x?2y)?上式可以表示为 v3?1.9?(2x?3y)???v1??l1??31??v???l???1?2??x? ?2??2????y?????v3????l3????2?3???即

?v1??l1??2.9??31???x??；L??l???0.9?； A??1?2?； XV??v??y?可得 ?2??2???????????1.9???2?3???v3???l3????x?X????C?1ATL?(ATA)?1ATL

?y??式中

C?1?(ATA)?1??31???1312???14?5??????????1?2?????514? 1?2?3???23??????????145?11?145??????14?5??145?171?145??514－1所以

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?x?X????C?1ATL?y???2.9?145312????1?0.9??????171?145??1?2?3?????1.9???2.9????1?47413 ?0.9????171?29?23?32???1.9???1?164.6??171??2.6??0.9626?＝???0.0152?即解得，

x?0.9626??

y?0.0152?将最佳估计值代入误差方程可得，

?v1??V?L?AX???v2???v3???l1??31????1?2??x? ??l?2????y?????l3????2?3????2.9??31????1?2??0.9626???0.9??0.0152????????1.9????2?3??

?0.0030??????0.0322????0.0204??将计算得到的数据代入式中

???v131.46?10－3??0.0382 n?t3-2

2i2)。 为求出估计量x，y的标准差，首先求出不定常数dij (i，j?1，由已知，不定常数dij的系数与正规方程的系数相同，因而dij是矩阵C?1中各元素，即

d??d1?145?C?1??1112???145? dd171???2122?

14

则

d11?可得估计量的标准差为

1414?0.0819 d22??0.0819 171171?x??d11?0.03820.0819?0.0109

?y??d22?0.03820.0819?0.0109

5-3 测力计示值与测量时的温度t的对应值独立测得如下表所示。 15 18 21 24 t/?C F/N 43.61 43.63 43.68 43.71 27 43.74 30 43.78 设t无误差，t值随t的变化呈线性关系F?k0?kt，试给出线性方程中系数k0和k的最小二乘估计及其相应精度。

解：利用矩阵求解，误差方程V?L?AX可写成

?v1??l1??1?v??l??1?2??2???v3??l3??1????????v4??l4??1?v5??l5??1??????v6????l6????1?15?18??21??k0??? 24???k?27??30???即

?v1??l1??43.61??1?v??l??43.63??122????????v??l??43.68??1； A?V??3?；L??3??????v4??l4??43.71??1?v5??l5??43.74??1???????vl43.78????????1??6??6??15?18???21??k0?X?；??k? 24???27??30?? 可得

?k?X??0??C?1ATL?(ATA)?1ATL

?k??式中

15

C?1?(ATA)?1??1??1????111111??1=??????151821242730??1??1?????1?135??6=???1353165??3195?135?１=6135?6???135?1353165=１?3195?135?6?945???135?－115???18???21????24??27????30?????1

所以

?k?X??0??C?1ATL?k???43.61??43.63???１?3195?135??111111??43.68????151821242730??43.71? ?1356945????????43.74???43.78?????43.4324?????0.01152?将最佳估计值代入误差方程V?L?AX得

?v1??l1??1?v??l??1?2??2???v??l??1V??3???3?－??v4??l4??1?v5??l5??1???????1?v6????l6????0.0048???0.00976????0.00568????0.00112????0.00344???0.002????

16

?15?18??21??43.4324???24???0.01152?27??30??