数值分析简明教程第二版(王超能)习题答案24页全解word版[1]

?(k?1)2(k)1(k)4?x2?x3??x1555?1(k)3(k)2?(k?1)x??x1?x3?，GJ?2555??(k?1)2(k)1(k)11?x1?x2??x3555?方程组（1）的高斯-赛德尔迭代：

??4?1，迭代收敛。 5?(k?1)2(k)1(k)4?x2?x3??x1555?2(k)16(k)6?(k?1)，GG?Sx??x2?x3??2252525?6(k)321?(k?1)18(k)x?x?x3?2?3125125125?

??18?1，迭代收敛。 256.1 高斯消元法

1、（p.198，题2）用选列主元高斯消元法求解下列方程组：

?x1?x2?x3??4?（1）?5x1?4x2?3x3??12

?2x?x?x?1123?1?2x1?3x2?5x3?5?（2）?3x1?4x2?7x3?6

?x?3x?3x?523?15?4?1?11?4??5?43?12??1r1?r2????r1?r2??51【解】 （1）?5?43?12???1?11?4???0?5??21111??21111???????21

3?12?28??? 55?111?????5?43?12??2r1?r3?5?43?12??5?43?12?5?r3??5???5?r3???0?12?8???0?12?8???0?12?8?

13179???21111???013?1790??????555??????5?43?12?1r2?r3?5?43?12?13r3?5?43?12?r2?r3??13???25???013?179???013?179???013?179?

2525???0?12?8???001?100??????1313???x?794x?3x3?124?6?3?(?1)?12?6,x1?2??3. 所以： x3??1，x2?31355

?2355??3476??2r1?r2?3???r1?r2??3（2）?3476???2355???0??1335??1335??1?????1?r1?r33413376??3476??1?3r2?1???0113? 3???133535??????3??0??0?1?r2?r3?3?5??0?0??476??343r3??113??0152??3??0533?476?5r3?3?3?529???0?03565???0?3476?76??r2?r3??13???0529?

?0113?29????476??529? 012??所以： x3?2，x2??2x3?9?4x2?7x3?6?4?1?7?2?6?1,x1????4. 535

2、（p.199，题9）计算下列三阶坡度阵的条件数：

??1?1（1）??2?1??31213141?3?1??。 4?1?5??1213141?3?1??，先求A-1。 4?1?5???00??1?10? 2?001???11?100?31?6120?

?41?01?453????1?1【解】令：A???2?1??3

??1?1??2?1??3121314111??100?1?1323?r1?r2??2111010???041212??1?111001???5??3451211411??100??1r1?r2?1323??1?6120?01??11?0001?5??12????112r2??0?1??3?

1?r2?r312111???11001??180r3?232????011?6120?01???11?00?00?11?1806??????1?100?31?6120?

?130?180180???1?1?2?r3?r2??01??0011???100??1r3?r1?10?960?60?323??0?36192?180?010?36192?180? ???130?180180??00130?180180????????12r2?r1?1009?3630??36???010?36192?180??9，所以 A?1???36192?130?180180????00???30?180

最后求得条件数为：cond(A)?A?1??A??116?408?748

??30??180?180?

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