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?(k?1)2(k)1(k)4?x2?x3??x1555?2(k)16(k)6?(k?1)x??x2?x3?，GG?S?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?3476??3476????r1?r2??3?11?3r2?（2）?3476???2355???01???0113?

33??1335??1335??1335???1335?????????1?r1?r33??3??0??0?41537123??3476?6??3476?3r3?r2?r3????3??0113???0529? ???0113?3??0529?????

?3476?5r3?3476???3????0529???0529?

?003565??0012??????2x3?9?4x2?7x3?6?4?1?7?2?6?1,x1????4. 所以： x3?2，x2?5351?r2?r35 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??10?

?01?????1?1【解】令：A???2?1??3

??1?1??2?1??312131413141511??100?11?123??2r1?r2?111010???0?12122??1?110001???345???1?12r2??0?1???31?r2?r3121211411??100??1r1?r2?1323??1?6120?01??11?0001?5????121211??100?1?3180r3?1?6120??0??11?00?11?1806????1?100?31?6120?

?41?01?453??12101?100?31?6120?

?130?180180?????1??0??0??1210??1?r3?r2??0??0??11???100??1r3?r1?10?960?60?323??0?36192?180?010?36192?180? ???130?180180??00130?180180???????

1?r2?r129?3630??3630??100?9?，所以 A?1???36192?180?

??010?36192?180???????00130?180180???30?180180???1?A?

最后求得条件数为：cond(A)?A

??11?408?748 6