《数值分析简明教程》第二版(王能超 编著)课后习题答案高等教育出版社.doc
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1、0.1算法 31、 (p.11,题1)用二分法求方程在1,2内的近似根,要求误差不x,x,1,0-3超过10. ,1ba*,3【解】 由二分法的误差估计式,得到|,|,10xxkkk,1,1223ln10k,1k,9.两端取自然对数得,因此取,即至少需k,1,8.962,1000ln2二分9次.求解过程见下表。 k 符号 bf(x)axkkkk0 1 2 1.5 + 1 2 3 4 5 6 7 8 9 x2、(p.11,题2) 证明方程在区间0,1内有唯一个实根;使用f(x),e,10x,21,2,10二分法求这一实根,要求误差不超过。 2x【解】 由于,则在区间0,1上连续,且f(x)f(x
2、),e,10x,201,即,f(0),f(1),0f(0),e,10,0,2,1,0f(1),e,10,1,2,e,8,0由连续函数的介值定理知,在区间0,1上至少有一个零点. f(x)x又,即在区间0,1上是单调的,故在区间0,1内f(x)f(x)f(x),e,10,0有唯一实根. ,11ba*,2k|,|,,102,100xx由二分法的误差估计式,得到.kkk,1,12222ln10k,7k,2,3.3219,6.6438两端取自然对数得,因此取,即至少需二分ln27次.求解过程见下表。 k bf(x)ax 符号 kkkk0 0 1 0.5 1 2 3 emotionalBecause o
3、f the guerrilla forces in the surrounding area have been withdrawn, activities in the Lakes Cheng Wanjun also refuge in Japanese, Zhu Xi forces became the puppet to attack key targets Eve puppet attacking in Wuzhen, face the problem of the retreat was to meet the enemy, in a debate within the leader
4、ship, some believe that the elves, concentrate a superior force, surely do, early transfer is the best way to reduce losses. Deputy Commander Wang Hesong impassioned statement: Wuzhen hold, is our militarys duty! I have win a series of victories, morale, Wuzhen battle to victory, fight this battle,
5、a Chinese miracle in the history of guerrilla warfare.Commander Zhu Xi did not have much, but the final word. On November 5, we flew over the town for several days, bomb bursts, trembling moan tranquil town in the war. People have the feeling of a catastrophe in the town. 6th reconnaissance around t
6、he command personnel brought an urgent intelligence, town-wide panic, in the face of news of the Japanese attack town, to avoid the scourge of war, the majority of people young and old fled to the countryside in the town. Also hard to find a boat in the river. Past the bustling town just seems a des
7、erted and desolate. 7th day in implementing low-level flights over the town, fired. That night, the Elves attack Wuzhen confirmed information. Night, bursts of gunfire coming from the distance, reportedly is Deputy Commander Wang Hesong Xu Miao came Nanxun sniper assault of the Elves. 8th, the situa
8、tion is more severe in the town. One afternoon, outside East Gate and South Gate there was intense gunfire. Enemy attacks by frustrated independent battalion confronted Chen qi. At this time, the command . 11th, as Commander of Zhu Xi-Kai Wen Yongzhi, Director of 4 5 6 7 0.2误差 1(p.12,题8)已知e=2.71828,
9、试问其近似值,x=2.71,x,2.718x,2.7x,2.712312各有几位有效数字,并给出它们的相对误差限。 【解】有效数字: 1,1 因为,所以有两位有效数字; |e,x|,0.01828?,0.05,,10x,2.71121,1 因为|e,x|,0.00828?,0.05,,10,所以亦有两位有效数字; x,2.712221,3 因为|e,x|,0.00028?,0.0005,,10,所以有四位有效数字; x,2.718332|e,x|0.051; ,1.85%1rx2.71|e,x|0.052,1.85%; 2rx2.712|e,x|0.00053,0.0184%。 3rx2.71
10、83评 (1)经四舍五入得到的近似数,其所有数字均为有效数字; (2)近似数的所有数字并非都是有效数字. 2(p.12,题9)设,x,0.0718均为经过四舍五入得出的近x,2.72x,2.71828312似值,试指明它们的绝对误差(限)与相对误差(限)。 ,0.005,31,1.84,10【解】 ,; ,0.005,r11x2.721,0.000005,62,1.84,10,0.000005,; ,r22x2.718282,0.00005,43,6.96,10,0.00005,; ,r33x0.07183评 经四舍五入得到的近似数,其绝对误差限为其末位数字所在位的半个单位. ,4x,1.42
11、x,0.0184x,184,103(p.12,题10)已知,的绝对误差限均为123emotionalBecause of the guerrilla forces in the surrounding area have been withdrawn, activities in the Lakes Cheng Wanjun also refuge in Japanese, Zhu Xi forces became the puppet to attack key targets Eve puppet attacking in Wuzhen, face the problem of the r
12、etreat was to meet the enemy, in a debate within the leadership, some believe that the elves, concentrate a superior force, surely do, early transfer is the best way to reduce losses. Deputy Commander Wang Hesong impassioned statement: Wuzhen hold, is our militarys duty! I have win a series of victo
13、ries, morale, Wuzhen battle to victory, fight this battle, a Chinese miracle in the history of guerrilla warfare.Commander Zhu Xi did not have much, but the final word. On November 5, we flew over the town for several days, bomb bursts, trembling moan tranquil town in the war. People have the feelin
14、g of a catastrophe in the town. 6th reconnaissance around the command personnel brought an urgent intelligence, town-wide panic, in the face of news of the Japanese attack town, to avoid the scourge of war, the majority of people young and old fled to the countryside in the town. Also hard to find a
15、 boat in the river. Past the bustling town just seems a deserted and desolate. 7th day in implementing low-level flights over the town, fired. That night, the Elves attack Wuzhen confirmed information. Night, bursts of gunfire coming from the distance, reportedly is Deputy Commander Wang Hesong Xu M
16、iao came Nanxun sniper assault of the Elves. 8th, the situation is more severe in the town. One afternoon, outside East Gate and South Gate there was intense gunfire. Enemy attacks by frustrated independent battalion confronted Chen qi. At this time, the command . 11th, as Commander of Zhu Xi-Kai We
17、n Yongzhi, Director of 【思路岛答案网】 整理提供 ,2,问它们各有几位有效数字, 0.5,10,2【解】 由绝对误差限均为知有效数字应从小数点后两位算起,故,有0.5,10x,1.421,4三位;有一位;而,也是有一位。 x,0.0184x,184,10,0.0184231.1泰勒插值和拉格朗日插值 1、(p.54,习题1)求作在节点的5次泰勒插值多项式,并计算x,0p(x)f(x),sinx05和估计插值误差,最后将有效数值与精确解进行比较。 p(0.3367)p(0.5)55(1)(2)(3)【解】由,求得;f(x),sinxf(x),cosxf(x),sinxf(
18、x),cosx(4)(5)(6);,所以 f(x),sinxf(x),cosxf(x),sinx(2)(5)f(x)f(x)(1)2500 ,f(x),f(x)(x,x),(x,x),?,(x,x)p(x)0000052!5!(2)(5)f(0)f(0)(1)25,f(0),f(0)x,x,?,x 2!5!1135,x,x,x 3!5!(6)|f()|sin()|1,666x,0.5,(x,x),(x,x),x插值误差:,若,则 R(x)0056!6!6!350.33670.3367,0.3367,,,0.3303742887,而p(0.3367)53!5!60.3367,6,5R(0.336
19、7),2.02,10,0.5,10,精度到小数点后5位, 56!故取p(0.3367),0.33037,与精确值相比f(0.3367),sin(0.3367),0.330374191?5较,在插值误差的精度内完全吻合 x,1,x,1,x,3,x,42、(p.55,题12)给定节点,试分别对下列函数导出拉格朗0123日余项: 3(1); f(x),4x,3x,243(2) f(x),x,2x(4)3f(),n,3【解】依题意,拉格朗日余项公式为 R(x),(x,x) i,34!i,0(4)R(x),0(1) ? ; f(x),03(4)(2)因为,所以 f(x),4!emotionalBecau
20、se of the guerrilla forces in the surrounding area have been withdrawn, activities in the Lakes Cheng Wanjun also refuge in Japanese, Zhu Xi forces became the puppet to attack key targets Eve puppet attacking in Wuzhen, face the problem of the retreat was to meet the enemy, in a debate within the le
21、adership, some believe that the elves, concentrate a superior force, surely do, early transfer is the best way to reduce losses. Deputy Commander Wang Hesong impassioned statement: Wuzhen hold, is our militarys duty! I have win a series of victories, morale, Wuzhen battle to victory, fight this batt
22、le, a Chinese miracle in the history of guerrilla warfare.Commander Zhu Xi did not have much, but the final word. On November 5, we flew over the town for several days, bomb bursts, trembling moan tranquil town in the war. People have the feeling of a catastrophe in the town. 6th reconnaissance arou
23、nd the command personnel brought an urgent intelligence, town-wide panic, in the face of news of the Japanese attack town, to avoid the scourge of war, the majority of people young and old fled to the countryside in the town. Also hard to find a boat in the river. Past the bustling town just seems a
24、 deserted and desolate. 7th day in implementing low-level flights over the town, fired. That night, the Elves attack Wuzhen confirmed information. Night, bursts of gunfire coming from the distance, reportedly is Deputy Commander Wang Hesong Xu Miao came Nanxun sniper assault of the Elves. 8th, the s
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