《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3、f the guerrilla forces in the surrounding area have been withdrawn, activities in the Lakes Cheng Wanjun also refuge in Japanese, Zhu Xi forces became the puppet to attack key targets Eve puppet attacking in Wuzhen, face the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

25、ituation 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)f(), R(x),(x，1)(x,1)(x,3)(x,4

26、),(x，1)(x,1)(x,3)(x,4)34!3、(p.55，题13)依据下列数据表，试用线性插值和抛物线插值分别计算的近sin(0.3367)似值并估计误差。 i0 1 2 x0.32 0.34 0.36 isin(x)0.314567 0.333487 0.352274 i(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(1) 线性插值 x,0.3367因为在节点和之间，先估计误差 xx01max()()x,xx,x()sin()f,01()()()()()Rx,x,xx,x,x,xx,x, 101012!2220.0114,，10;须保留到

27、小数点后4为，计算过程多余两位。 22y2(x-x)/410y=(x-x)(x-x)01x0xx01 x,xx,x101,sin(x)，sin(x),(x,x)sin(x)，(x,x)sin(x)P(x) 0101101x,xx,xx,x0110101,(0.3367,0.32)sin(0.34)，(0.34,0.3367)sin(0.32)P(x) 10.021,0.0167，sin(0.34)，0.0033，sin(0.32) 0.02,0.3304 emotionalBecause of the guerrilla forces in the surrounding area have

28、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 leadership, some believe that the elves, concentrate a sup

29、erior 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, a Chinese miracle in the history of guerrilla warfar

30、e.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 the command personnel brought an urgent intelligence,

31、 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 deserted and desolate. 7th day in implementing low-leve

32、l 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 situation is more severe in the town. One afternoon, outs

33、ide 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 【思路岛答案网】 整理提供 (2) 抛物线插值 插值误差: f(),cos(), ,(x,x)(x,x)(x,x),(x,x)(x,x)(x,x)R(x)01201223!63

34、max(x,x)(x,x)(x,x)3，0.011,6012 ,，10662yy=(x-x)(x-x)(x-x)0123Max=3(x-x)/810x0xxx012 抛物线插值公式为: P(x)2(x,x)(x,x)(x,x)(x,x)(x,x)(x,x)021012,sin(x)，sin(x)，sin(x) 012(x,x)(x,x)(x,x)(x,x)(x,x)(x,x)010210122120(,)(,)(,)(,)xxxxxxxx1，1012,sin()，(,)(,)sin(),sin()xxxxxxx002122,220.02，P(0.3367)25,10,3.8445，sin(0.

35、32)，38.911，sin(0.34),2.7555，sin(0.36) 20.025,10,0.33037439？,3.8445，sin(0.32)，38.911，sin(0.34),2.7555，sin(0.36) 20.02经四舍五入后得:P(0.3367),0.330374，与精确值相比sin(0.3367),0.330374191？2较，在插值误差范围内完全吻合 1.3分段插值与样条函数 32,x，x0,x,1S(x),1、(p.56，习题33)设分段多项式 ,32xbxcxx2，,11,2,是以0,1,2为节点的三次样条函数，试确定系数b，c的值. 【解】依题意，要求S(x)在x

36、=1节点 emotionalBecause 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 d

37、ebate 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 victories, morale, Wuzhen battle to victo

38、ry, 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 feeling of a catastrophe in the town. 6th

39、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 boat in the river. Past the bustlin

40、g 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 Miao came Nanxun sniper assault of th

41、e 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 Wen Yongzhi, Director of 3232函数值连续: ，

42、S(1),1，1,2，1，b，1，c，1,1,S(1),，即: b，c,1(1)22一阶导数连续: ， S(1),3，1，2，1,6，1，2，b，1，c,S(1) ,，即: 2b，c,1(2)解方程组(1)和(2)，得，即 b,2,c,332,x，x0,x,1 S(x),32xxxx2,2，3,11,2, 由于，所以S(x) 在x=1节点的二阶S(1),3，2，1，2,6，2，1,2，2,S(1),，导数亦连续。 12、 已知函数y, 的一组数据，和，x,0,x,1,x,2y,1,y,0.5,y,0.201201221，x(1)求其分段线性插值函数; (2)计算的近似值，并根据余项表达式估计误

43、差。 f(1.5)【解】(1)依题意，将x分为0,1和1,2两段，对应的插值函数为，利用S(x)和S(x)12拉格朗日线性插值公式，求得 x,xx,xx,1x,001 ; S(x),y，y,，1，0.5,0.5x，1101x,xx,x0,11,00110x,xx,xx,2x,121 S(x),y，y,，0.5，0.2,0.3x，0.8 212x,xx,x1,22,112211f(1.5),0.30769230769？(2)，而 ，S(1.5),0.3，1.5，0.8,0.35221，1.5实际误差为:。 |f(1.5),S(1.5)|,0.0423？,0.05222,2x,2(1,3x)24x

44、(1,x)(1)(2)(3)f(x),f(x),f(x),由,可222324(1，x)(1，x)(1，x)(2)知，则余项表达式 M,f(1),0.52(2)M|f()|,242R(x),|(x,1)(x,2)|,，0.5,0.5,0.0625,0.5 2!2!1.4 曲线拟合 1、(p.57，习题35)用最小二乘法解下列超定方程组: emotionalBecause of the guerrilla forces in the surrounding area have been withdrawn, activities in the Lakes Cheng Wanjun also ref

45、uge 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 leadership, some believe that the elves, concentrate a superior force, surely do, early transfer is the best way to red

46、uce 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, a Chinese miracle in the history of guerrilla warfare.Commander Zhu Xi did not have much, but the final word. On

47、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 the command personnel brought an urgent intelligence, town-wide panic, in the face of news of the Japanese attack

48、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 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