《数值计算方法》试题集及答案解析

()]∫−求积公式为199222134当f(x)=x时,公式显然精确成立;当f(x)=x时,左=5,右=3。所以代数精度为3。21t=2x−311111811dx=dt≈[+]+[+]∫1x∫−1t+39−1+31+39−1/2+312+397=≈0.692861403、已知xi1345f(x)i2654分别用拉格朗日插值法和牛顿插值法求f(x)的三次插值多项式P3(x),并求f(2)的近似值(保留四位小数)。(x−3)(x−4)(x−5)(x−1)(x−4)(x−5)L3(x)=2+6答案:(1−3)(1−4)(1−5)(3−1)(3−4)(3−5)(x−1)(x−3)(x−5)(x−1)(x−3)(x−4)+5+4(4−1)(4−3)(4−5)(5−1)(5−3)(5−4)差商表为xy一阶均差二阶均差三阶均差ii12936245-1-154-10141P3(x)=N3(x)=2+2(x−1)−(x−1)(x−3)+(x−1)(x−3)(x−4)4≈=f(2)P3(2)5.54、取步长h=0.2,用预估-校正法解常微分方程初值问题y′=2x+3yy(0)=1≤≤(0x1)(0)yn+1=yn+0.2×(2xn+3yn)=+×+++(0)答案:解:yn+1yn0.1[(2xn3yn)(2xn+13yn+1)]=++即yn+10.52xn1.78yn0.04n012345xn00.20.40.60.81.0y11.825.879610.713719.422435.0279n5、已知xi-2-1012f(x)i42135′求f(x)的二次拟合曲线p2(x),并求f(0)的近似值。答案:解:2342ixyxxxxyxyiiiiiiiii0-244-816-8161-121-11-2220100000313111334254816102001510034341∑10