请设计二分算法，求x^3-x-1=0在区间【1，1.5】内的解（精确度为0.1）

Question

如题所述

Answer 1

解：令f(x) = x3– x – 1，求方程x3– x – 1 = 0在[1，1.5]内的解，由二分法可得：f(1) = 13– 1 – 1 = -1 < 0①，f(3/2) = (3/2)3 – (3/2) – 1 = 7/8 >0②，(1 + 3/2)/2= 5/4 = 1.25，f(5/4) = (5/4)3– (5/4) – 1 = -19/64 < 0③，所以方程x3 – x – 1 = 0在(5/4，3/2)内有解，(5/4 + 3/2)/2 = 11/8 = 1.375，f(11/8) = (11/8)3– (11/8) – 1 = 115/512 > 0④，所以方程x3 – x – 1 = 0在(5/4，11/8)内有解，(5/4 + 11/8)/2 = 21/16 = 1.3125，f(21/16) = (21/16)3– (21/16) – 1 = -211/4096 < 0⑤，所以方程x3 – x – 1 = 0在(21/16，11/8)内有解，(21/16 + 11/8)/2= 43/32 = 1.34375，f(43/32) = (43/32)3 – (43/32) – 1 = 2707/32768> 0⑥，所以方程x3– x – 1 = 0在(21/16，43/32)内有解，即原方程在(1.3125，1.34375)内有解，所以精确到0.1，本题的近似解为1.3 ；
解：4x + 5y + 2z= 30①，5x – 2y + 4z= 21②，①*2 + ②*5，可得33x + 24z =165 => 11x + 8z = 55 => z = (55 – 11x)/8 ≥ 0 => 55 – 11x ≥ 0 => 55 ≥ 11x => 0 ≤ x ≤ 5，检验可得：
1）x = 0时，z = (55 –11x)/8 = 55/8不是非负整数，舍去；
2）x = 1时，z = (55 –11x)/8 = 44/8 = 11/2不是非负整数，舍去；
3）x = 2时，z = (55 –11x)/8 = 33/8不是非负整数，舍去；
4）x = 3时，z = (55 –11x)/8 = 22/8 = 11/4不是非负整数，舍去；
5）x = 4时，z = (55 –11x)/8 = 11/8不是非负整数，舍去；
6）x = 5时，z = (55 –11x)/8 = 0，代入4x + 5y + 2z= 30，可得20 + 5y = 30=> 5y = 10 => y = 2，符合题意；
综上所述，原方程组的非负整数解为(x，y，z) = (5，2，0) 。