第一章 绪论 (部分习题答案).docVIP

第1周部分作业答案 定理1.2.1 设是的近似值，且 ， 其中是0到9之间的自然数，，为整数。 1 如果具有 位有效数字，那么的相对误差限为. 2 如果的相对误差限为，那么至少具有位有效数字。4. 1 已知近似值有5位有效数字，试求其相对误差限. 2 已知近似值的相对误差限是，问至少有几位有效数字？ 设近似值，且，其中是0到9之间的自然数，，为整数。因为有5位有效数字1.2.1 1 知： , 即的相对误差限为.，且的相对误差限为, 故由定义1.2.3知：即至少有3位有效数字。Theorem 1.2.1. Suppose , is the approximation to x. 1 If has significant digits, then the relative error bound of is . 2 If the relative error bound of is , where , then has at least significant digits. 3. 1 Suppose the approximation has 5 significant digits, try to find the relative error bound of . 2 Determine the minimal number of significant digits of the approximation such that the relative error bound of is 0.03%. Solution. Suppose , is the approximation to x. 1 Since has 5 significant digits, part 1 of Theorem 1.2.1 gives , that is, the relative error bound of is from Definition 1.2.3. 2 Since , and the relative error bound of is , we have , that is, retains at least 3 significant digits from Definition 1.2.5.