最优化方法课程设计-斐波那契法分析与实现-完整版(新)

所谓的光辉岁月，并不是以后，闪耀的日子，而是无人问津时，你对梦想的偏执。

最优化方法

题目：斐波那契法分析与实现

院系：信息与计算科学学院

专业：统计学

姓名学号：小熊熊 11071050137

指导教师：大胖胖

日期： 2014 年 01 月 10 日

摘要

科学的数学化是当代科学发展的一个主要趋势，最优化理论与算法是一个重要的数学分支，它所研究的问题是讨论在众多的方案中什么样的方案最优以及怎样找出最优方案.

一维搜索是指寻求一元函数在某个区间上的最优点的方法.这类方法不仅有实用价值，而且大量多维最优化方法都依赖于一系列的一维最优化.本文就斐波那契法的一维搜索进行了详细的分析，并且成功的用 MATLAB 实现了斐波那契法求解单峰函数的极小值问题.

斐波那契法的一维搜索过程是建立在一个被称为斐波那契数列的基础上进行的，斐波那契法成功地实现了单峰函数极值范围的缩减.从理论上来说，斐波那契法的精度比黄金分割法要高.但由于斐波那契法要事先知道计算函数值的次数，故相比之下，黄金分割法更为简单一点，它不需要事先知道计算次数，并且当n 7 时，黄金分割法的收敛速率与斐波那契法越来越接近.因此，在实际应用中，常常采用黄金分割法. 斐波那契法也是一种区间收缩算法，和黄金分割法不同的是：黄金分割法每次收缩只改变搜索区间的一个端点，即它是单向收缩法. 而斐波那契法同时改变搜索区间的两个端点，是一种双向收缩法.

关键字：一维搜索斐波那契法单峰函数黄金分割法MATLAB

Abstract

Mathematical sciences is a major trend in contemporary scientific development, optimization theory and algorithms is an important branch of mathematics, the problems it was discussed in numerous research programs in the best of what programs and how to find the optimal solution .

One-dimensional search is the best method of seeking functions of one variable on the merits of a certain interval. Such methods not only have practical value, but also a large number of multi-dimensional optimization methods rely on a series of one-dimensional optimization article on Fibonacci the one-dimensional search method carried out a detailed analysis, and successful in MATLAB Fibonacci method for solving unimodal function minimization problem.

Fibonacci method of one-dimensional search process is based on the Fibonacci sequence is called a Fibonacci conducted on, Fibonacci method successfully achieved a unimodal function extreme range reduction. Theory , Fibonacci method accuracy is higher than the golden section method, but the number of times due to the Fibonacci method to calculate function values to know in advance, so the contrast, the golden section method is more simply, it does not need to know in advance the number of calculations and at that time, the rate of convergence of golden section and the Fibonacci method getting closer, so in practical applications, often using the golden section method. Fibonacci method is also a range contraction algorithm, and the golden section method the difference is: golden section each contraction only one endpoint to change the search range that it is unidirectional shrinkage law Fibonacci search method while changing the two endpoints of the range, is a two-way contraction method.

Key words: one-dimensional search Fibonacci method unimodal function Golden Section function MATLAB

目录

1.前言 (1)

1.1 一维搜索 (1)

1.2 单峰函数 (1)

1.3 单峰函数的性质 (1)

2.斐波那契法分析 (2)

2.1 区间缩短率 (2)

2.2 斐波那契数列 (3)

2.3 斐波那契法原理 (3)

2.4 斐波那契法与黄金分割法的关系 (6)

3.斐波那契法实现 (7)

3.1 斐波那契算法步骤 (7)

3.2 斐波那契法的MATLAB 程序 (8)

3.3 斐波那契算法举例 (10)

4.课程设计总结 (12)

4.1 概述 (12)

4.2 个人心得体会 (12)

5.参考文献 (13)