科學(xué)和工程計(jì)算基礎(chǔ)

編者的話                  
 前言                  
 第1章 緒論                  
 1.1 課程的內(nèi)容. 意義和特點(diǎn)                  
 1.2 誤差的基本概念                  
 1.2.1 誤差和有效數(shù)字                  
 1.2.2 函數(shù)求值的誤差估計(jì)                  
 1.2.3 計(jì)算機(jī)中數(shù)的表示和舍入誤差                  
 1.3 數(shù)值穩(wěn)定性和病態(tài)問(wèn)題                  
 1.3.1 算法的穩(wěn)定性                  
 1.3.2 病態(tài)效學(xué)問(wèn)題和條件數(shù)                  
 1.4 算法的實(shí)現(xiàn)                  
 習(xí)題1                  
 數(shù)值試驗(yàn)題1                  
 第2章 預(yù)備知識(shí)                  
 2.1 微積分若干基本概念和基本定理                  
 2.1.1 數(shù)列極限和函數(shù)極限                  
 2.1.2 閉區(qū)間上的連續(xù)函數(shù)                  
 2.1.3 函數(shù)序列的一致收斂性                  
 2.1.4 中值定理                  
 2.1.5 變參數(shù)積分求導(dǎo)公式                  
 2.2 常微分方程的基本概念和有關(guān)理論                  
 2.2.1 基本概念                  
 2.2.2 初值問(wèn)題解的存在唯一性                  
 2.2.3 初值問(wèn)題的適定性. 條件                  
 2.2.4 兩點(diǎn)邊值問(wèn)題                  
 2.3 線性代數(shù)的有關(guān)概念和結(jié)論                  
 2.3.1 線性空間                  
 2.3.2 矩陣和矩陣變換                  
 2.3.3 初等矩陣                  
 2.3.4 矩陣的特征值和譜半徑                  
 2.3.5 Jordan標(biāo)準(zhǔn)形                  
 2.3.6 矩陣特征值估計(jì)——Gerschgorin圓盤(pán)定理                  
 2.3.7 對(duì)角占優(yōu)陣                  
 2.3.8 對(duì)稱正定陣                  
 2.3.9 分塊矩陣                  
 2.3.10 向量和連續(xù)函數(shù)的內(nèi)積                  
 2.3.11 向量范數(shù), 矩陣范數(shù)和連續(xù)函數(shù)的范數(shù)                  
 習(xí)題2                  
 第3章 線性代數(shù)方程組的數(shù)值解法                  
 3.1 引言                  
 3.2 高斯消去法                  
 3.2.1 順序消去過(guò)程和矩陣的LU三角分解                  
 3.2.2 可行性和計(jì)算量                  
 3.2.3 數(shù)值穩(wěn)定性：選主元                  
 3.3 矩陣的直接三角分解法                  
 3.3.1 三角形方程組的追趕法                  
 3.3.2 對(duì)稱正定的Cholesky分解法                  
 3.4 方程組的性態(tài). 條件數(shù)                  
 3.4.1 病態(tài)方程組和矩陣的條件數(shù)                  
 3.4.2 條件數(shù)的應(yīng)用：方程組誤差估計(jì)                  
 3.5 大型方程組的迭代方法                  
 3.5.1 Jacobi迭代和Gauss－Seidel迭代法                  
 3.5.2 迭代法的收斂性和收斂速度                  
 3.5.3 Jacobi迭代法和Gauss－Seidel迭代法的收斂性判定                  
 3.5.4 分塊迭代法                  
 3.6 應(yīng)用例題                  
 評(píng)注                  
 習(xí)題3                  
 數(shù)值試驗(yàn)題3                  
 第4章 插值和擬合                  
 4.1 引言                  
 4.1.1 函數(shù)的插值                  
 4.1.2 離散數(shù)據(jù)的擬臺(tái)                  
 4.2 插值                  
 4.2.1 拉格朗日插值法                  
 4.2.2 插值的余項(xiàng)                  
 4.2.3 均差和牛頓插值法                  
 4.3 分段低次插值                  
 4.3.1 龍格現(xiàn)象和分段線性插值                  
 4.3.2 分段埃爾米特三次插值                  
 4.3.3 附注：二重埃爾米特插值                  
 4.4 三次樣條插值                  
 4.4.1 樣條插值的背景和定義                  
 4.4.2 三次樣條插值的定解條件                  
 4.4.3 三彎矩算法                  
 4.4.4 例題和一致收斂性                  
 4.5 正交多項(xiàng)式                  
 4.5.1 連續(xù)函數(shù)空間                  
 4.5.2 離散點(diǎn)列上的正交多項(xiàng)式                  
 4.5.3 連續(xù)區(qū)間上的正交多項(xiàng)式                  
 4.6 離散數(shù)據(jù)的曲線擬臺(tái)                  
 4.6.1 線性模型和最小二乘擬臺(tái)                  
 4.6.2 正規(guī)方程和解的存在唯一性                  
 4.6.3 多項(xiàng)式擬合和例題                  
 4.6.4 正規(guī)方程的病態(tài)和正交多項(xiàng)式擬合                  
 評(píng)注                  
 習(xí)題4                  
 數(shù)值試驗(yàn)題4                  
 第5章 數(shù)值積分和數(shù)值微分                  
 5.1 引言                  
 5.2 梯形公式和Simpson求積公式                  
 5.2.1 梯形公式和Simpson公式                  
 5.2.2 復(fù)化梯形公式和復(fù)化Simpson公式                  
 5.3 Gauss求積公式                  
 5.3.1 Gauss點(diǎn)與正交多項(xiàng)式零點(diǎn)的關(guān)系                  
 5.3.2 常用的Gauss型求積公式                  
 5.3.3 Gauss公式的余項(xiàng)                  
 5.3.4 Gauss求積公式的數(shù)值穩(wěn)定性和收斂性                  
 5.4 數(shù)值微分                  
 5.4.1 Taylor展開(kāi)法                  
 5.4.2 插值型求導(dǎo)公式                  
 5.4.3 三次樣條求導(dǎo)                  
 5.5 外推技巧和自適應(yīng)技術(shù)                  
 5.5.1 外推原理                  
 5.5.2 數(shù)值微分的外推算法                  
 5.5.3 數(shù)值積分的Romberg算法                  
 5.5.4 自動(dòng)變步長(zhǎng)Simpson方法和自適應(yīng)Simpson方法                  
 5.6 應(yīng)用例題                  
 評(píng)注                  
 習(xí)題5                  
 數(shù)值試驗(yàn)題5                  
 第6章 常微分方程的數(shù)值解法                  
 6.1 引言                  
 6.2 切值問(wèn)題的數(shù)值解法                  
 6.2.1 Euler方法及其截?cái)嗾`差和階                  
 6.2.2 Runge－Kutta法                  
 6.2.3 單步法的穩(wěn)定性                  
 6.2.4 線性多步法                  
 6.2.5 預(yù)測(cè)-校正技術(shù)和外推技巧                  
 6.3 一階常微分方程組的數(shù)值方法                  
 6.3.1 一階方程組和高階方程                  
 6.3.2 剛性方程（組）                  
 6.4 邊值問(wèn)題的打靶法和差分法                  
 6.4.1 打靶法                  
 6.4.2 差分法                  
 6.5 *有限元方法                  
 6.5.1 泛函極值和Euler方程                  
 6.5.2 兩點(diǎn)邊值問(wèn)題的變分原理                  
 6.5.3 變分近似法——Ritz－Galerkin方法                  
 6.5.4 有限元方法                  
 評(píng)注                  
 習(xí)題6                  
 數(shù)值試驗(yàn)題6                  
 第7章 非線性方程和方程組的解法                  
 7.1 引言                  
 7.1.1 問(wèn)題的背景和內(nèi)容概要                  
 7.1.2 一元方程的搜索法                  
 7.2 一元方程的基本迭代法                  
 7.2.1 基本迭代法及其收斂性                  
 7.2.2 局部收斂性和收斂階                  
 7.2.3 收斂性的改善                  
 7.3 一元方程的牛頓迭代法                  
 7.3.1 牛頓迭代法及其收斂性                  
 7.3.2 重根時(shí)的牛頓迭代改善                  
 7.3.3 離散牛頓法                  
 7.4 非線性方程組的解法                  
 7.4.1 不動(dòng)點(diǎn)迭代法                  
 7.4.2 牛頓迭代法                  
 7.4.3 擬牛頓法                  
 附錄7.1 某些定理的證明                  
 附錄7.2 延拓法                  
 評(píng)注                  
 習(xí)題7                  
 數(shù)值試驗(yàn)題7                  
 第8章 最優(yōu)化方法                  
 8.1 引言                  
 8.2 線性規(guī)劃及其解法                  
 8.2.1 標(biāo)準(zhǔn)形式和基本性質(zhì)                  
 8.2.2 單純形算法                  
 8.2.3 單純形方法的初始化                  
 8.2.4 線性規(guī)劃的對(duì)偶性質(zhì)                  
 8.2.5 對(duì)偶變尺度算法                  
 8.3 無(wú)約束最優(yōu)化方法                  
 8.3.1 基本概念和下降法                  
 8.3.2 一維搜索                  
 8.3.3 下降方向和收斂性                  
 8.3.4 非線性最小二乘問(wèn)題                  
 8.4 約束最優(yōu)化方法                  
 8.4.1 引言                  
 8.4.2 罰函數(shù)法                  
 8.4.3 下降法                  
 8.4.4 凸二次規(guī)劃的內(nèi)點(diǎn)算法                  
 評(píng)注                  
 習(xí)題8                  
 數(shù)值試驗(yàn)題8                  
 第9章 矩陣特征值問(wèn)題的數(shù)值解法                  
 9.1 引言                  
 9.1.1 問(wèn)題的背景和內(nèi)容概要                  
 9.1.2 特征值的擾動(dòng)和條件數(shù)                  
 9.2 冪法及其變形                  
 9.2.1 冪法和外推加速                  
 9.2.2 反冪法和原點(diǎn)位移                  
 9.2.3 對(duì)稱矩陣的修正冪法                  
 9.3 矩陣的兩種正交變換                  
 9.3.1 平面旋轉(zhuǎn)變換和鏡面反射變換                  
 9.3.2 化矩陣為Hessenberg形                  
 9.3.3 矩陣的QR分解                  
 9.4 QR算法                  
 9.4.1 QR算法及其收斂性                  
 9.4.2 QR算法的改善                  
 9.4.3 雙步隱式QR算法                  
 評(píng)注                  
 習(xí)題9                  
 數(shù)值試驗(yàn)題9                  
 附錄 Matlab語(yǔ)言簡(jiǎn)介                  
 f.1 Matlab語(yǔ)言的特點(diǎn)                  
 f.2 環(huán)境窗口. 語(yǔ)言結(jié)構(gòu)和編程方法                  
 f.3 主要語(yǔ)法和符號(hào)                  
 f.4 矩陣的操作和運(yùn)算                  
 f.5 庫(kù)函數(shù)                  
 f.6 若干算法的Matlab程序                  
 參考文獻(xiàn)                  
 習(xí)題答案                  
 索引