<?xml version="1.0" encoding="utf-8" ?><rss version="2.0"><channel><title><![CDATA[Liang_Ling的博客]]></title><description><![CDATA[]]></description><link>https://blog.youkuaiyun.com/Liang_Ling</link><language>zh-cn</language><generator>https://blog.youkuaiyun.com/</generator><copyright><![CDATA[Copyright &copy; Liang_Ling]]></copyright><item><title><![CDATA[Quotient Space]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/79452332</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/79452332</guid><author>Liang_Ling</author><pubDate>Mon, 05 Mar 2018 22:56:17 +0800</pubDate><description><![CDATA[Quotient Space

Suppose that XXX is a normed space, and YYY is a closed subspace of XXX, we can define an equivalent relation on XXX by x1∼x2x1∼x2x_1\sim x_2 if and only if x1−x2∈Yx1−x2∈Yx_1-x_2\in Y....]]></description><category></category></item><item><title><![CDATA[Quotient Space]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/79452322</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/79452322</guid><author>Liang_Ling</author><pubDate>Mon, 05 Mar 2018 22:54:55 +0800</pubDate><description><![CDATA[Quotient Space

Suppose that XXX is a normed space, and YYY is a closed subspace of XXX, we can define an equivalent relation on XXX by x1∼x2x1∼x2x_1\sim x_2 if and only if x1−x2∈Yx1−x2∈Yx_1-x_2\in Y....]]></description><category></category></item><item><title><![CDATA[Complex analysis review 6]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/78831837</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/78831837</guid><author>Liang_Ling</author><pubDate>Mon, 18 Dec 2017 12:42:41 +0800</pubDate><description><![CDATA[Laurent SeriesThe special properties of a complex function is much more determined by its singularity, to study the singularity of a function, we first give a useful theorem that does not hold for real]]></description><category></category></item><item><title><![CDATA[Complex analysis review 5]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/78831826</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/78831826</guid><author>Liang_Ling</author><pubDate>Mon, 18 Dec 2017 12:39:46 +0800</pubDate><description><![CDATA[Maximum modulus principle and Schwarz lemmaAverage Vaule Propertiesf(z0)=12πi∫∂D(z0,r)f(ξ)ξ−z0dξ=12πi∫2π0f(z0+reit)ireitreitdt=12π∫2π0f(z0+reit)dt.
\begin{align*}
f(z_0) &= \frac{1}{2\pi i}\int_{\parti]]></description><category></category></item><item><title><![CDATA[Complex analysis review 4]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/78831070</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/78831070</guid><author>Liang_Ling</author><pubDate>Mon, 18 Dec 2017 11:12:31 +0800</pubDate><description><![CDATA[The zeros of analytic functionsUse Liouville theorem, we can prove the fundamental theorem of algebra.Theorem 1If p(z)p(z) is a polynomial, then there is at least one z0z_0 such that p(z0)=0p(z_0)=0. L]]></description><category></category></item><item><title><![CDATA[Complex analysis review 3]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/78814150</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/78814150</guid><author>Liang_Ling</author><pubDate>Fri, 15 Dec 2017 16:09:25 +0800</pubDate><description><![CDATA[Taylor series and Liouville theoremMany interesting results can be obtained by the Cauchy Integral Theory. Theorem 1Suppose that f(z)f(z) is analytic on U⊂CU\subset C, and continuous on U¯\bar{U}. Then]]></description><category></category></item><item><title><![CDATA[Complex analysis review 2(Cauchy Integral Theory)]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/78811839</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/78811839</guid><author>Liang_Ling</author><pubDate>Fri, 15 Dec 2017 13:15:08 +0800</pubDate><description><![CDATA[Theorem 1(Cauchy-Green formula, Pompeiu formula)Suppose that U⊂CU\subset C is a bounded domain, having C1C^1 boundary. f(z)=u(x,y)+iv(x,y)∈C1(U¯)f(z) = u(x,y)+iv(x,y)\in C^1(\bar{U}), then  
f(z)=12πi∫]]></description><category></category></item><item><title><![CDATA[Complex analysis review 1]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/78786782</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/78786782</guid><author>Liang_Ling</author><pubDate>Tue, 12 Dec 2017 22:13:46 +0800</pubDate><description><![CDATA[ArgumentArga=arga+2πZArg \,a=arg\, a +2\pi \mathbb{Z}Stereographic Projectionx1=z+z¯1+|z|2x2=z−z¯1+|z|2x3=|z|2−1|z|2+1
x_1 = \frac{z+\bar{z}}{1+|z|^2}\\
x_2 =  \frac{z-\bar{z}}{1+|z|^2}\\
x_3 = \frac{|]]></description><category></category></item><item><title><![CDATA[共轭梯度法(Conjugate Gradient Method)]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/78552788</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/78552788</guid><author>Liang_Ling</author><pubDate>Thu, 16 Nov 2017 16:33:52 +0800</pubDate><description><![CDATA[我们要求解线性方程组 
Ax=bAx=b 
其中AA 是对称正定矩阵（spd），给定RnR^n中n个线性无关的向量r0,r1,⋯,rn−1r_0,r_1,\cdots,r_{n-1}, 我们想要构造另外的一组基p0,p1,⋯,pn−1p_0,p_1,\cdots,p_{n-1}, 并满足 
⟨pi,pj⟩=0,∀i≠j\langle p_i, p_j\rangle=0,\quad \forall i]]></description><category></category></item><item><title><![CDATA[范数(Norm)和谱半径(Spectral Radii)]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/78156851</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/78156851</guid><author>Liang_Ling</author><pubDate>Tue, 03 Oct 2017 23:28:55 +0800</pubDate><description><![CDATA[向量(vector) 
x=[x1,x2,⋯,xn]Tx=[x_1,x_2,\cdots,x_n]^T 
共轭转置(conjugate transpose) 
x∗=[x1¯,x2¯,⋯,xn¯]x^*=[\bar{x_1},\bar{x_2},\cdots,\bar{x_n}]向量2范数(vector norm) 
||x||2=(∑i=1n|xi|2)1/2||x||_2=(\sum_{i=1}]]></description><category></category></item><item><title><![CDATA[Moreau-Yosida 正则化和近似算子(proximal operator)]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/78147702</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/78147702</guid><author>Liang_Ling</author><pubDate>Sun, 01 Oct 2017 15:59:50 +0800</pubDate><description><![CDATA[强凸函数（strongly convex)：ff 凸函数且存在常数c>0c>0,使得 
f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y)−12λ(1−λ)||x−y||2f(\lambda x+(1-\lambda)y)\leq \lambda f(x)+(1-\lambda)f(y)-\frac{1}{2}\lambda(1-\lambda)||x-y||^2  
定义函数 
ϕ(x,y)]]></description><category></category></item><item><title><![CDATA[线性方程组迭代法--一个综述]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/77773514</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/77773514</guid><author>Liang_Ling</author><pubDate>Fri, 01 Sep 2017 15:17:39 +0800</pubDate><description><![CDATA[下面我们给出求解线性方程组现有的众多迭代法的一个综述，简单介绍它们适用的情况等。稳定方法
Jacobi 
它基本思想是单独求解某一个独立的变量；一次迭代仅仅对一个变量有作用，它便于理解但是收敛速度慢。
Gauss-Seidel 
和Jacobi方法很像，但每次都利用当前的最新信息，一般情况下， 如果Jacobi方法收敛，那么Gauss-Seidel方法会收敛更快，尽管速度仍然慢。
SOR 
SOR是]]></description><category></category></item><item><title><![CDATA[数值积分]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/76895677</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/76895677</guid><author>Liang_Ling</author><pubDate>Tue, 08 Aug 2017 12:46:11 +0800</pubDate><description><![CDATA[数值积分是应用函数在给定区间上的定义，计算出在该区间 上的积分(近似)值。为什么需要数值积分? 
从数学上说，有些初等函数的原函数不是初等函数 ； 
有些时候我们只知道函数在给定区间上有限个点处的函数值 ； 
虽然有些函数的原函数可以得到，但是求值过于复杂；数值积分的策略是用另一个函数替代原来的被积函数，而前者的积分是很容易计算的。 
多项式(来自于多项式插值或逼近)以及样条函数是常用的选择。目标是]]></description><category></category></item><item><title><![CDATA[标准正交基]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/76862332</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/76862332</guid><author>Liang_Ling</author><pubDate>Mon, 07 Aug 2017 22:29:58 +0800</pubDate><description><![CDATA[计算最佳逼近元的时候，为什么要选取标准正交基？ 
1 根据幂基计算最佳逼近元，计算过程的稳定性不好； 
2 下面的定理说明标准正交基的优势：定理1 
设G的标准正交基为{g1,g2,⋯,gn},f∈E\{g_1,g_2,\cdots,g_n\},f\in E,则g=∑ni=1cigig=\sum_{i=1}^nc_ig_i为f在E中的最佳逼近当且仅当ci=⟨f,gi⟩c_i=\langle f,g_]]></description><category></category></item><item><title><![CDATA[离散内积与最小二乘]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/76768926</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/76768926</guid><author>Liang_Ling</author><pubDate>Sun, 06 Aug 2017 14:14:24 +0800</pubDate><description><![CDATA[在数据拟合时，通常要根据已知节点来构造离散范数，并在新的范数的1意义下拟合。离散内积： 
函数f，g的关于离散点列{xi}ni=0\{x_i\}_{i=0}^n的离散内积为： 
(f,g)h=∑i=0nf(xi)g(xi)(f,g)_h=\sum_{i=0}^nf(x_i)g(x_i)由此可以定义离散范数： 
函数f的离散范数为： 
||f||h=(f,f)h‾‾‾‾‾‾√||f||_h = \sq]]></description><category></category></item><item><title><![CDATA[最佳节点的选取--第一类Tchebyshev多项式]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/76718540</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/76718540</guid><author>Liang_Ling</author><pubDate>Sat, 05 Aug 2017 14:17:55 +0800</pubDate><description><![CDATA[最近在准备复习qualify，打算时不时更一些复习的重要知识点。 
在数值分析中，通过多项式逼近函数有如下的多项式插值误差定理： 
定理1 
设f∈Cn+1[a,b]f\in C^{n+1}[a,b]，多项式pp是ff在不同节点x0,x1,⋯,xnx_0,x_1,\cdots, x_n上的插值多项式，degp≤ndeg\, p \le n。则对[a,b][a,b]中每个xx,都有ξx∈(a,b)\x]]></description><category></category></item><item><title><![CDATA[Ekeland Variational Principles]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/73056585</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/73056585</guid><author>Liang_Ling</author><pubDate>Sun, 11 Jun 2017 17:46:23 +0800</pubDate><description><![CDATA[Basic Forms 
{Theorem 1.1} 
Let (X,d)(X,d) be a complete metric space and let f:X→R⋃{+∞}f:X\rightarrow R \bigcup \{+\infty\} be a lsc function bounded from below. Suppose that ϵ>0\epsilon>0 and z∈Xz\in]]></description><category></category></item><item><title><![CDATA[正则表达式]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/71750742</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/71750742</guid><author>Liang_Ling</author><pubDate>Fri, 12 May 2017 17:53:12 +0800</pubDate><description><![CDATA[literal     匹配文本字符串的字面值literal  root  匹配字符串rootre1|re2     匹配正则表达式re1或re2 root|admin  匹配字符串root或admin.           匹配除换行符外的任意字符  a.c  匹配ac中间有任意一个字符的选项^           匹配字符串的起始部分  ^root  匹配以root开头的结果匹配字符串的结束部分]]></description><category></category></item><item><title><![CDATA[Sherman-Morrison-Woodburg 定理]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/70036818</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/70036818</guid><author>Liang_Ling</author><pubDate>Mon, 10 Apr 2017 20:59:08 +0800</pubDate><description><![CDATA[接触到这个定理的缘由是碰到矩阵秩一校正的问题，下面给出定理的内容。 
定理 
设AA是n×nn\times n非奇异矩阵，UU是n×mn\times m矩阵，CC是m×mm\times m非奇异矩阵，VV是m×nm\times n矩阵，如果C−1+UA−1VC^{-1}+UA^{-1}V非奇异，那么A+UCVA+UCV也是非奇异的，且 
(A+UCV)−1=A−1−A−1U(C−1+UA−1V)−1]]></description><category></category></item><item><title><![CDATA[线性方程组的直接解法（python）]]></title><link>https://blog.youkuaiyun.com/Liang_Ling/article/details/65447955</link><guid>https://blog.youkuaiyun.com/Liang_Ling/article/details/65447955</guid><author>Liang_Ling</author><pubDate>Thu, 23 Mar 2017 19:28:32 +0800</pubDate><description><![CDATA[1.高斯顺序消去法 
设detA≠0det\,A\neq0,此时方程组有唯一解。 
(A1b1)=(Ab)=⎛⎝⎜⎜⎜⎜a111a121⋯a1n1a112a122⋯a1n2⋯⋯⋯⋯a11na12n⋯a1nnb11b12⋯b1n⎞⎠⎟⎟⎟⎟(A^1\,b^1)=(A\,b)=
\left(
\begin{matrix}
a_{11}^1&a_{12}^1&\cdots&a_{1n}^1&b_1^1\\]]></description><category></category></item></channel></rss>