Questions in category: 计算数学 (Computational mathematics)

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## 1. 求 $x^5-4x^3-5=0$ 的根

Posted by haifeng on 2023-06-03 19:33:56 last update 2023-06-03 20:14:59 | Answers (0) | 收藏

$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$

>> :mode polyn
Switch into polynomial mode.

>> diff(x^5-4x^3-5)
out> 5x^4-12x^2

------------------------

>> (x^5-4x^3-5)/(5x^4-12x^2)
in> (x^5-4x^3-5)/(5x^4-12x^2)

out>
quotient> q(x) = 1|5x
remainder> r(x) = -8|5x^3-5

1|5x^1
------------------------

>> printRecursiveSeries(4/5*x_n+(8/5*x_n^3+5)/(5*x_n^4-12*x_n^2),x_n,1,100,\n,linenumber)
[1]     1
[2]     -0.1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571429
[3]     -20.68684146042636608674344523401127174712080372457730948296986032835089438863023768684146042636608673092
[4]     -16.565023688617589109177537826433863432338260623780573240602345316225300420691945262445633989624803938736
[5]     -13.2714938151392127327086618324645366416606984746301979545605997238256662746354987884645019492889088259888
[6]     -10.64160729106856435253004280224457254617986844856116673479583215765055870684575566382988446238459617659104
[7]     -8.543927904421098082103266940566651336467393310841794835891510921669912488167846412120893765286900283972832
[8]     -6.8736750146567822836834747098409137278847103761224278023001309485647375068881413772811833775034090294782656
[9]     -5.54751385154459757117959853911308426694533117603318730701466364274880256207172022811686144943490408568261248
[10]    -4.499428383849649530577659730552688681850614839831792743978496600619449792627229845215124665110187212646089984
...  ...
[95]    -1.7523882613158145756642021439965730423292471867922368214477915180260352744599353088675504105795106405000000039951342135619708337669405749916685322147717924146834921482445434074138489452088000512
[96]    -1.75238826131581457566420214399657304232924718679223682144779151802603527445993530886755041057951064050000000319610737084957666701355245999333482577181743393174679371859563472593107915616704004096
[97]    -1.752388261315814575664202143996573042329247186792236821447791518026035274459935308867550410579510640500000002556885896679661333610841967994667860617453947145397434974876507780744863324933632032768
[98]    -1.7523882613158145756642021439965730423292471867922368214477915180260352744599353088675504105795106405000000020455087173437290668886735743957342884939631577163179479799012062245958906599469056262144
[99]    -1.75238826131581457566420214399657304232924718679223682144779151802603527445993530886755041057951064050000000163640697387498325351093885951658743079517052617305435838392096497967671252795752450097152
[100]   -1.752388261315814575664202143996573042329247186792236821447791518026035274459935308867550410579510640500000001309125579099986602808751087613269944636136420938443486707136771983741370022366019600777216

>> x=-1.752388261315814575664202143996573042329247186792236821447791518026035274459935308867550410579510640500000001309125579099986602808751087613269944636136420938443486707136771983741370022366019600777216
----------------------------
type: double
name: x
value: -0
--------------------
>> (x)^5-4*(x)^3-5
in> (-1.752388261315814575664202143996573042329247186792236821447791518026035274459935308867550410579510640500000001309125579099986602808751087613269944636136420938443486707136771983741370022366019600777216)^5-4*(-1.752388261315814575664202143996573042329247186792236821447791518026035274459935308867550410579510640500000001309125579099986602808751087613269944636136420938443486707136771983741370022366019600777216)^3-5

out> 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100246145459086898585802895870188576837095998950884918298641158723559282188775705998581874986989258205424303464763848840790191389632164176548802253105448816681594005165936343156722773516711893034676728450651453570649817816469002947814009596245777511620864799261738696376838799747784277730251509134065093393799300694265837562131421519346343974503745560305120305614651107308855715555811229082159622948908292166108381373370499489108074739974508281096479430654358092447625637625382610938780421530534478487833251837308685695313740517610641168059282723155215195454469839074609502737089508716321411361434807371548709509131679916412796742165441812738587766584562366496393623176677562279651152507508747918974460653536570760308781440856424720183327725525430635283070792546407203728655490798266409749217358117945684603185503201696956723823148920764841223782621217436808496957051984681847936038599655424
------------------------

## 2. [By A.Singer] From graph to manifold Laplacians: The convergence rate.

Posted by haifeng on 2023-03-01 09:23:37 last update 2023-03-30 10:00:14 | Answers (0) | 收藏

# From graph to manifold Laplacians: The convergence rate

## Author:  A. Singer

Appl. Comput. Harmon. Anal. 21(2006)  128--134

## 1. 介绍

$W_{ij}=k\Bigl(\|x_i-x_j\|^2/(2\varepsilon)\Bigr),$

$D=(D_{ij})_{N\times N},\quad D_{ij}=0, \text{若}\ i\neq j;\quad D_{ii}=\sum_{j=1}^{N}W_{ij}.$

$L=D^{-1}W-I,$

$\frac{1}{\varepsilon}\lim_{N\rightarrow\infty}\sum_{j=1}^{N}L_{ij}f(x_j)=\frac{1}{2}\Delta_M f(x_i)+O(\varepsilon^{1/2}).\tag{1.4}$

$\frac{1}{\varepsilon}\sum_{j=1}^{N}L_{ij}f(x_j)=\frac{1}{2}\Delta_M f(x_i)+O\biggl(\frac{1}{N^{1/2}\varepsilon^{1+d/4}},\varepsilon^{1/2}\biggr).\tag{1.5}$

Footnote: 这里记号 $O(\cdot,\cdot)$ 意思是存在(不依赖于 $N$ 和 $\varepsilon$ 的)正常数 $C_1$, $C_2$, 使得

$\Biggl|O\biggl(\frac{1}{N^{1/2}\varepsilon^{1+d/4}},\varepsilon^{1/2}\biggr)\Biggr|\leqslant C_1\frac{1}{N^{1/2}\varepsilon^{1+d/4}}+C_2\varepsilon^{1/2},$

$\varepsilon=\frac{C(\mathcal{M})}{N^{1/(3+d/2)}},$

$\frac{1}{\varepsilon}\sum_{j=1}^{N}L_{ij}f(x_j)=\frac{1}{2}\Delta_M f(x_i)+O\biggl(\frac{1}{N^{1/2}\varepsilon^{1/2+d/4}},\varepsilon\biggr).\tag{1.7}$

(11条消息) 一文读懂 Bias（偏差）、Error（误差）、Variance（方差）_bias偏差_Suprit的博客-CSDN博客

## 3. 国内计算软件

Posted by haifeng on 2022-08-06 08:13:15 last update 2022-08-29 07:24:03 | Answers (0) | 收藏

## 七维高科

http://www.7d-soft.com/

1. “领先世界、当今最强大、最易于使用的数值优化分析计算软件平台：最强大的全局优化算法令优化拟合不再成为难题，独特的ODE求解器轻松应对任意边值问题。”
2. “非线性拟合、非线性方程组、参数估算反演、方程求解、微分方程求解、微分方程拟合 、非线性规划、混合整数规划...，令你忘却Matlab、Origin、Lingo、Gams等世界品牌使用时的繁琐、低效与不足。”
3. “遍布世界数十万名科技工作者的选择！”

## 元计算

http://www.yuanjisuan.cn/

“FELAC.IDE采用元件化思想自主定制有限元计算的基本工序，使用有限元语言来书写偏微分方程及多种算法，高效、高质量地生成有限元求解器的通用有限元软件集成开发平台。”

## 中科大ABACUS

http://abacus.ustc.edu.cn/
http://ooe.ustc.edu.cn/news8.html

## MMP

http://www.mmrc.iss.ac.cn/mmp/index.htm

(已停止研发)

## maTHμ

https://github.com/maTHmU

## 北太天元数值计算通用软件Numerical Computation Software

“北太天元数值计算通用软件提供科学计算、可视化、交互式程序设计，具备强大的底层数学函数库，支持数值计算、数据分析、数据可视化、数据优化、算法开发等工作，并通过SDK与API接口，扩展支持各类学科与行业场景，为各领域科学家与工程师提供优质、可靠的科学计算环境。”

## 4. Chudnovsky 公式

Posted by haifeng on 2019-03-09 11:44:12 last update 2021-04-18 11:23:00 | Answers (0) | 收藏

$\frac{1}{\pi}=12\sum_{k=0}^{+\infty}\frac{(-1)^k\cdot(6k)!(13591409+545140134k)}{(3k)!(k!)^3\cdot 640320^{3k+\frac{3}{2}}}.$

$\frac{(640320)^{\frac{3}{2}}}{12\pi}=\frac{426880\sqrt{10005}}{\pi}=\sum_{k=0}^{+\infty}\frac{(6k)!(545140134k+13591409)}{(3k)!(k!)^3(-262537412640768000)^k}$

$\frac{640320^{3/2}}{12\pi}=\frac{426880\sqrt{10005}}{\pi}=\sum^\infty_{k=0}\frac{(6k)!(545140134k+13591409)}{(3k)!(k!)^3\left(-640320\right)^{3k}}$

References:

https://en.wikipedia.org/wiki/Chudnovsky_algorithm

http://numbers.computation.free.fr/Constants/PiProgram/pifast.html

http://www.numberworld.org/y-cruncher/internals/binary-splitting-library.html#pi_chudnovsky

## 5. Gamma 函数

Posted by haifeng on 2015-09-19 16:48:27 last update 2015-09-19 17:11:06 | Answers (0) | 收藏

$\Gamma(s)=\int_0^{+\infty}e^{-t}t^s\cdot\frac{1}{t}dt$

GNU GSL

http://www.gnu.org/software/gsl/manual/html_node/Gamma-Functions.html#index-gsl_005fsf_005flngamma-583

References:

http://stackoverflow.com/questions/15472803/gamma-or-log-gamma-function-in-c-or-c

http://www.nr.com/

## 6. Riemann Zeta 函数的编程

Posted by haifeng on 2015-09-16 17:54:16 last update 2015-09-19 16:36:18 | Answers (1) | 收藏

http://www.cplusplus.com/forum/general/93834/

## 7. Lagrange 插值多项式

Posted by haifeng on 2015-06-07 23:29:39 last update 2015-06-07 23:54:34 | Answers (0) | 收藏

Lagrange 插值多项式 (或称 Lagrange 多项式)

Lagrange polynomial (interpolation polynomial in the Lagrange form )

$P(x):=\sum_{j=1}^{n}P_j(x),$

$P_j(x)=y_j\prod_{k=1,k\neq j}^{n}\frac{x-x_k}{x_j-x_k},$

$\begin{split} P(x)=&\frac{(x-x_2)(x-x_3)\cdots (x-x_n)}{(x_1-x_2)(x_1-x_3)\cdots (x_1-x_n)}y_1\\ &+\frac{(x-x_1)(x-x_3)\cdots (x-x_n)}{(x_2-x_1)(x_2-x_3)\cdots (x_2-x_n)}y_2\\ &+\cdots +\frac{(x-x_1)(x-x_2)\cdots (x-x_{n-1})}{(x_n-x_1)(x_n-x_2)\cdots (x_n-x_{n-1})}y_n. \end{split}$

$A_i:=\prod_{j=1,j\neq i}^{k}\frac{1}{\lambda_i-\lambda_j}(A-\lambda_j I),\quad i=1,2,\ldots, k.$

## 8. 3n+1循环的等价问题(Open problem)

Posted by haifeng on 2011-07-01 13:05:43 last update 2011-07-01 13:08:34 | Answers (0) | 收藏

$n=S(m)= \begin{cases} 2m&n\text{ is even}\\ (2m-1)/3&n\text{ is odd and }n>1. \end{cases}$ 问是否可由数字1在变换S下生成所有自然数. 一个很简单的事实是, 在S下由1生成了一棵树. 现在的问题是这棵树是否包含了所有自然数?

## 9. $\sqrt{3}$ 的数值求解

Posted by haifeng on 2011-06-29 16:45:15 last update 2011-06-29 16:47:27 | Answers (1) | 收藏

$x_n=\frac{1}{2}(x_{n-1}+\frac{3}{x_{n-1}}),\quad n=1,2,\ldots,$ 给定初值 $x_0>0$. 证明 $\lim\limits_{n\rightarrow\infty}x_n=\sqrt{3}$.