之前写的 用MATHEMATICA计算一元高次方程 内容较复杂,探讨了 Solve 的局限性和 FindRoot 的使用。

用Mathematica计算一元高次方程

如果只要计算增长率,用以下代码即可:

解释:

StartYear是起始年份,EndYear是结束年份,Year是一共经历了多少年。

StartNumber 就是初始年份的值,EndNumber是结束年份的值。

最后一行是计算,因为增长率不可能超出[-1,1],所以直接寻找值。只要数据正确,就不会报错了。

上面的例子的运行结果:

我遇到这个问题后在网上找到了答案,网上原文如下:

Question: DSolve returns true

Using Mathematica 7, I got a solution

The only time I was able to reproduce the “DSolve[True….” message was when I had the sequence of commands as follows,

What’s happening is that y'[x] is known to be x- y[x], and the DSolve is doing a compare of y-y[x] == y-y[x], returning True. Then it can’t solve True for y[x] in x, resulting in the question being returned.

Would you try exiting the kernel, restart fresh, and re-typing (* commands 1*) only.

 

(内容出自 Wolfram Community http://community.wolfram.com/groups/-/m/t/332674

意思就是,在DSolve解方程时,已事先给出了 y‘[x] 解析表达式,后来又来 y'[x] == x – y[x] 后,相当于在这里比较真假,因此返回 True。

解决办法:删去事先给出的f[x]表达式,清理内存,或者关掉从头重新计算,建议采用后一种方法,比较快速稳妥。

以下内容来自 wolfram 帮助页面:http://reference.wolfram.com/language/guide/ControlObjects.html


控件对象

Wolfram 语言提供一系列完整的控件对象,所有对象以便捷的符号形式表示. Manipulate 自动调用这些对象;您也可以在一般输入时直接调用它们,或在您自己的动态界面内调用它们.

Control 一个自动选择类型的交互控制

Slider, VerticalSlider Slider2D

Checkbox RadioButton Opener

IntervalSlider 使用下界和上界设值

ColorSlider 交互选择一个颜色

Setter 设置按下鼠标时的值: Setter Setter

Toggler 点击时切换值

SetterBar, RadioButtonBar, CheckboxBar, TogglerBar 按钮列表

PopupMenu 选择菜单

ListPicker 选项列表

InputField 可接受字符串、数字或任何表达式的输入栏

FormControl 含有释义明确的字段的任意表单

LocatorPane 图形或其它背景上可移动的定位器

Animator  ▪  Manipulator  ▪  Locator  ▪  ProgressIndicator  ▪  Trigger  ▪  ClickPane

Button 可以设置任意外观和行为的按钮

PasteButton 复制表达式的按钮

ActionMenu, ButtonBar 选择行为的菜单

EventHandler 低层次事件处理

MousePosition  ▪  ControlActive  ▪  ControllerState  ▪  CurrentValue

GestureHandler, TouchPosition 触碰控件

FileNameSetter 系统文件浏览

ColorSetter 系统颜色选择

SystemDialogInput 文件、字体、声音等

Hyperlink 超链接到单元和网址

FormObject 指定格式结构

FormFunction 设置活动的可提交格式

注释 »

Tooltip  ▪  Mouseover  ▪  Overlay  ▪  StatusArea  ▪  PopupWindow  ▪  Labeled  ▪ 

动态交互式语言 »

Dynamic  ▪  DynamicModule  ▪  MousePosition  ▪  Setting  ▪  DynamicSetting  ▪ 

用户界面的结构与布局 »

Grid  ▪  Row  ▪  Column  ▪  Panel  ▪  TabView  ▪  Pane  ▪  Deploy  ▪ 

控件选项 »

ImageSize  ▪  Appearance  ▪  Deployed  ▪  Enabled  ▪  AutoAction  ▪ 

构建用户界面 »

ChoiceDialog  ▪  CreatePalette  ▪  CreateDialog  ▪  CreateDocument  ▪ 

相关教程

张量计算比较繁琐,尤其是广义相对论和黎曼几何结合起来的计算更是冗繁。好在现在已经出现了很多计算张量的工具包。吴老师使用Maple的 GRTenser 计算,我打算看看针对 Mathematica 的 EDC and RGTC。

EDC and RGTC,即 Riemannian Geometry & Tensor Calculus @ Mathematica,链接:http://www.inp.demokritos.gr/~sbonano/RGTC/

Download RGTC (Version 3.8.9 – May 2013)

  • Download all files – compressed: .sit format (100 KB), .zip format (135 KB)
  • Uncompressed files (~1000 KB):  RGTC.nb  —  OperatorPLT.nb  —  NPsymbolPLT.nb  —  EDCRGTCcode.m. (Only the combined matrixEDC and RGTC code in package format is included — it must be placed in an appropriate directory).
  • Note: RGTC cannot be used for calculations with abstract tensors (manipulation of tensor expressions with abstract indices). It only operates on explicit tensors (nested lists of components which are functions of the coordinates). For abstract calculations try the package xTensor.

Additional Examples can be found here.

英文版维基百科的介绍如下(来自 Tensor software https://en.wikipedia.org/wiki/Tensor_software),红色字体是和广义相对论计算有关的工具,我专门标注了出来。


Standalone software

  • SPLATT[1] is an open source software package for high-performance sparse tensor factorization. SPLATT ships a stand-alone executable, C/C++ library, and Octave/MATLABAPI.
  • Cadabra[2] is a computer algebra system (CAS) designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor polynomial simplification including multi-term symmetries, fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many more. The input format is a subset of TeX. Both a command-line and a graphical interface are available.
  • Tela[3] is a software package similar to Matlab and (GNU) Octave, but designed specifically for tensors.

Software for use with Mathematica

  • Tensor[4] is a tensor package written for the Mathematica system. It provides many functions relevant for General Relativity calculations in general Riemann-Cartan geometries.
  • Ricci[5] is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free.
  • TTC[6] Tools of Tensor Calculus is a Mathematica package for doing tensor and exterior calculus on differentiable manifolds.
  • EDC and RGTC,[7] “Exterior Differential Calculus” and “Riemannian Geometry & Tensor Calculus,” are free Mathematica packages for tensor calculus especially designed but not only for general relativity.
  • Tensorial[8] “Tensorial 4.0” is a general purpose tensor calculus package for Mathematica.
  • xAct:[9] Efficient Tensor Computer Algebra for Mathematica. xAct is a collection of packages for fast manipulation of tensor expressions.
  • GREAT[10] is a free package for Mathematica that computes the Christoffel connection and the basic tensors of General Relativity from a given metric tensor.
  • Atlas 2 for Mathematica[11] is a powerful Mathematica toolbox which allows to do a wide range of modern differential geometry calculations
  • GRTensorM[12] is a computer algebra package for performing calculations in the general area of differential geometry.
  • MathGR[13] is a package to manipulate tensor and GR calculations with either abstract or explicit indices, simplify tensors with permutational symmetries, decompose tensors from abstract indices to partially or completely explicit indices and convert partial derivatives into total derivatives.
  • TensoriaCalc[14] is a tensor calculus package written for Mathematica 9 and higher, aimed at providing user-friendly functionality and a smooth consistency with the Mathematica language itself. As of January 2015, given a metric and the coordinates used, TensoriaCalc can compute Christoffel symbols, the Riemann curvature tensor, and Ricci tensor/scalar; it allows for user-defined tensors and is able to perform basic operations such as taking the covariant derivatives of tensors. TensoriaCalc is continuously under development due to time constraints faced by its inventor/developer.

Software for use with Maple

  • GRTensorII[15] is a computer algebra package for performing calculations in the general area of differential geometry.
  • Atlas 2 for Maple[16] is a modern differential geometry for Maple.
  • DifferentialGeometry[17] is a package which performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, General Relativity, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. It is included with Maple.

Software for use with Matlab

Software for use with Maxima

Maxima[23] is a free open source general purpose computer algebra system which includes several packages for tensor algebra calculations in its core distribution. It is particularly useful for calculations with abstract tensors, i.e., when one wishes to do calculations without defining all components of the tensor explicitly. It comes with three tensor packages:[24]

  • itensor for abstract (indicial) tensor manipulation,
  • ctensor for component-defined tensors, and
  • atensor for algebraic tensor manipulation.

Software for use with R

  • Tensor[25] is an R package for basic tensor operations.
  • rTensor[26] provides several tensor decomposition approaches.
  • tensorBF[27] is an R package for Bayesian Tensor decomposition.
  • MTF[28] Bayesian Multi-Tensor Factorization for data fusion and Bayesian versions of Tensor PCA and Tensor CCA. Software: MTF

Libraries

  • Redberry[29] is an open source computer algebra system designed for symbolic tensor manipulation. Redberry provides common tools for expression manipulation, generalized on tensorial objects, as well as tensor-specific features: indices symmetries, LaTeX-style input, natural dummy indices handling, multiple index types etc. The HEP package includes tools for Feynman diagrams calculation: Dirac and SU(N) algebra, Levi-Civita simplifications, tools for calculation of one-loop counterterms etc. Redberry is written in Java and provides extensive Groovy-based programming language.
  • libxm[30] is a lightweight distributed-parallel tensor library written in C.
  • FTensor[31] is a high performance tensor library written in C++.
  • TL[32] is a multi-threaded tensor library implemented in C++ used in Dynare++. The library allows for folded/unfolded, dense/sparse tensor representations, general ranks (symmetries). The library implements Faa Di Bruno formula and is adaptive to available memory. Dynare++ is a standalone package solving higher order Taylor approximations to equilibria of non-linear stochastic models with rational expectations.
  • vmmlib[33] is a C++ linear algebra library that supports 3-way tensors, emphasizing computation and manipulation of several tensor decompositions.
  • Spartns[34] is a Sparse Tensor framework for Common Lisp.
  • FAstMat[35] is a thread-safe general tensor algebra library written in C++ and specially designed for FEM/FVM/BEM/FDM element/edge wise computations.
  • Cyclops Tensor Framework [36] is a distributed memory library for efficient decomposition of tensors of arbitrary type and parallel MPI+OpenMP execution of tensor contractions/functions.
  • TiledArray[37] is a scalable, block-sparse tensor library that is designed to aid in rapid composition of high-performance algebraic tensor equation. It is designed to scale from a single multicore computer to a massively-parallel, distributed-memory system.
  • libtensor [38] is a set of performance linear tensor algebra routines for large tensors found in post-Hartree-Fock methods in quantum chemistry.
  • ITensor [39] features automatic contraction of matching tensor indices. It is written in C++ and has higher-level features for quantum physics algorithms based on tensor networks.
  • Fastor [40] is a high performance C++ tensor algebra library that supports tensors of any arbitrary dimensions and all their possible contraction and permutation thereof. It employs compile-time graph search optimisations to find the optimal contraction sequence between arbitrary number of tensors in a network. It has high level domain specific features for solving nonlinear multiphysics problem using FEM.
  • Xerus [41] is a C++ tensor algebra library for tensors of arbitrary dimensions and tensor decomposition into general tensor networks (focusing on matrix product states). It offers Einstein notation like syntax and optimizes the contraction order of any network of tensors at runtime so that dimensions need not be fixed at compile-time.

我对广义相对论的很多计算并不是很清楚,基本上也没怎么计算过度规、张量、四维电磁势等等东西,只是在现成的度规下开始做黑洞解,然后算一些温度、熵,深一些就做一些泰勒展开,或者用一下留数定理等,对张量的完整计算并不熟悉。

吴老师使用Maple的GRTenser张量包来计算黑洞相关的解,效果非常好,只可惜我没用过Maple,也不是很懂这个软件。对于Mathematica用的稍微多一点的我,在网上找到了一些书,这些书基本上是外文图书,但是内容都很不错。突然感叹,外国人虽少,但对某一点是真专注,再小众也有人做的非常深非常好,佩服。

这是在亚马逊搜索到的图书:查看链接

我下载了几本电子书,感觉还可以,放到这里可以下载:

MATHEMATICA全书(第4版).pdf
Mathmatica for theoretical physics I.pdf
Mathmatica for theoretical physics II.pdf