PhysicsCircus

使用Matlab來作數值分析

1. 數值分析
2. Moler數值分析
3. 輔大的數值分析
4. 台大的數值分析

format long e; 將表示式以科學記號表示(e: scientific, f: fixed工程用)

Taylor Expansion

為了要能看histogram分布，所以就想要學習root，抓取了cvs snapshot，安裝時發現缺少了libXpm, libXext, libXmu，所以就安裝了libXpm4, libXpm-dev, libxau-dev, libxext-dev, libxi-dev, x11proto-input-dev, x11proto-xext-dev, libxmu-dev, libxmu-headers。 利用

./configure
make
make install

來安裝root到系統中。 設定export ROOTSYS=~/root

>> help symbolic

  Symbolic Math Toolbox.
  Version 3.0.1 (R13SP1) 27-Dec-2002

  Calculus.
    diff        - Differentiate.
    int         - Integrate.
    limit       - Limit.
    taylor      - Taylor series.
    jacobian    - Jacobian matrix.
    symsum      - Summation of series.

  Linear Algebra.
    diag        - Create or extract diagonals.
    triu        - Upper triangle.
    tril        - Lower triangle.
    inv         - Matrix inverse.
    det         - Determinant.
    rank        - Rank.
    rref        - Reduced row echelon form.
    null        - Basis for null space.
    colspace    - Basis for column space.
    eig         - Eigenvalues and eigenvectors.
    svd         - Singular values and singular vectors.
    jordan      - Jordan canonical (normal) form.
    poly        - Characteristic polynomial.
    expm        - Matrix exponential.

  Simplification.
    simplify    - Simplify.
    expand      - Expand.
    factor      - Factor.
    collect     - Collect.
    simple      - Search for shortest form.
    numden      - Numerator and denominator.
    horner      - Nested polynomial representation.
    subexpr     - Rewrite in terms of subexpressions.
    subs        - Symbolic substitution.

  Solution of Equations.
    solve       - Symbolic solution of algebraic equations.
    dsolve      - Symbolic solution of differential equations.
    finverse    - Functional inverse.
    compose     - Functional composition.

  Variable Precision Arithmetic.
    vpa         - Variable precision arithmetic.
    digits      - Set variable precision accuracy.

  Integral Transforms.
    fourier     - Fourier transform.
    laplace     - Laplace transform.
    ztrans      - Z transform.
    ifourier    - Inverse Fourier transform.
    ilaplace    - Inverse Laplace transform.
    iztrans     - Inverse Z transform.

  Conversions.
    double      - Convert symbolic matrix to double.
    poly2sym    - Coefficient vector to symbolic polynomial.
    sym2poly    - Symbolic polynomial to coefficient vector.
    char        - Convert sym object to string.

  Basic Operations.
    sym         - Create symbolic object.
    syms        - Short-cut for constructing symbolic objects.
    findsym     - Determine symbolic variables.
    pretty      - Pretty print a symbolic expression.
    latex       - LaTeX representation of a symbolic expression.
    ccode       - C code representation of a symbolic expression.
    fortran     - FORTRAN representation of a symbolic expression.

  Special Functions.
    sinint      - Sine integral.
    cosint      - Cosine integral.
    zeta        - Riemann zeta function.
    lambertw    - Lambert W function.
    gcd         - Greatest common divisor.
    lcm         - Least common multiple.

  String handling utilities.
    isvarname   - Check for a valid variable name (MATLAB Toolbox).
    vectorize   - Vectorize a symbolic expression.

  Pedagogical and Graphical Applications.
    rsums       - Riemann sums.
    ezcontour   - Easy to use contour plotter.
    ezcontourf  - Easy to use filled contour plotter.
    ezmesh      - Easy to use mesh (surface) plotter.
    ezmeshc     - Easy to use combined mesh/contour plotter.
    ezplot      - Easy to use function, implicit, and parametric curve plotter.
    ezplot3     - Easy to use spatial curve plotter.
    ezpolar     - Easy to use polar coordinates plotter.
    ezsurf      - Easy to use surface plotter.
    ezsurfc     - Easy to use combined surface/contour plotter.
    funtool     - Function calculator.
    taylortool  - Taylor series calculator.

  Demonstrations.
    symintro    - Introduction to the Symbolic Toolbox.
    symcalcdemo - Calculus demonstration.
    symlindemo  - Demonstrate symbolic linear algebra.
    symvpademo  - Demonstrate variable precision arithmetic
    symrotdemo  - Study plane rotations.
    symeqndemo  - Demonstrate symbolic equation solving.

  Access to Maple. (Not available with Student Version.)
    maple       - Access Maple kernel.
    mfun        - Numeric evaluation of Maple functions.
    mfunlist    - List of functions for MFUN.
    mhelp       - Maple help.
    procread    - Install a Maple procedure. (Requires Extended Toolbox.)

Hydrogen Molecules是2個Proton各別擁有1個electron。 IBM Account Info: 140.110.13.2/u00sam00/慣用密碼

syms r a lam mu positive
Phi=2*lam^(3/2)*exp(-lam*r)
PhiA=2*lam^(3/2)*exp(-lam*(r^2+a^2-2*r*a*mu)^(1/2))
Phi2A=2*lam^(3/2)*exp(-lam*(r^2+4*a^2-4*r*a*mu)^(1/2))
Saa=int(Phi*Phi*r*r,r,0,inf)
PhiABr=int(Phi*Phi2A, mu, -1, 1)/2
PhiABr=simplify(Phi2Ar)
Sab=int(PhiABr*r*r, r, 0, inf)
Sab=simplify(Sab)
N=1/2/(1+Sab)

dPhi=diff(Phi,r)
Kaa=int(dPhi*dPhi*r*r,r,0,inf)
dPhi2A=diff(Phi2A,r)
dPhi2A=simplify(dPhi2A)

Kab=int(dPhi*dPhi2A,mu,-1,1)/2
Kab=simplify(Kab)
Kab=int(Kab*r*r,r,0,inf)

K=Kaa+Kab

1s orbital

Atomic orbitals: 1s equations

The symbols used in the following are:

    * r = radius expressed in atomic units (1 Bohr radius = 52.9 pm)
    * π = 3.14159 approximately
    * e = 2.71828 approximately
    * Z = effective nuclear charge for that orbital in that atom.
    * ρ = 2Zr/n where n is the principal quantum number (1 for the 1s orbital)

Table of equations for the 1s orbital. Function  Equation
Radial wave function, R1s  = 2 × Z3/2 × e-ρ/2
Angular wave function, Y1s  = 1 × (1/4π)1/2
Wave function, ψ1s  = R1s × Y1s
 = 2Z3/2e-ρ/2 × (1/4π)1/2
Electron density  = ψ1s2
Radial distribution function  = 4πr2ψ1s2

Quantum Hall Effect and Adiabatic Transport Landau Level