Numerical Analysis – When, What, (but not how)  Once you understand the Math & know C++, Numerical Methods are basically blocks of iterative & conditional math code. I found the real trick was seeing the forest for the trees – knowing which method to use for which situation. Its pretty easy to get lost in the details – so I’ve tried to organize these methods in a way that I can quickly look this up.   I’ve included links to detailed explanations and to C++ code examples.  I’ve tried to classify Numerical methods in the following broad categories:     Solving Systems of Linear Equations    Solving Non-Linear Equations Iteratively    Interpolation    Curve Fitting    Optimization    Numerical Differentiation & Integration    Solving ODEs    Boundary Problems    Solving EigenValue problems   Enjoy – I did !  Solving Systems of Linear Equations  Overview  Solve sets of algebraic equations with x unknowns  The set is commonly in matrix form  Gauss-Jordan Elimination  http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination  C++: http://www.codekeep.net/snippets/623f1923-e03c-4636-8c92-c9dc7aa0d3c0.aspx  Produces solution of the equations & the coefficient matrix  Efficient, stable  2 steps:  · Forward Elimination – matrix decomposition: reduce set to triangular form (0s below the diagonal) or row echelon form. If degenerate, then there is no solution  · Backward Elimination –write the original matrix as the product of ints inverse matrix & its reduced row-echelon matrix à reduce set to row canonical form & use back-substitution to find the solution to the set   Elementary ops for matrix decomposition:  · Row multiplication  · Row switching  · Add multiples of rows to other rows  Use pivoting to ensure rows are ordered for achieving triangular form  LU Decomposition  http://en.wikipedia.org/wiki/LU_decomposition  C++: http://ganeshtiwaridotcomdotnp.blogspot.co.il/2009/12/c-c-code-lu-decomposition-for-solving.html    Represent the matrix as a product of lower & upper triangular matrices  A modified version of GJ Elimination  Advantage – can easily apply forward & backward elimination to solve triangular matrices  Techniques:  · Doolittle Method – sets the L matrix diagonal to unity  · Crout Method - sets the U matrix diagonal to unity  Note: both the L & U matrices share the same unity diagonal & can be stored compactly in the same matrix  Gauss-Seidel Iteration   http://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method  C++:  http://www.nr.com/forum/showthread.php?t=722  Transform the linear set of equations into a single equation & then use numerical integration (as integration formulas have Sums, it is implemented iteratively).  an optimization of Gauss-Jacobi: 1.5 times faster, requires 0.25 iterations to achieve the same tolerance  Solving Non-Linear Equations Iteratively  find roots of polynomials – there may be 0, 1 or n solutions for an n order polynomial  use iterative techniques  Iterative methods  · used when there are no known analytical techniques  · Requires set functions to be continuous & differentiable  · Requires an initial seed value – choice is critical to convergence à conduct multiple runs with different starting points & then select best result  · Systematic - iterate until diminishing returns, tolerance or max iteration conditions are met  · bracketing techniques will always yield convergent solutions, non-bracketing methods may fail to converge  Incremental method   if a nonlinear function has opposite signs at 2 ends of a small interval x1 & x2, then there is likely to be a solution in their interval – solutions are detected by evaluating a function over interval steps, for a change in sign, adjusting the step size dynamically.  Limitations – can miss closely spaced solutions in large intervals, cannot detect degenerate (coinciding) solutions, limited to functions that cross the x-axis, gives false positives for singularities  Fixed point method  http://en.wikipedia.org/wiki/Fixed-point_iteration  C++: http://books.google.co.il/books?id=weYj75E_t6MC&pg=PA79&lpg=PA79&dq=fixed+point+method++c%2B%2B&source=bl&ots=LQ-5P_taoC&sig=lENUUIYBK53tZtTwNfHLy5PEWDk&hl=en&sa=X&ei=wezDUPW1J5DptQaMsIHQCw&redir_esc=y#v=onepage&q=fixed%20point%20method%20%20c%2B%2B&f=false    Algebraically rearrange a solution to isolate a variable then apply incremental method  Bisection method  http://en.wikipedia.org/wiki/Bisection_method  C++: http://numericalcomputing.wordpress.com/category/algorithms/    Bracketed - Select an initial interval, keep bisecting it ad midpoint into sub-intervals and then apply incremental method on smaller & smaller intervals – zoom in  Adv: unaffected by function gradient à reliable  Disadv: slow convergence  False Position Method  http://en.wikipedia.org/wiki/False_position_method  C++: http://www.dreamincode.net/forums/topic/126100-bisection-and-false-position-methods/  Bracketed - Select an initial interval , & use the relative value of function at interval end points to select next sub-intervals (estimate how far between the end points the solution might be & subdivide based on this)  Newton-Raphson method  http://en.wikipedia.org/wiki/Newton's_method  C++: http://www-users.cselabs.umn.edu/classes/Summer-2012/csci1113/index.php?page=./newt3    Also known as Newton's method  Convenient, efficient  Not bracketed – only a single initial guess is required to start iteration – requires an analytical expression for the first derivative of the function as input.  Evaluates the function & its derivative at each step.   Can be extended to the Newton MutiRoot method for solving multiple roots  Can be easily applied to an of n-coupled set of non-linear equations – conduct a Taylor Series expansion of a function, dropping terms of order n, rewrite as a Jacobian matrix of PDs & convert to simultaneous linear equations !!!  Secant Method  http://en.wikipedia.org/wiki/Secant_method  C++: http://forum.vcoderz.com/showthread.php?p=205230    Unlike N-R, can estimate first derivative from an initial interval (does not require root to be bracketed) instead of inputting it  Since derivative is approximated, may converge slower. Is fast in practice as it does not have to evaluate the derivative at each step.  Similar implementation to False Positive method  Birge-Vieta Method  http://mat.iitm.ac.in/home/sryedida/public_html/caimna/transcendental/polynomial%20methods/bv%20method.html  C++: http://books.google.co.il/books?id=cL1boM2uyQwC&pg=SA3-PA51&lpg=SA3-PA51&dq=Birge-Vieta+Method+c%2B%2B&source=bl&ots=QZmnDTK3rC&sig=BPNcHHbpR_DKVoZXrLi4nVXD-gg&hl=en&sa=X&ei=R-_DUK2iNIjzsgbE5ID4Dg&redir_esc=y#v=onepage&q=Birge-Vieta%20Method%20c%2B%2B&f=false  combines Horner's method of polynomial evaluation (transforming into lesser degree polynomials that are more computationally efficient to process) with Newton-Raphson to provide a computational speed-up  Interpolation  Overview  Construct new data points for as close as possible fit within range of a discrete set of known points (that were obtained via sampling, experimentation)   Use Taylor Series Expansion of a function f(x) around a specific value for x  Linear Interpolation  http://en.wikipedia.org/wiki/Linear_interpolation  C++: http://www.hamaluik.com/?p=289    Straight line between 2 points à concatenate interpolants between each pair of data points   Bilinear Interpolation  http://en.wikipedia.org/wiki/Bilinear_interpolation  C++: http://supercomputingblog.com/graphics/coding-bilinear-interpolation/2/    Extension of the linear function for interpolating functions of 2 variables – perform linear interpolation first in 1 direction, then in another.  Used in image processing – e.g. texture mapping filter. Uses 4 vertices to interpolate a value within a unit cell.  Lagrange Interpolation   http://en.wikipedia.org/wiki/Lagrange_polynomial  C++: http://www.codecogs.com/code/maths/approximation/interpolation/lagrange.php    For polynomials   Requires recomputation for all terms for each distinct x value – can only be applied for small number of nodes  Numerically unstable  Barycentric Interpolation  http://epubs.siam.org/doi/pdf/10.1137/S0036144502417715  C++: http://www.gamedev.net/topic/621445-barycentric-coordinates-c-code-check/    Rearrange the terms in the equation of the Legrange interpolation by defining weight functions that are independent of the interpolated value of x  Newton Divided Difference Interpolation  http://en.wikipedia.org/wiki/Newton_polynomial  C++: http://jee-appy.blogspot.co.il/2011/12/newton-divided-difference-interpolation.html  Hermite Divided Differences:    Interpolation polynomial approximation for a given set of data points in the NR form - divided differences are used to approximately calculate the various differences.  For a given set of 3 data points , fit a quadratic interpolant through the data  Bracketed functions allow Newton divided differences to be calculated recursively  Difference table   Cubic Spline Interpolation  http://en.wikipedia.org/wiki/Spline_interpolation  C++: https://www.marcusbannerman.co.uk/index.php/home/latestarticles/42-articles/96-cubic-spline-class.html    Spline is a piecewise polynomial  Provides smoothness – for interpolations with significantly varying data  Use weighted coefficients to bend the function to be smooth & its 1st & 2nd derivatives are continuous through the edge points in the interval  Curve Fitting  A generalization of interpolating whereby given data points may contain noise à the curve does not necessarily pass through all the points  Least Squares Fit  http://en.wikipedia.org/wiki/Least_squares  C++: http://www.ccas.ru/mmes/educat/lab04k/02/least-squares.c    Residual – difference between observed value & expected value  Model function is often chosen as a linear combination of the specified functions   Determines:  A) The model instance in which the sum of squared residuals has the least value  B) param values for which model best fits data  Straight Line Fit  Linear correlation between independent variable and dependent variable  Linear Regression   http://en.wikipedia.org/wiki/Linear_regression  C++: http://www.oocities.org/david_swaim/cpp/linregc.htm    Special case of statistically exact extrapolation  Leverage least squares  Given a basis function, the sum of the residuals is determined and the corresponding gradient equation is expressed as a set of normal linear equations in matrix form that can be solved (e.g. using LU Decomposition)  Can be weighted - Drop the assumption that all errors have the same significance –-> confidence of accuracy is different for each data point. Fit the function closer to points with higher weights  Polynomial Fit - use a polynomial basis function  Moving Average  http://en.wikipedia.org/wiki/Moving_average  C++: http://www.codeproject.com/Articles/17860/A-Simple-Moving-Average-Algorithm    Used for smoothing (cancel fluctuations to highlight longer-term trends & cycles), time series data analysis, signal processing filters  Replace each data point with average of neighbors.   Can be simple (SMA), weighted (WMA), exponential (EMA). Lags behind latest data points – extra weight can be given to more recent data points. Weights can decrease arithmetically or exponentially according to distance from point.  Parameters: smoothing factor, period, weight basis  Optimization  Overview  Given function with multiple variables, find Min (or max by minimizing –f(x))  Iterative approach  Efficient, but not necessarily reliable  Conditions: noisy data, constraints, non-linear models  Detection via sign of first derivative - Derivative of saddle points will be 0  Local minima  Bisection method  Similar method for finding a root for a non-linear equation  Start with an interval that contains a minimum  Golden Search method  http://en.wikipedia.org/wiki/Golden_section_search  C++: http://www.codecogs.com/code/maths/optimization/golden.php    Bisect intervals according to golden ratio 0.618..  Achieves reduction by evaluating a single function instead of 2  Newton-Raphson Method  Brent method  http://en.wikipedia.org/wiki/Brent's_method  C++: http://people.sc.fsu.edu/~jburkardt/cpp_src/brent/brent.cpp  Based on quadratic or parabolic interpolation – if the function is smooth & parabolic near to the minimum, then a parabola fitted through any 3 points should approximate the minima – fails when the 3 points are collinear , in which case the denominator is 0  Simplex Method  http://en.wikipedia.org/wiki/Simplex_algorithm  C++: http://www.codeguru.com/cpp/article.php/c17505/Simplex-Optimization-Algorithm-and-Implemetation-in-C-Programming.htm    Find the global minima of any multi-variable function  Direct search – no derivatives required  At each step it maintains a non-degenerative simplex – a convex hull of n+1 vertices.  Obtains the minimum for a function with n variables by evaluating the function at n-1 points, iteratively replacing the point of worst result with the point of best result, shrinking the multidimensional simplex around the best point.  Point replacement involves expanding & contracting the simplex near the worst value point to determine a better replacement point  Oscillation can be avoided by choosing the 2nd worst result  Restart if it gets stuck  Parameters: contraction & expansion factors  Simulated Annealing  http://en.wikipedia.org/wiki/Simulated_annealing  C++: http://code.google.com/p/cppsimulatedannealing/    Analogy to heating & cooling metal to strengthen its structure  Stochastic method – apply random permutation search for global minima - Avoid entrapment in local minima via hill climbing   Heating schedule - Annealing schedule params: temperature, iterations at each temp, temperature delta  Cooling schedule – can be linear, step-wise or exponential  Differential Evolution  http://en.wikipedia.org/wiki/Differential_evolution  C++: http://www.amichel.com/de/doc/html/  More advanced stochastic methods analogous to biological processes: Genetic algorithms, evolution strategies  Parallel direct search method against multiple discrete or continuous variables  Initial population of variable vectors chosen randomly – if weighted difference vector of 2 vectors yields a lower objective function value then it replaces the comparison vector   Many params: #parents, #variables, step size, crossover constant etc  Convergence is slow – many more function evaluations than simulated annealing  Numerical Differentiation  Overview  2 approaches to finite difference methods:   · A) approximate function via polynomial interpolation then differentiate  · B) Taylor series approximation – additionally provides error estimate  Finite Difference methods  http://en.wikipedia.org/wiki/Finite_difference_method  C++: http://www.wpi.edu/Pubs/ETD/Available/etd-051807-164436/unrestricted/EAMPADU.pdf    Find differences between high order derivative values - Approximate differential equations by finite differences at evenly spaced data points  Based on forward & backward Taylor series expansion of f(x) about x plus or minus multiples of delta h.   Forward / backward difference - the sums of the series contains even derivatives and the difference of the series contains odd derivatives – coupled equations that can be solved.   Provide an approximation of the derivative within a O(h^2) accuracy  There is also central difference & extended central difference which has a O(h^4) accuracy  Richardson Extrapolation  http://en.wikipedia.org/wiki/Richardson_extrapolation  C++: http://mathscoding.blogspot.co.il/2012/02/introduction-richardson-extrapolation.html  A sequence acceleration method applied to finite differences  Fast convergence, high accuracy O(h^4)   Derivatives via Interpolation  Cannot apply Finite Difference method to discrete data points at uneven intervals – so need to approximate the derivative of f(x) using the derivative of the interpolant via 3 point Lagrange Interpolation  Note: the higher the order of the derivative, the lower the approximation precision  Numerical Integration  Estimate finite & infinite integrals of functions  More accurate procedure than numerical differentiation  Use when it is not possible to obtain an integral of a function analytically or when the function is not given, only the data points are  Newton Cotes Methods  http://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas  C++: http://www.siafoo.net/snippet/324    For equally spaced data points  Computationally easy – based on local interpolation of n rectangular strip areas that is piecewise fitted to a polynomial to get the sum total area  Evaluate the integrand at n+1 evenly spaced points – approximate definite integral by Sum  Weights are derived from Lagrange Basis polynomials  Leverage Trapezoidal Rule for default 2nd formulas, Simpson 1/3 Rule for substituting 3 point formulas, Simpson 3/8 Rule for 4 point formulas. For 4 point formulas use Bodes Rule. Higher orders obtain more accurate results  Trapezoidal Rule uses simple area, Simpsons Rule replaces the integrand f(x) with a quadratic polynomial p(x) that uses the same values as f(x) for its end points, but adds a midpoint  Romberg Integration  http://en.wikipedia.org/wiki/Romberg's_method  C++: http://code.google.com/p/romberg-integration/downloads/detail?name=romberg.cpp&can=2&q=  Combines trapezoidal rule with Richardson Extrapolation  Evaluates the integrand at equally spaced points  The integrand must have continuous derivatives  Each R(n,m) extrapolation uses a higher order integrand polynomial replacement rule (zeroth starts with trapezoidal) à a lower triangular matrix set of equation coefficients where the bottom right term has the most accurate approximation. The process continues until the difference between 2 successive diagonal terms becomes sufficiently small.   Gaussian Quadrature  http://en.wikipedia.org/wiki/Gaussian_quadrature  C++: http://www.alglib.net/integration/gaussianquadratures.php  Data points are chosen to yield best possible accuracy – requires fewer evaluations  Ability to handle singularities, functions that are difficult to evaluate   The integrand can include a weighting function determined by a set of orthogonal polynomials.  Points & weights are selected so that the integrand yields the exact integral if f(x) is a polynomial of degree <= 2n+1   Techniques (basically different weighting functions):  · Gauss-Legendre Integration w(x)=1  · Gauss-Laguerre Integration w(x)=e^-x  · Gauss-Hermite Integration w(x)=e^-x^2  · Gauss-Chebyshev Integration w(x)= 1 / Sqrt(1-x^2)  Solving ODEs  Use when high order differential equations cannot be solved analytically  Evaluated under boundary conditions  RK for systems – a high order differential equation can always be transformed into a coupled first order system of equations  Euler method  http://en.wikipedia.org/wiki/Euler_method  C++: http://rosettacode.org/wiki/Euler_method    First order Runge–Kutta method.   Simple recursive method – given an initial value, calculate derivative deltas.  Unstable & not very accurate (O(h) error) – not used in practice  A first-order method - the local error (truncation error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size   In evolving solution between data points xn & xn+1, only evaluates derivatives at beginning of interval xn à asymmetric at boundaries  Higher order Runge Kutta  http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods  C++: http://www.dreamincode.net/code/snippet1441.htm    2nd & 4th order RK - Introduces parameterized midpoints for more symmetric solutions à accuracy at higher computational cost  Adaptive RK – RK-Fehlberg – estimate the truncation at each integration step & automatically adjust the step size to keep error within prescribed limits. At each step 2 approximations are compared – if in disagreement to a specific accuracy, the step size is reduced  Boundary Value Problems  Where solution of differential equations are located at 2 different values of the independent variable x à more difficult, because cannot just start at point of initial value – there may not be enough starting conditions available at the end points to produce a unique solution  An n-order equation will require n boundary conditions – need to determine the missing n-1 conditions which cause the given conditions at the other boundary to be satisfied  Shooting Method  http://en.wikipedia.org/wiki/Shooting_method  C++: http://ganeshtiwaridotcomdotnp.blogspot.co.il/2009/12/c-c-code-shooting-method-for-solving.html    Iteratively guess the missing values for one end & integrate, then inspect the discrepancy with the boundary values of the other end to adjust the estimate  Given the starting boundary values u1 & u2 which contain the root u, solve u given the false position method (solving the differential equation as an initial value problem via 4th order RK), then use u to solve the differential equations.  Finite Difference Method  For linear & non-linear systems  Higher order derivatives require more computational steps – some combinations for boundary conditions may not work though  Improve the accuracy by increasing the number of mesh points  Solving EigenValue Problems  An eigenvalue can substitute a matrix when doing matrix multiplication à convert matrix multiplication into a polynomial EigenValue   For a given set of equations in matrix form, determine what are the solution eigenvalue & eigenvectors  Similar Matrices - have same eigenvalues. Use orthogonal similarity transforms to reduce a matrix to diagonal form from which eigenvalue(s) & eigenvectors can be computed iteratively    Jacobi method  http://en.wikipedia.org/wiki/Jacobi_method  C++: http://people.sc.fsu.edu/~jburkardt/classes/acs2_2008/openmp/jacobi/jacobi.html  Robust but Computationally intense – use for small matrices < 10x10  Power Iteration  http://en.wikipedia.org/wiki/Power_iteration  For any given real symmetric matrix, generate the largest single eigenvalue & its eigenvectors  Simplest method – does not compute matrix decomposition à suitable for large, sparse matrices  Inverse Iteration  Variation of power iteration method – generates the smallest eigenvalue from the inverse matrix  Rayleigh Method  http://en.wikipedia.org/wiki/Rayleigh's_method_of_dimensional_analysis  Variation of power iteration method   Rayleigh Quotient Method  Variation of inverse iteration method   Matrix Tri-diagonalization Method  Use householder algorithm to reduce an NxN symmetric matrix to a tridiagonal real symmetric matrix vua N-2 orthogonal transforms        Whats Next  Outside of Numerical Methods there are lots of different types of algorithms that I’ve learned over the decades:      Data Mining – (I covered this briefly in a previous post: http://geekswithblogs.net/JoshReuben/archive/2007/12/31/ssas-dm-algorithms.aspx )    Search & Sort    Routing    Problem Solving    Logical Theorem Proving    Planning    Probabilistic Reasoning    Machine Learning    Solvers (eg MIP)    Bioinformatics (Sequence Alignment, Protein Folding)    Quant Finance (I read Wilmott’s books – interesting)   Sooner or later, I’ll cover the above topics as well.