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  • Wrapping a paragraph inside a div?

    - by LOD121
    How do I make my paragraph wrap inside my div? It is currently overflowing outside of the div and I have no idea how to stop the paragraph from overflowing over the edge. I do not want a scroll bar with: overflow: scroll; and the other overflow options don't seem to help here either... I have the following code: div { width: 1200px; margin: 0 auto; } .container { overflow: hidden; } .content { width: 1000px; float: left; margin-left: 0; text-align: left; } .rightpanel { width: 190px; float: right; margin-right: 0; } <div class="container"> <div class="content"> <p>Some content flowing over more than one line</p> </div> <div class="rightpanel"> <!-- content --> </div> </div> Edit: <div class="container"> <div class="content"> <div class="leftcontent"> </div> <div class="newsfeed"> <div class="newsitem"> <p>Full age sex set feel her told. Tastes giving in passed direct me valley as supply. End great stood boy noisy often way taken short. Rent the size our more door. Years no place abode in no child my. Man pianoforte too solicitude friendship devonshire ten ask. Course sooner its silent but formal she led. Extensive he assurance extremity at breakfast. Dear sure ye sold fine sell on. Projection at up connection literature insensible motionless projecting.<br><br>Be at miss or each good play home they. It leave taste mr in it fancy. She son lose does fond bred gave lady get. Sir her company conduct expense bed any. Sister depend change off piqued one. Contented continued any happiness instantly objection yet her allowance. Use correct day new brought tedious. By come this been in. Kept easy or sons my it done.</p> </div> </div> </div> <div class="rightpanel"> </div>

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  • SQLAuthority News – Job Interviewing the Right Way (and for the Right Reasons) – Guest Post by Feodor Georgiev

    - by pinaldave
    Feodor Georgiev is a SQL Server database specialist with extensive experience of thinking both within and outside the box. He has wide experience of different systems and solutions in the fields of architecture, scalability, performance, etc. Feodor has experience with SQL Server 2000 and later versions, and is certified in SQL Server 2008. Feodor has written excellent article on Job Interviewing the Right Way. Here is his article in his own language. A while back I was thinking to start a blog post series on interviewing and employing IT personnel. At that time I had just read the ‘Smart and gets things done’ book (http://www.joelonsoftware.com/items/2007/06/05.html) and I was hyped up on some debatable topics regarding finding and employing the best people in the branch. I have no problem with hiring the best of the best; it’s just the definition of ‘the best of the best’ that makes things a bit more complicated. One of the fundamental books one can read on the topic of interviewing is the one mentioned above. If you have not read it, then you must do so; not because it contains the ultimate truth, and not because it gives the answers to most questions on the subject, but because the book contains an extensive set of questions about interviewing and employing people. Of course, a big part of these questions have different answers, depending on location, culture, available funds and so on. (What works in the US may not necessarily work in the Nordic countries or India, or it may work in a different way). The only thing that is valid regardless of any external factor is this: curiosity. In my belief there are two kinds of people – curious and not-so-curious; regardless of profession. Think about it – professional success is directly proportional to the individual’s curiosity + time of active experience in the field. (I say ‘active experience’ because vacations and any distractions do not count as experience :)  ) So, curiosity is the factor which will distinguish a good employee from the not-so-good one. But let’s shift our attention to something else for now: a few tips and tricks for successful interviews. Tip and trick #1: get your priorities straight. Your status usually dictates your priorities; for example, if the person looking for a job has just relocated to a new country, they might tend to ignore some of their priorities and overload others. In other words, setting priorities straight means to define the personal criteria by which the interview process is lead. For example, similar to the following questions can help define the criteria for someone looking for a job: How badly do I need a (any) job? Is it more important to work in a clean and quiet environment or is it important to get paid well (or both, if possible)? And so on… Furthermore, before going to the interview, the candidate should have a list of priorities, sorted by the most importance: e.g. I want a quiet environment, x amount of money, great helping boss, a desk next to a window and so on. Also it is a good idea to be prepared and know which factors can be compromised and to what extent. Tip and trick #2: the interview is a two-way street. A job candidate should not forget that the interview process is not a one-way street. What I mean by this is that while the employer is interviewing the potential candidate, the job seeker should not miss the chance to interview the employer. Usually, the employer and the candidate will meet for an interview and talk about a variety of topics. In a quality interview the candidate will be presented to key members of the team and will have the opportunity to ask them questions. By asking the right questions both parties will define their opinion about each other. For example, if the candidate talks to one of the potential bosses during the interview process and they notice that the potential manager has a hard time formulating a question, then it is up to the candidate to decide whether working with such person is a red flag for them. There are as many interview processes out there as there are companies and each one is different. Some bigger companies and corporates can afford pre-selection processes, 3 or even 4 stages of interviews, small companies usually settle with one interview. Some companies even give cognitive tests on the interview. Why not? In his book Joel suggests that a good candidate should be pampered and spoiled beyond belief with a week-long vacation in New York, fancy hotels, food and who knows what. For all I can imagine, an interview might even take place at the top of the Eifel tower (right, Mr. Joel, right?) I doubt, however, that this is the optimal way to capture the attention of a good employee. The ‘curiosity’ topic What I have learned so far in my professional experience is that opinions can be subjective. Plus, opinions on technology subjects can also be subjective. According to Joel, only hiring the best of the best is worth it. If you ask me, there is no such thing as best of the best, simply because human nature (well, aside from some physical limitations, like putting your pants on through your head :) ) has no boundaries. And why would it have boundaries? I have seen many curious and interesting people, naturally good at technology, though uninterested in it as one  can possibly be; I have also seen plenty of people interested in technology, who (in an ideal world) should have stayed far from it. At any rate, all of this sums up at the end to the ‘supply and demand’ factor. The interview process big-bang boils down to this: If there is a mutual benefit for both the employer and the potential employee to work together, then it all sorts out nicely. If there is no benefit, then it is much harder to get to a common place. Tip and trick #3: word-of-mouth is worth a thousand words Here I would just mention that the best thing a job candidate can get during the interview process is access to future team members or other employees of the new company. Nowadays the world has become quite small and everyone knows everyone. Look at LinkedIn, look at other professional networks and you will realize how small the world really is. Knowing people is a good way to become more approachable and to approach them. Tip and trick #4: Be confident. It is true that for some people confidence is as natural as breathing and others have to work hard to express it. Confidence is, however, a key factor in convincing the other side (potential employer or employee) that there is a great chance for success by working together. But it cannot get you very far if it’s not backed up by talent, curiosity and knowledge. Tip and trick #5: The right reasons What really bothers me in Sweden (and I am sure that there are similar situations in other countries) is that there is a tendency to fill quotas and to filter out candidates by criteria different from their skill and knowledge. In job ads I see quite often the phrases ‘positive thinker’, ‘team player’ and many similar hints about personality features. So my guess here is that discrimination has evolved to a new level. Let me clear up the definition of discrimination: ‘unfair treatment of a person or group on the basis of prejudice’. And prejudice is the ‘partiality that prevents objective consideration of an issue or situation’. In other words, there is not much difference whether a job candidate is filtered out by race, gender or by personality features – it is all a bad habit. And in reality, there is no proven correlation between the technology knowledge paired with skills and the personal features (gender, race, age, optimism). It is true that a significantly greater number of Darwin awards were given to men than to women, but I am sure that somewhere there is a paper or theory explaining the genetics behind this. J This topic actually brings to mind one of my favorite work related stories. A while back I was working for a big company with many teams involved in their processes. One of the teams was occupying 2 rooms – one had the team members and was full of light, colorful posters, chit-chats and giggles, whereas the other room was dark, lighted only by a single monitor with a quiet person in front of it. Later on I realized that the ‘dark room’ person was the guru and the ultimate problem-solving-brain who did not like the chats and giggles and hence was in a separate room. In reality, all severe problems which the chatty and cheerful team members could not solve and all emergencies were directed to ‘the dark room’. And thus all worked out well. The moral of the story: Personality has nothing to do with technology knowledge and skills. End of story. Summary: I’d like to stress the fact that there is no ultimately perfect candidate for a job, and there is no such thing as ‘best-of-the-best’. From my personal experience, the main criteria by which I measure people (co-workers and bosses) is the curiosity factor; I know from experience that the more curious and inventive a person is, the better chances there are for great achievements in their field. Related stories: (for extra credit) 1) Get your priorities straight. A while back as a consultant I was working for a few days at a time at different offices and for different clients, and so I was able to compare and analyze the work environments. There were two different places which I compared and recently I asked a friend of mine the following question: “Which one would you prefer as a work environment: a noisy office full of people, or a quiet office full of faulty smells because the office is rarely cleaned?” My friend was puzzled for a while, thought about it and said: “Hmm, you are talking about two different kinds of pollution… I will probably choose the second, since I can clean the workplace myself a bit…” 2) The interview is a two-way street. One time, during a job interview, I met a potential boss that had a hard time phrasing a question. At that particular time it was clear to me that I would not have liked to work under this person. According to my work religion, the properly asked question contains at least half of the answer. And if I work with someone who cannot ask a question… then I’d be doing double or triple work. At another interview, after the technical part with the team leader of the department, I was introduced to one of the team members and we were left alone for 5 minutes. I immediately jumped on the occasion and asked the blunt question: ‘What have you learned here for the past year and how do you like your job?’ The team member looked at me and said ‘Nothing really. I like playing with my cats at home, so I am out of here at 5pm and I don’t have time for much.’ I was disappointed at the time and I did not take the job offer. I wasn’t that shocked a few months later when the company went bankrupt. 3) The right reasons to take a job: personality check. A while back I was asked to serve as a job reference for a coworker. I agreed, and after some weeks I got a phone call from the company where my colleague was applying for a job. The conversation started with the manager’s question about my colleague’s personality and about their social skills. (You can probably guess what my internal reaction was… J ) So, after 30 minutes of pouring common sense into the interviewer’s head, we finally agreed on the fact that a shy or quiet personality has nothing to do with work skills and knowledge. Some years down the road my former colleague is taking the manager’s position as the manager is demoted to a different department. Reference: Feodor Georgiev, Pinal Dave (http://blog.SQLAuthority.com) Filed under: PostADay, Readers Contribution, SQL, SQL Authority, SQL Query, SQL Server, SQL Tips and Tricks, T SQL, Technology

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  • A Taxonomy of Numerical Methods v1

    - by JoshReuben
    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.

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