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  • How to make sprint planning fun

    - by Jacob Spire
    Not only are our sprint planning meetings not fun, they're downright dreadful. The meetings are tedious, and boring, and take forever (a day, but it feels like a lot longer). The developers complain about it, and dread upcoming plannings. Our routine is pretty standard (user story inserted into sprint backlog by priority story is taken apart to tasks tasks are estimated in hours repeat), and I can't figure out what we're doing wrong. How can we make the meetings more enjoyable? ... Some more details, in response to requests for more information: Why are the backlog items not inserted and prioritized before sprint kickoff? User stories are indeed prioritized; we have no idea how long they'll take until we break them down into tasks! From the (excellent) answers here, I see that maybe we shouldn't estimate tasks at all, only the user stories. The reason we estimate tasks (and not stories) is because we've been getting story-estimates terribly wrong -- but I guess that's the subject for an altogether different question. Why are developers complaining? Meetings are long. Meetings are monotonous. Story after story, task after task, struggling (yes, struggling) to estimate how long it will take and what it involves. Estimating tasks makes user-story-estimation seem pointless. The longer the meeting, the less focus in the room. The less focused colleagues are, the longer the meeting takes. A recursive hate-spiral develops. We've considered splitting the meeting into two days in order to keep people focused, but the developers wouldn't hear of it. One day of planning is bad enough; now we'll have two?! Part of our problem is that we go into very small detail (in order to get more accurate estimations). But when we estimate roughly, we go way off the mark! To sum up the question: What are we doing wrong? What additional ways are there to make the meeting generally more enjoyable?

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  • Estimating compressed file size using a list parameter

    - by Sai
    I am currently compressing a list of files from a directory in the following format: tar -cvjf test_1.tar.gz -T test_1.lst --no-recursion The above command will compress only those files mentioned in the list. I am doing this because this list is generated such that it fits a DVD. However, during compression the compression rate decreases the estimated file size and there is abundant space left in the DVD. This is something like a Knapsack algorithm. I would like to estimate the compressed file size and add some more files to the list. I found that it is possible to estimate file size using the following command: tar -cjf - Folder/ | wc -c This command does not take a list parameter. Is there a way to estimate compressed file size? I am also looking into options like perl scripts etc. Edit: I think I should provide more information since I have been doing a lot of web search. I came across a perl script(Link)that sort of emulates the Knapsack algorithm. The current problem with the above mentioned script is that it splits the files in their original state. When I compress the files after splitting them, there are opportunities for adding more files which I consider to be inefficient. There are 2 ways I could resolve the inefficiency: a) Compress individual files and save them in a directory using a script. The compressed file could provide a best estimate. I could generate a script using a folder of compressed files and use them on the uncompressed ones. b) Check whether the compressed file's size is less than the required size. If so, I should keep adding files until I meet the requirement. However, the addition of new files to the compressed file is an optimization problem by itself.

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  • Heaps of Trouble?

    - by Paul White NZ
    If you’re not already a regular reader of Brad Schulz’s blog, you’re missing out on some great material.  In his latest entry, he is tasked with optimizing a query run against tables that have no indexes at all.  The problem is, predictably, that performance is not very good.  The catch is that we are not allowed to create any indexes (or even new statistics) as part of our optimization efforts. In this post, I’m going to look at the problem from a slightly different angle, and present an alternative solution to the one Brad found.  Inevitably, there’s going to be some overlap between our entries, and while you don’t necessarily need to read Brad’s post before this one, I do strongly recommend that you read it at some stage; he covers some important points that I won’t cover again here. The Example We’ll use data from the AdventureWorks database, copied to temporary unindexed tables.  A script to create these structures is shown below: CREATE TABLE #Custs ( CustomerID INTEGER NOT NULL, TerritoryID INTEGER NULL, CustomerType NCHAR(1) COLLATE SQL_Latin1_General_CP1_CI_AI NOT NULL, ); GO CREATE TABLE #Prods ( ProductMainID INTEGER NOT NULL, ProductSubID INTEGER NOT NULL, ProductSubSubID INTEGER NOT NULL, Name NVARCHAR(50) COLLATE SQL_Latin1_General_CP1_CI_AI NOT NULL, ); GO CREATE TABLE #OrdHeader ( SalesOrderID INTEGER NOT NULL, OrderDate DATETIME NOT NULL, SalesOrderNumber NVARCHAR(25) COLLATE SQL_Latin1_General_CP1_CI_AI NOT NULL, CustomerID INTEGER NOT NULL, ); GO CREATE TABLE #OrdDetail ( SalesOrderID INTEGER NOT NULL, OrderQty SMALLINT NOT NULL, LineTotal NUMERIC(38,6) NOT NULL, ProductMainID INTEGER NOT NULL, ProductSubID INTEGER NOT NULL, ProductSubSubID INTEGER NOT NULL, ); GO INSERT #Custs ( CustomerID, TerritoryID, CustomerType ) SELECT C.CustomerID, C.TerritoryID, C.CustomerType FROM AdventureWorks.Sales.Customer C WITH (TABLOCK); GO INSERT #Prods ( ProductMainID, ProductSubID, ProductSubSubID, Name ) SELECT P.ProductID, P.ProductID, P.ProductID, P.Name FROM AdventureWorks.Production.Product P WITH (TABLOCK); GO INSERT #OrdHeader ( SalesOrderID, OrderDate, SalesOrderNumber, CustomerID ) SELECT H.SalesOrderID, H.OrderDate, H.SalesOrderNumber, H.CustomerID FROM AdventureWorks.Sales.SalesOrderHeader H WITH (TABLOCK); GO INSERT #OrdDetail ( SalesOrderID, OrderQty, LineTotal, ProductMainID, ProductSubID, ProductSubSubID ) SELECT D.SalesOrderID, D.OrderQty, D.LineTotal, D.ProductID, D.ProductID, D.ProductID FROM AdventureWorks.Sales.SalesOrderDetail D WITH (TABLOCK); The query itself is a simple join of the four tables: SELECT P.ProductMainID AS PID, P.Name, D.OrderQty, H.SalesOrderNumber, H.OrderDate, C.TerritoryID FROM #Prods P JOIN #OrdDetail D ON P.ProductMainID = D.ProductMainID AND P.ProductSubID = D.ProductSubID AND P.ProductSubSubID = D.ProductSubSubID JOIN #OrdHeader H ON D.SalesOrderID = H.SalesOrderID JOIN #Custs C ON H.CustomerID = C.CustomerID ORDER BY P.ProductMainID ASC OPTION (RECOMPILE, MAXDOP 1); Remember that these tables have no indexes at all, and only the single-column sampled statistics SQL Server automatically creates (assuming default settings).  The estimated query plan produced for the test query looks like this (click to enlarge): The Problem The problem here is one of cardinality estimation – the number of rows SQL Server expects to find at each step of the plan.  The lack of indexes and useful statistical information means that SQL Server does not have the information it needs to make a good estimate.  Every join in the plan shown above estimates that it will produce just a single row as output.  Brad covers the factors that lead to the low estimates in his post. In reality, the join between the #Prods and #OrdDetail tables will produce 121,317 rows.  It should not surprise you that this has rather dire consequences for the remainder of the query plan.  In particular, it makes a nonsense of the optimizer’s decision to use Nested Loops to join to the two remaining tables.  Instead of scanning the #OrdHeader and #Custs tables once (as it expected), it has to perform 121,317 full scans of each.  The query takes somewhere in the region of twenty minutes to run to completion on my development machine. A Solution At this point, you may be thinking the same thing I was: if we really are stuck with no indexes, the best we can do is to use hash joins everywhere. We can force the exclusive use of hash joins in several ways, the two most common being join and query hints.  A join hint means writing the query using the INNER HASH JOIN syntax; using a query hint involves adding OPTION (HASH JOIN) at the bottom of the query.  The difference is that using join hints also forces the order of the join, whereas the query hint gives the optimizer freedom to reorder the joins at its discretion. Adding the OPTION (HASH JOIN) hint results in this estimated plan: That produces the correct output in around seven seconds, which is quite an improvement!  As a purely practical matter, and given the rigid rules of the environment we find ourselves in, we might leave things there.  (We can improve the hashing solution a bit – I’ll come back to that later on). Faster Nested Loops It might surprise you to hear that we can beat the performance of the hash join solution shown above using nested loops joins exclusively, and without breaking the rules we have been set. The key to this part is to realize that a condition like (A = B) can be expressed as (A <= B) AND (A >= B).  Armed with this tremendous new insight, we can rewrite the join predicates like so: SELECT P.ProductMainID AS PID, P.Name, D.OrderQty, H.SalesOrderNumber, H.OrderDate, C.TerritoryID FROM #OrdDetail D JOIN #OrdHeader H ON D.SalesOrderID >= H.SalesOrderID AND D.SalesOrderID <= H.SalesOrderID JOIN #Custs C ON H.CustomerID >= C.CustomerID AND H.CustomerID <= C.CustomerID JOIN #Prods P ON P.ProductMainID >= D.ProductMainID AND P.ProductMainID <= D.ProductMainID AND P.ProductSubID = D.ProductSubID AND P.ProductSubSubID = D.ProductSubSubID ORDER BY D.ProductMainID OPTION (RECOMPILE, LOOP JOIN, MAXDOP 1, FORCE ORDER); I’ve also added LOOP JOIN and FORCE ORDER query hints to ensure that only nested loops joins are used, and that the tables are joined in the order they appear.  The new estimated execution plan is: This new query runs in under 2 seconds. Why Is It Faster? The main reason for the improvement is the appearance of the eager Index Spools, which are also known as index-on-the-fly spools.  If you read my Inside The Optimiser series you might be interested to know that the rule responsible is called JoinToIndexOnTheFly. An eager index spool consumes all rows from the table it sits above, and builds a index suitable for the join to seek on.  Taking the index spool above the #Custs table as an example, it reads all the CustomerID and TerritoryID values with a single scan of the table, and builds an index keyed on CustomerID.  The term ‘eager’ means that the spool consumes all of its input rows when it starts up.  The index is built in a work table in tempdb, has no associated statistics, and only exists until the query finishes executing. The result is that each unindexed table is only scanned once, and just for the columns necessary to build the temporary index.  From that point on, every execution of the inner side of the join is answered by a seek on the temporary index – not the base table. A second optimization is that the sort on ProductMainID (required by the ORDER BY clause) is performed early, on just the rows coming from the #OrdDetail table.  The optimizer has a good estimate for the number of rows it needs to sort at that stage – it is just the cardinality of the table itself.  The accuracy of the estimate there is important because it helps determine the memory grant given to the sort operation.  Nested loops join preserves the order of rows on its outer input, so sorting early is safe.  (Hash joins do not preserve order in this way, of course). The extra lazy spool on the #Prods branch is a further optimization that avoids executing the seek on the temporary index if the value being joined (the ‘outer reference’) hasn’t changed from the last row received on the outer input.  It takes advantage of the fact that rows are still sorted on ProductMainID, so if duplicates exist, they will arrive at the join operator one after the other. The optimizer is quite conservative about introducing index spools into a plan, because creating and dropping a temporary index is a relatively expensive operation.  It’s presence in a plan is often an indication that a useful index is missing. I want to stress that I rewrote the query in this way primarily as an educational exercise – I can’t imagine having to do something so horrible to a production system. Improving the Hash Join I promised I would return to the solution that uses hash joins.  You might be puzzled that SQL Server can create three new indexes (and perform all those nested loops iterations) faster than it can perform three hash joins.  The answer, again, is down to the poor information available to the optimizer.  Let’s look at the hash join plan again: Two of the hash joins have single-row estimates on their build inputs.  SQL Server fixes the amount of memory available for the hash table based on this cardinality estimate, so at run time the hash join very quickly runs out of memory. This results in the join spilling hash buckets to disk, and any rows from the probe input that hash to the spilled buckets also get written to disk.  The join process then continues, and may again run out of memory.  This is a recursive process, which may eventually result in SQL Server resorting to a bailout join algorithm, which is guaranteed to complete eventually, but may be very slow.  The data sizes in the example tables are not large enough to force a hash bailout, but it does result in multiple levels of hash recursion.  You can see this for yourself by tracing the Hash Warning event using the Profiler tool. The final sort in the plan also suffers from a similar problem: it receives very little memory and has to perform multiple sort passes, saving intermediate runs to disk (the Sort Warnings Profiler event can be used to confirm this).  Notice also that because hash joins don’t preserve sort order, the sort cannot be pushed down the plan toward the #OrdDetail table, as in the nested loops plan. Ok, so now we understand the problems, what can we do to fix it?  We can address the hash spilling by forcing a different order for the joins: SELECT P.ProductMainID AS PID, P.Name, D.OrderQty, H.SalesOrderNumber, H.OrderDate, C.TerritoryID FROM #Prods P JOIN #Custs C JOIN #OrdHeader H ON H.CustomerID = C.CustomerID JOIN #OrdDetail D ON D.SalesOrderID = H.SalesOrderID ON P.ProductMainID = D.ProductMainID AND P.ProductSubID = D.ProductSubID AND P.ProductSubSubID = D.ProductSubSubID ORDER BY D.ProductMainID OPTION (MAXDOP 1, HASH JOIN, FORCE ORDER); With this plan, each of the inputs to the hash joins has a good estimate, and no hash recursion occurs.  The final sort still suffers from the one-row estimate problem, and we get a single-pass sort warning as it writes rows to disk.  Even so, the query runs to completion in three or four seconds.  That’s around half the time of the previous hashing solution, but still not as fast as the nested loops trickery. Final Thoughts SQL Server’s optimizer makes cost-based decisions, so it is vital to provide it with accurate information.  We can’t really blame the performance problems highlighted here on anything other than the decision to use completely unindexed tables, and not to allow the creation of additional statistics. I should probably stress that the nested loops solution shown above is not one I would normally contemplate in the real world.  It’s there primarily for its educational and entertainment value.  I might perhaps use it to demonstrate to the sceptical that SQL Server itself is crying out for an index. Be sure to read Brad’s original post for more details.  My grateful thanks to him for granting permission to reuse some of his material. Paul White Email: [email protected] Twitter: @PaulWhiteNZ

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  • Estimating cost of labor for a controlled experiment

    - by Lorin Hochstein
    Let's say you are a software engineering researcher and you are designing a controlled experiment to compare two software technologies or techniques (e.g., TDD vs. non-TDD, Python vs. Go) with respect to some qualities of interest (e.g., quality of resulting code, programmer productivity). According to your study design, participants will work alone to implement a non-trivial software system. You estimate it should take about six months for a single programmer to complete the task. You also estimate via power analysis that you will need around sixty study participants to obtain statistically significant results, assuming the technologies actually do yield different outcomes. To maximize external validity, you want to use professional programmers as study participants. Unfortunately, it isn't possible to find professional programmers who can volunteer for several months to work full-time on implementing a software system. You decide to go the simplest route and contract with a large IT consulting firm to obtain access to programmers to participate in the study. What is a reasonable estimate of the cost range, per person-month, for the programming labor? Assume you are constrained to work with a U.S.-based firm, but it doesn't matter where in the U.S. the firm itself or the programmers or located. Note: I'm looking for a reasonable order-of-magnitude range suitable for back-of-the-envelope calculations so that when people say "Why doesn't somebody just do a study to measure X", I can say, "Because running that study properly would cost $Y", and have a reasonable argument for the value of $Y.

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  • Should the number of developers be considered when estimating a task?

    - by Ludwig Magnusson
    I am pretty inexperienced with working in agile projects but I have tried it a few times and I always run into this problem when estimating a task. Do we bring into the estimate the number of developers that will work on the task? Let me explain: Task A is estimated to one time unit and developer 1 will work on it. Task B is also estimated to one time unit and developer 2 and 3 will work on it together. I.e. if developer 1 begins to work on task A at the same time developer 2 and 3 begins to work on task B they will all finish at the same time according to the estimate. Should the estimate for task B be twice of that for task A or the same? The problem as I see it is that when a task is received and estimated, it is not always possible to know how many people will work on it. And if you assumed that two developers would work on the task for one time unit but it turns out that only one developer will actually do it, this will not automatically mean that that developer will work on it for two time units. Is there any standard practice for this?

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  • Emacs org-mode: how to avoid duplicate lines in agenda, when items is scheduled AND has deadline

    - by Martin
    Many of my TODO items in Emacs org-mode have a DEADLINE defined in the future (e. g. Friday) and are at the same time SCHEDULED today so that I already know I have to start working on this task. Then, this task will appear twice in my agenda. That's not nice but not necessarily a problem yet, but if then the task has assigned a time estimate for its duration and I go to column view with C-C C-X C-C to see how much time my tasks today will need, the time estimate for this task is counted twice, so e. g. if the time effort estimate is 2 hours, I'll have 4 hours in my daily agenda, as the item appears as well as scheduled today (or in the past) as also with its deadline in 3 days. How can I avoid counting an item twice?

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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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  • Validating User Stories: How much change is too much?

    - by David Kaczynski
    While the core of requirements development and acceptance criteria would ideally take place during the planning meeting in order to create a better estimate, Scrum encourages continuous interaction with the product owner throughout the sprint to validate and refine user stories. What kind of criteria is used to judge if there is too much change being imposed on a user story mid-sprint? When is it appropriate to change the requirements of the user story? When is it appropriate to cancel the user story / sprint in order to re-evaluate and re-estimate a user story in question?

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  • Software cost estimation

    - by David Conde
    I've seen on my work place (a University) most students making the software estimation cost of their final diploma work using COCOMO. My guessing is that this way of estimating costs is somewhat old (COCOMO dates of 1981), hence my question: How do you estimate costs in your software? I've seen things like : Cost = ( HoursOfWork + EstimatedIddle ) * HourlyRate That's not what I want, I'm looking for a properly (scientifically) defined cost model EDIT I've found some related questions on SO: What are some of the software cost estimation methods and models? How do you estimate the cost of developing software requirements?

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  • Why are software schedules so hard to define?

    - by 0A0D
    It seems that, in my experience, getting us engineers to accurately estimate and determine tasks to be completed is like pulling teeth. Rather than just giving a swag estimate of 2-3 weeks or 3-6 months... what is the simplest way to define software schedules so they are not so painful to define? For instance, customer A wants a feature by 02/01/2011. How do you schedule time to implement this feature knowing that other bug fixes may be needed along the way and take up additional engineering time?

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  • Why are software schedules so hard to define?

    - by 0A0D
    It seems that, in my experience, getting us engineers to accurately estimate and determine tasks to be completed is like pulling teeth. Rather than just giving a swag estimate of 2-3 weeks or 3-6 months... what is the simplest way to define software schedules so they are not so painful to define? For instance, customer A wants a feature by 02/01/2011. How do you schedule time to implement this feature knowing that other bug fixes may be needed along the way and take up additional engineering time?

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  • Delivering estimates and client expectations?

    - by FishOrDie
    When a client asks for an estimate on how long it would take to develop different sections of an app, is it best to give them a total amount or what it would take for each section? Is it better/more common to give a range of hours/days or just a single number? Do you think most clients feel that if a programmer says it should take 50 hours that they should be billed for 50 hours? If I say it would take 50 and it actually takes 60, do I tell them in advance that I'm going over on my estimate or just charge what was originally quoted?

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  • Estimating costs in a GOAP system

    - by fullwall
    I'm currently developing a GOAP system in Java. An explanation of GOAP can be found at http://web.media.mit.edu/~jorkin/goap.html. Essentially, it's using A* to plot between Actions that mutate the world state. To provide a fair chance for all Actions and Goals to execute, I'm using a heuristic function to estimate the cost of doing something. What is the best way to estimate this cost so that it is comparable to all the other costs? As an example, estimating the cost of running away from an enemy versus attacking it - how should the cost be calculated to be comparable?

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  • Is 500 million lines of code even remotely possible? [on hold]

    - by kmote
    The New York Times is reporting that the Healthcare.gov website contains "about 500 million lines of software code." This number, attributed to "one specialist", and widely repeated across the interwebs, seems incredibly far-fetched (even assuming a large fraction of that number includes standard libraries). If this is an accurate estimate, it would truly be staggering (as this fascinating infographic vividly reveals). I realize StackExchange:Programmers isn't Snopes.com, but I'd like to find out if anyone here believes this is even remotely possible. I'd like to know if there is a plausible system of accounting (using examples from publicly available data, if possible) that could lead someone to conclude that such an estimate is within the realm of reason. How could a codebase (by any measure) sum up to such an exhorbitant number of code lines?

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  • Estimating file transfer time over network?

    - by rocko
    I am transferring file from one server to another. So, to estimate the time it would take to transfer some GB's of file over the network, I am pinging to that IP and taking the average time. For ex: i ping to 172.26.26.36 I get the average round trip time to be x ms, since ping send 32 bytes of data each time. I estimate speed of network to be 2*32*8(bits)/x = y Mbps -- multiplication with 2 because its average round trip time. So transferring 5GB of data will take 5000/y seconds Am I correct in my method of estimating time. If you find any mistake or any other good method please share.

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  • probability and relative frequency

    - by Alexandru
    If I use relative frequency to estimate the probability of an event, how good is my estimate based on the number of experiments? Is standard deviation a good measure? A paper/link/online book would be perfect. http://en.wikipedia.org/wiki/Frequentist

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  • Best techniques for estimation

    - by viswanathan
    What are the possible techniques to arrive at a good estimate? We use Delphi estimation technique for estimating tasks. What are the other better ways to do so? Also what would be the do's and dont's while giving an estimate.

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  • Why does this JavaScript not correctly update input values?

    - by dmanexe
    I have two input fields, and without changing their names (i.e., I have to use the name with the brackets), how do I make this javascript code work? <script> function calc_concreteprice(mainform) { var oprice; var ototal; oprice = (eval(mainform.'estimate[concrete][sqft]'.value) * eval(mainform.'estimate[concrete][price]'.value)); ototal = (oprice); mainform.'estimate[concrete][quick_total]'.value = ototal; } </script> Here's the HTML of the input area. <tr> <td>Concrete Base Price</td> <td><input type="text" name="concrete[concrete][price]" value="" class="item_mult" onBlur="calc_concreteprice(document.forms.mainform);" /> per SF <strong>times</strong> <input type="text" name="concrete[concrete][sqft]" value="" class="item_mult" onBlur="calc_concreteprice(document.forms.mainform);" /> SF</td> <td> = <input type="text" name="concrete[concrete][quick_total]" value="" /></td> </tr> I know I can get it working by changing_the_input_name_with_underscores but I need to have the names with the brackets (storing the form contents in an array).

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  • Creating new table entry when updating another entry of another table - Ruby on Rails

    - by Michaël
    Hi, I have written this code in my "show" view of estimates and I want that, when submitting (update), it creates a new repairs (Repair.new(???)) with some parameters. I don't know where to write the "new repairs" code, in this view or the controller (in update part). I need that the Repair is created one time, not each time the @estimate is updated. <% form_for @estimate, :url => {:controller => "estimates", :action => "update"} do |f| %> <%= f.error_messages %> <select id="estimate_accept" name="estimate[accept]"> <option value="1" selected="selected">accept</option> <option value="2">refuse</option> </select> <%= f.submit "Update" %> <% end %> Thank you for your help, I hope my explanations are clear!

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  • How to find "y" values of the already estimated monotone function of the non-monotone regression curve corresponding to the original "x" points?

    - by parenthesis
    The title sounds complicated but that is what I am looking for. Focus on the picture. ## data x <- c(1.009648,1.017896,1.021773,1.043659,1.060277,1.074578,1.075495,1.097086,1.106268,1.110550,1.117795,1.143573,1.166305,1.177850,1.188795,1.198032,1.200526,1.223329,1.235814,1.239068,1.243189,1.260003,1.262732,1.266907,1.269932,1.284472,1.307483,1.323714,1.326705,1.328625,1.372419,1.398703,1.404474,1.414360,1.415909,1.418254,1.430865,1.431476,1.437642,1.438682,1.447056,1.456152,1.457934,1.457993,1.465968,1.478041,1.478076,1.485995,1.486357,1.490379,1.490719) y <- c(0.5102649,0.0000000,0.6360097,0.0000000,0.8692671,0.0000000,1.0000000,0.0000000,0.4183691,0.8953987,0.3442624,0.0000000,0.7513169,0.0000000,0.0000000,0.0000000,0.0000000,0.1291901,0.4936121,0.7565551,1.0085108,0.0000000,0.0000000,0.1655482,0.0000000,0.1473168,0.0000000,0.0000000,0.0000000,0.1875293,0.4918018,0.0000000,0.0000000,0.8101771,0.6853480,0.0000000,0.0000000,0.0000000,0.0000000,0.4068802,1.1061434,0.0000000,0.0000000,0.0000000,0.0000000,0.0000000,0.0000000,0.0000000,0.0000000,0.0000000,0.6391678) fit1 <- c(0.5102649100,0.5153380934,0.5177234836,0.5255544980,0.5307668662,0.5068087080,0.5071001179,0.4825657520,0.4832969250,0.4836378194,0.4842147729,0.5004039310,0.4987301366,0.4978800742,0.4978042478,0.4969807064,0.5086987191,0.4989497612,0.4936121200,0.4922210302,0.4904593166,0.4775197108,0.4757040857,0.4729265271,0.4709141776,0.4612406896,0.4459316517,0.4351338346,0.4331439717,0.4318664278,0.3235179189,0.2907908968,0.1665721429,0.1474035158,0.1443999345,0.1398517097,0.1153991839,0.1142140393,0.1022584672,0.1002410843,0.0840033244,0.0663669309,0.0629119398,0.0627979240,0.0473336492,0.0239237481,0.0238556876,0.0084990298,0.0077970954,0.0000000000,-0.0006598571) fit2 <- c(-0.0006598571,0.0153328298,0.0228511733,0.0652889427,0.0975108758,0.1252414661,0.1270195143,0.1922510501,0.2965234797,0.3018551305,0.3108761043,0.3621749370,0.4184150225,0.4359301495,0.4432114081,0.4493565757,0.4510158144,0.4661865431,0.4744926045,0.4766574718,0.4796937554,0.4834718810,0.4836125426,0.4839450098,0.4841092849,0.4877317306,0.4930561638,0.4964939389,0.4970089201,0.4971376528,0.4990394601,0.5005881678,0.5023814257,0.5052125977,0.5056691690,0.5064254338,0.5115481820,0.5117259449,0.5146054557,0.5149729419,0.5184178197,0.5211542908,0.5216215426,0.5216426533,0.5239797875,0.5273573222,0.5273683002,0.5293994824,0.5295130266,0.5306236672,0.5307303109) ## picture plot(x, y) ## red regression curve points(x, fit1, col=2); lines(x, fit1, col=2) ## blue monotonic curve to the regression points(min(x) + cumsum(c(0, rev(diff(x)))), rev(fit2), col="blue"); lines(min(x) + cumsum(c(0, rev(diff(x)))), rev(fit2), col="blue") ## "x" original point matches with the regression estimated point ## but not with the estimated (fit2=estimate) monotonic curve abline(v=1.223329, lty=2, col="grey") Focus on the dashed grey line. The idea is to get y value of the monotonic blue curve corresponding to x original value. The grey line should cross three points (the original one "black", the regression estimate "red", the adjusted regression estimate "blue"). Can we do this? Methodology: The object "fit2" is the output of the function rearrangement(). It is always monotonically increasing. library(Rearrangement) fit2 <- rearrangement(x=as.data.frame(x), y=fit1)

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  • Python or Ruby for freelance?

    - by Sophia
    Hello, I'm Sophia. I have an interest in self-learning either Python, or Ruby. The primary reason for my interest is to make my life more stable by having freelance work = $. It seems that programming offers a way for me to escape my condition of poverty (I'm on the edge of homelessness right now) while at the same time making it possible for me to go to uni. I intend on being a math/philosophy major. I have messed with Python a little bit in the past, but it didn't click super well. The people who say I should choose Python say as much because it is considered a good first language/teaching language, and that it is general-purpose. The people who say I should choose Ruby point out that I'm a very right-brained thinker, and having multiple ways to do something will make it much easier for me to write good code. So, basically, I'm starting this thread as a dialog with people who know more than I do, as an attempt to make the decision. :-) I've thought about asking this in stackoverflow, but they're much more strict about closing threads than here, and I'm sort of worried my thread will be closed. :/ TL;DR Python or Ruby for freelance work opportunities ($) as a first language? Additional question (if anyone cares to answer): I have a personal feeling that if I devote myself to learning, I'd be worth hiring for a project in about 8 weeks of work. I base this on a conservative estimate of my intellectual capacities, as well as possessing motivation to improve my life. Is my estimate necessarily inaccurate? random tidbit: I'm in Portland, OR I'll answer questions that are asked of me, if I can help the accuracy and insight contained within the dialog.

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  • how much more memcache memory do i need to get 95% hit ratio? [on hold]

    - by OneSolitaryNoob
    I have a memcache instance running that has a 90% hit ratio. How can I estimate how much more memory it needs to get to a 95% hit ratio? edit: This question was blocked, but I do not think this is impossible to answer. After all, anyone that's used a caching system HAS answered this question, most likely with trial&error&luck. I can look at my usage patterns. I can increase or decrease memory and see how hit rate changes. Both of these provide data that informs an estimate. But what's a good/better/best way to do this?

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