On 4/23/2010 11:32:18 AM, waynereid wrote:
>To all, thank you.
...
>I agree with a previous
>comment that if I have a long
>file that I need to make
>changes to, I open and go to
>the bottom so that the whole
>sheet calculates and then go
>back through.
>
>I was hoping to be able to
>provide a clear, concise,
>technical answer as to why
>this works the best for me,
>But looks like not so easy.
>
>Wayne
____________________________
You got the point.
CAS don't compile in the sense of like FORTRAN IV [1970] did, therefore compile is the wrong word. At best, Mathcad like Mathematica creates the "kernel", collecting the functions. The other point is that the functions aren't necessarily executed in IPN and Mathcad versions based on Maple did organise the suite of calculations, in some sort then : compiled. MuPad is another gear that may not do the same "pseudo-compile".
However, read more.
Compile is useful in situations where you have to evaluate a particular numerical or logical expression many times. By taking the time to call Compile, you can get a compiled function which can be executed more quickly than an ordinary Mathematica function.
For simple expressions such as x Sin[x], there is usually little difference between the execution speed for ordinary and compiled functions. However, as the size of the expressions involved increases, the advantage of compilation also increases. For large expressions, compilation can speed up execution by a factor as large as 20.
Compilation makes the biggest difference for expressions containing a large number of simple, say arithmetic, functions. For more complicated functions, such as BesselK or Eigenvalues, most of the computation time is spent executing internal Mathematica algorithms, on which compilation has no effect.
Even though you can use compilation to speed up numerical functions that you write, you should still try to use built�in Mathematica functions whenever possible. Built�in functions will usually run faster than any compiled Mathematica programs you can create. In addition, they typically use more extensive algorithms, with more complete control over numerical precision and so on.
You should realize that built�in Mathematica functions quite often themselves use Compile. Thus, for example, NIntegrate by default automatically uses Compile on the expression you tell it to integrate. Similarly, functions like Plot and Plot3D use Compile on the expressions you ask them to plot. Built�in functions that use Compile typically have the option Compiled. Setting Compiled -> False tells the functions not to use Compile.
Compile works by making assumptions about the types of objects that occur in evaluating the expression you give. The default assumption is that all variables in the expression are approximate real numbers.
Compile nevertheless also allows integers, complex numbers and logical variables (True or False), as well as arrays of numbers. You can specify the type of a particular variable by giving a pattern which matches only values that have that type. Thus, for example, you can use the pattern _Integer to specify the integer type. Similarly, you can use True | False to specify a logical variable that must be either True or False.
The types that Compile handles correspond essentially to the types that computers typically handle at a machine�code level. Thus, for example, Compile can handle approximate real numbers that have machine precision, but it cannot handle arbitrary�precision numbers. In addition, if you specify that a particular variable is an integer, Compile generates code only for the case when the integer is of �machine size�, typically between a.
When the expression you ask to compile involves only standard arithmetic and logical operations, Compile can deduce the types of objects generated at every step simply from the types of the input variables. However, if you call other functions, Compile will typically not know what type of value they return. If you do not specify otherwise, Compile assumes that any other function yields an approximate real number value. You can, however, also give an explicit list of patterns, specifying what type to assume for an expression that matches a particular pattern.
The idea of Compile is to create a function which is optimized for certain types of arguments. Compile is nevertheless set up so that the functions it creates work with whatever types of arguments they are given. When the optimization cannot be used, a standard Mathematica expression is evaluated to find the value of the function.
The compiled code generated by Compile must make assumptions not only about the types of arguments you will supply, but also about the types of all objects that arise during the execution of the code. Sometimes these types depend on the actual values of the arguments you specify. Thus, for example, Sqrt[x] yields a real number result for real x if x is not negative, but yields a complex number if x is negative.
Compile always makes a definite assumption about the type returned by a particular function. If this assumption turns out to be invalid in a particular case when the code generated by Compile is executed, then Mathematica simply abandons the compiled code in this case, and evaluates an ordinary Mathematica expression to get the result.
An important feature of Compile is that it can handle not only mathematical expressions, but also various simple Mathematica programs. Thus, for example, Compile can handle conditionals and control flow structures.
In all cases, Compile[vars, expr] holds its arguments unevaluated. This means that you can explicitly give a �program� as the expression to compile.
If you make a definition like f[x_] := x Sin[x], Mathematica will store the expression x Sin[x] in a form that can be evaluated for any x. Then when you give a particular value for x, Mathematica substitutes this value into x Sin[x], and evaluates the result. The internal code that Mathematica uses to perform this evaluation is set up to work equally well whether the value you give for x is a number, a list, an algebraic object, or any other kind of expression.
Having to take account of all these possibilities inevitably makes the evaluation process slower. However, if Mathematica could assume that x will be a machine number, then it could avoid many steps, and potentially evaluate an expression like x Sin[x] much more quickly.
Using Compile, you can construct compiled functions in Mathematica, which evaluate Mathematica expressions assuming that all the parameters which appear are numbers (or logical variables). Compile[aa, a, � a, expr] takes an expression expr and returns a �compiled function� which evaluates this expression when given arguments a, a, � .
In general, Compile creates a CompiledFunction object which contains a sequence of simple instructions for evaluating the compiled function. The instructions are chosen to be close to those found in the machine code of a typical computer, and can thus be executed quickly.
One of the common motivations for converting Mathematica expressions into C or Fortran is to try to make them faster to evaluate numerically. But the single most important reason that C and Fortran can potentially be more efficient than Mathematica is that in these languages one always specifies up front what type each variable one uses will be�integer, real number, array, and so on.
The Mathematica function Compile makes such assumptions within Mathematica, and generates highly efficient internal code. Usually this code runs not much if at all slower than custom C or Fortran.
It is important to realize that since Mathematica can only sample your function at a limited number of points, it can always miss features of the function. By increasing PlotPoints, you can make Mathematica sample your function at a larger number of points. Of course, the larger you set PlotPoints to be, the longer it will take Mathematica to plot any function, even a smooth one.
Since Plot needs to evaluate your function many times, it is important to make each evaluation as quick as possible. As a result, Mathematica usually compiles your function into a low�level pseudocode that can be executed very efficiently. One potential problem with this, however, is that the pseudocode allows only machine�precision numerical operations. If the function you are plotting requires higher�precision operations, you may have to switch off compilation in Plot. You can do this by setting the option Compiled -> False. Note that Mathematica can only compile �inline code�; it cannot for example compile functions that you have defined. As a result, you should, when possible, use Evaluate as described in Section�1.9.1 to evaluate any such definitions and get a form that the Mathematica compiler can handle.
jmG
