Some functions return more than one piece of information, or give you a choice of formats in which to obtain the results.  By using the output= option, you can specify what information you would like a routine to return.  

For example, when using LUDecomposition, you can specify which components of the decomposition you want returned.  The examples below demonstrate the LUDecomposition command, but this functionality is also available in others, such as BidiagonalForm, QRDecomposition, and HessenbergForm.

For this example, we also demonstrate the use of software floats.

Set up a sample Matrix:

The default output is to return 'P' (permutation), 'L' (lower triangular), and 'U' (upper triangular) components:

The upper triangular component can also be automatically factored into a square upper triangular Matrix ('U1') and the row-reduced echelon form of M:

Usually you will want the whole decomposition, but if you only want part of it, you can ask for just the part you want:

You can also request 'NAG' format.  This special format is particularly space efficient:  the permutation Matrix is compressed into a pivot vector, and the L and U Matrices are combined into a single Matrix.  This format is understood by other commands, such as LinearSolve (note that the order of the vector and matrix must be preserved when passing to other routines).

Functions that return Matrices allow you to specify Matrix characteristics via the outputoptions option.  This may involve making a copy of the Matrix that was originally produced by the function.  (To determine if such a copy has been made, set infolevel[ LinearAlgebra ] := 2.)

The following examples create a random Matrix with a particular shape and an updatable scalar Matrix.

There are two main ways to minimize storage costs:

- use the 'inplace' feature

- create your Matrices with the right properties so that copies do not have to be made

Note that the first of these requires you to use the named version of the function (for example, Multiply instead of the '.' operator).  The second applies to both named functions and to functions called via operators.

Many LinearAlgebra commands can be carried out in more than one way.  Maple looks at the type of data it has received to determine the "best" algorithm to use.

For example, LUDecomposition has a number of techniques at its disposal, including Gaussian elimination, fraction-free Gaussian elimination, and Cholesky.  If you know which algorithm is most appropriate for your Matrix, you can save the LUDecomposition function the task of having to decide which one to use.  Simply specify the desired method with the method= option.

Note:  When you specify a particular method, no other method will be tried, even if problems are encountered.  For example, consider the following Matrix which is symmetric.  It is not positive-definite, but it has mistakenly been given that attribute.  When you call LinearSolve without specifying a method, it first tries the Cholesky method.  When that fails, it tries LU and gets the right answer.  When the Cholesky method is explicitly requested, it simply returns with an error.

Some variants of the LUDecomposition function can be carried out inplace (reducing storage costs) when the input Matrix has a float datatype.  

When you request NAG format output with the inplace option, the LU component is returned in the input Matrix.  The pivot vector is returned separately.

When you request the Cholesky method with the inplace option, the L component is returned in the original Matrix.  For efficiency, the upper triangle is not necessarily replaced with zeroes.

The LinearSolve command allows you to solve a linear system with one or more right-hand sides.  Each time you execute the LinearSolve command, it has to factor the input Matrix.  You can save time by providing all the right-hand sides in a single command so that it only has to factor the Matrix once:

If you do not have all the right-hand sides in advance, use LUDecomposition to factor the input Matrix, and pass this factorization (in a list) to LinearSolve.  You can use any decomposition, but for maximum speed with floating-point data, use 'NAG' format.  To minimize storage costs, use the inplace option (if available), as described in the previous section.

Note:  Using 'NAG' format with non-numeric entries saves storage, but it has an execution time comparable to other formats.