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Efficient Numerical Linear Algebra \302\251 Maplesoft, a division of Waterloo Maple Inc., 2005

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Computations on large Matrices and Vectors that contain floating-point data -- both hardware float data and arbitrary precision software float data -- can be performed very efficiently in Maple by taking advantage of a built-in library of numeric linear algebra routines. A number of these routines are provided through the alliance between Maplesoft and the Numerical Algorithms Group (NAG). In addition, parts of the CLAPACK and optimized ATLAS libraries have been integrated. In order to get the most benefit from these routines, it is important that you provide Maple with as much information about the structure of your problem as possible. Maple will almost certainly solve your problem without this additional information, and in some cases that is all you need. This Tips and Techniques explains the steps you can take when it is vital that the computations are done as efficiently as possible.

Numeric routines only work with floats (including complex floats), and integer matrices. (NAG routines all use floats). If you know you are using floats or integers, tell Maple so by using the Datatype= option in the function call. Otherwise, Maple spends more time inspecting the entries to make sure before proceeding with the computation. QyQ+SSJNRzYiLUknTWF0cml4R0YlNiU7IiIiIiImRikvSSlkYXRhdHlwZUdGJSZJJmZsb2F0RyUqcHJvdGVjdGVkRzYjIiIpRio=

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKj9Qd10i

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Provide shape and storage to give the most information you have about your matrix. The more you tell Maple, the more specialized routine Maple will pick. The specialized routines give you faster responses, and in some cases, reduce or eliminate the round-off error. QyQ+SSJNRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiY7IiIiIiImRiwvSSZzaGFwZUdGJUkqc3ltbWV0cmljR0YlL0koc3RvcmFnZUdGJUknc3BhcnNlR0YlISIi Set the flag UseHardwareFloats=true (if the problem allows). Hardware float computations are faster than computations with software floats. The LinearAlgebra package looks at the value of this flag, rather than the value of Digits, to determine if hardware floats can be used. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnUseHardwareFloats := true; The LinearAlgebra package provides commands for interactive use of the package, and a programming layer interface for efficient programming. The interactive layer does more type-checking, and sets defaults for optional parameters. These things are important for easy interactive use, but reduce the efficiency of a program. By using the programming layer commands, the efficiency of the code is improved by stripping away much of the error checking and insisting that all parameters are specified in a particular order. QyQ+SSJBRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMvSSQlaWRHRiUiKilvO0E6ISIi Here is the interactive version of the Transpose command: QyQtX0kuTGluZWFyQWxnZWJyYUc2IkkqVHJhbnNwb3NlR0YmNiNJIkFHRiYiIiI=

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKiVbMDI6

And here is the programmer version, contained inside the LA_Main sub-package of the LinearAlgebra package. QyQtX19JLkxpbmVhckFsZ2VicmFHNiRJKF9zeXNsaWJHNiIlKnByb3RlY3RlZEdJKExBX01haW5HRilJKlRyYW5zcG9zZUdGKTYlSSJBR0YpL0koaW5wbGFjZUdGKUkldHJ1ZUdGKi9JKG91dG9wdHNHRik3IiIiIg==

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKilvO0E6

To invoke the efficient numerical NAG routines in this example, you need to include the option outputoptions=[datatype=float[8]] in the construction of the Matrix. Note the difference in running times between the next two calls of the LinearSolve command. QyQtSSV3aXRoRzYiNiNJLkxpbmVhckFsZ2VicmFHRiUhIiI= QyQ+SSJuRzYiIiQrJCEiIg== QyQ+SSJBRzYiLUktUmFuZG9tTWF0cml4R0YlNiVJIm5HRiVGKS9JKmdlbmVyYXRvckdGJTsiIiEkIiM1ISIiRjA= QyQ+SSJiRzYiLUktUmFuZG9tVmVjdG9yR0YlNiRJIm5HRiUvSSpnZW5lcmF0b3JHRiU7IiIhJCIjNSEiIkYw QyQ+SSJ0RzYiLUkldGltZUclKnByb3RlY3RlZEdGJSEiIg== PkkieEc2Ii1JLExpbmVhclNvbHZlR0YkNiRJIkFHRiRJImJHRiQ=

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqOy4jZjk=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn QyQsJi1JJXRpbWVHJSpwcm90ZWN0ZWRHNiIiIiJJInRHRichIiJGKA==

JCIkNCIhIiQ=

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LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqX0xNWSI=

QyQsJi1JJXRpbWVHJSpwcm90ZWN0ZWRHNiIiIiJJInRHRichIiJGKA==

JCIjSiEiJA==

Now try increasing the dimension of the problems from 300 to 1000 and note again the difference. QyQ+SSJuRzYiIiUrNSEiIg== QyQ+SSJBRzYiLUktUmFuZG9tTWF0cml4R0YlNiRJIm5HRiUvSSpnZW5lcmF0b3JHRiU7IiIhJCIjNSEiIkYw QyQ+SSJiRzYiLUktUmFuZG9tVmVjdG9yR0YlNiRJIm5HRiUvSSpnZW5lcmF0b3JHRiU7IiIhJCIjNSEiIkYw QyQ+SSJ0RzYiLUkldGltZUclKnByb3RlY3RlZEdGJSEiIg== PkkieEc2Ii1JLExpbmVhclNvbHZlR0YkNiRJIkFHRiRJImJHRiQ=

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QyQsJi1JJXRpbWVHJSpwcm90ZWN0ZWRHNiIiIiJJInRHRichIiJGKA==

JCIlXTwhIiQ=

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LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqX1Y5WSI=

QyQsJi1JJXRpbWVHJSpwcm90ZWN0ZWRHNiIiIiJJInRHRichIiJGKA==

JCIkPighIiQ=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft. 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