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Matrix Computation

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Introduction

 

Maple has many tools for linear algebra. Its capabilities include

 

• 

symbolic and numeric computations, and hybrid matrices

• 

eigenvalues and eigenvectors, both classical and generalized.

• 

linear algebra over finite fields.

• 

matrix factorizations and system solvers

• 

numerical methods for dense and sparse systems with a high degree of user control

• 

hardware float and arbitrary precision software float data

• 

numeric routines from CLAPACK and optimized vendor BLAS (ATLAS and MKL) libraries, called automatically when appropriate.

• 

automatically parallelized numeric computation, when appropriate

 

Symbolic Matrix Computation

 

Here, we derive the Denavit-Hartenberg matrix for a robotic serial manipulator. These matrices were entered using the Matrix palette (other methods are described here) and a period is used for matrix multiplication.

 

B110000100001d__i0001.cosθ__isinθ__i00sinθ__icosθ__i0000100001:

B210000cosα__isinα__i00sinα__icosα__i00001.100a__i010000100001:

HB1.B2

cosθ__isinθ__icosα__isinθ__isinα__icosθ__ia__isinθ__icosθ__icosα__icosθ__isinα__isinθ__ia__i0sinα__icosα__id__i0001

(1)

Maple will handle arbitrarily large symbolic matrices.

 

Numeric Matrix Computation

 

Here we solve the linear system M.x = v for a sparse system. Numerical data is randomly generated

 

withLinearAlgebra:

MRandomMatrix1000,1000,density=0.001,datatype=float8;

_rtable18446744074781829830

(2)

vRandomVector1000,density=0.001,datatype=float8

_rtable18446744074808337222

(3)

for i from 1 to 1000 do Mi,ii:end do:

xLinearSolveM,v

_rtable18446744074781850310

(4)

To test the accuracy of the numeric solution, the following quantity must be zero or a very small number

NormM.xv

0.

(5)

 

Applications

Code Generation for a Robot Arm