lapack.ijt

LABTITLE=: 'LAPACK'
LABDEPENDS=: 'math/lapack'

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Lab Section LAPACK
LAPACK (Linear Algebra Package) is a set of routines for solving systems of simultaneous linear equations, least-squares solutions of linear systems of equations, eigenvalue problems, and singular value problems.

The associated matrix factorizations, LU, Cholesky, QR, SVD, Schur, generalized Schur, are also provided, as are related computations such as reordering of the Schur factorizations and estimating condition numbers.
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The LAPACK routines are accessed through the dll call mechanism as documented in User Manual chapter "Dlls and Memory Management".
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All definitions for LAPACK are in locale jlapack.

The scripts and other files for LAPACK are in directory:
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jpath '~addons/math/lapack'

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The first step in using LAPACK is to load the lapack.ijs script to define the jlapack locale.
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load 'math/lapack'

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There are hundreds of LAPACK routines but you only load interfaces for the ones you need. The interface is defined in a script with the name of the routine.

geev is a LAPACK routine for computing eigenvalues of a real or complex matrix.

If you want to use geev you need to load the geev script to make the necessary definitions in the jlapack locale.
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load 'math/lapack/geev'

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m=: ?4 4$100
geev_jlapack_ m  NB. compute eigenvalues and eigenvectors

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>1{geev_jlapack_ m  NB. just the eigenvalues

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If you are going to be using geev a lot, you could define it in the base locale so that you do not have to use the full name.
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geev=: geev_jlapack_
>1{geev m

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The LAPACK routines are documented in the doc subdirectory with a file with the same name, but a "lap" suffix.

This documentation is directly from the LAPACK library and is for both the mathematician and the programmer. Unfortunately the programmer is assumed to be a Fortran programmer. This means you will have to put in some extra effort to understand the programming interface details.

docs_jlapack_ shows a window with a list of the lap files.
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PREPARE
3 : 0''
if. 0 e. $ 1!:0 <jpath '~addons/math/lapack/doc' do.
  smoutput 'Note: on this machine, the LAPACK documentation has not'
  smoutput 'been found in the doc subdirectory.'
end.
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PREPARE

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docs_jlapack_ ''  NB. display list of lap files

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load 'math/lapack/geev'
>1{geev_jlapack_ 1j3+m        NB. eigenvalues of complex array

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gesvd computes the singular value decomposition (SVD) and left and right singular vectors of a real or complex matrix,
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load 'math/lapack/gesvd'
gesvd_jlapack_ m           NB. SVD

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The LAPACK routines are quite fast. For example:
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PREPARE
3 : 0''
smoutput '   mat=: ?100 100$1000'
mat__=: ?100 100$1000
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PREPARE

time=: 6!:2

NB. time in seconds to compute SVD:
time 'gesvd_jlapack_ mat'

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load 'math/lapack/potrf'
mp=: +/ . *
potrf_jlapack_ m mp |: m   NB. Cholesky decomposition

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load 'math/lapack/geqrf'
geqrf_jlapack_ m           NB. QR factorization

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load 'math/lapack/gesv'
m1=: ?.4 3$30
gesv_jlapack_ m;m1         NB. solves m * x = m1

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load 'math/lapack/getrf'
getrf_jlapack_ m           NB. LU decomposition

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routines_jlapack_ lists all the scripts in the lapack directory.
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routines_jlapack_ ''

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As you can see, interfaces are defined for only a few LAPACK routines.

Compare the list of available routines in docs_jlapack_ with the list of implemented interfaces in routines_jlapack_ .

With a little effort you should be able to add an interface for the LAPACK routine you require. You need to study the lap documentation for interfaces already provided, and by following the pattern in their interfaces, write a script for the interface you want.

If you write LAPACK interfaces, please send them to us so they can be included in the package.
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The calculation of eigenvalues using LAPACK is an integral part of the math/eigenpic demo. This shows the action of a matrix on unit vectors, and highlight the eigenvalues, which appear as a scaling factor times the eigenvectors.

Install the math/eigenpic package, and load as:

load 'math/eigenpic'
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For more information on LAPACK: http://www.netlib.org
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