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Given a matrix X and its vector form I am after the most efficient way to build the matrices L and U which extracts the lower and upper triangle from X.

So in MATLAB code it would be something like that:

clear();

numRows = 3;

numCols = numRows;

mX = randn(numRows, numCols);

vX = mX(:);

% Lower Triangle are indices 2, 3, 6

mL = [ 0, 1, 0, 0, 0, 0, 0, 0, 0 ; ...

0, 0, 1, 0, 0, 0, 0, 0, 0 ; ...

0, 0, 0, 0, 0, 1, 0, 0, 0 ];

% Upper Triangle are indices 4, 7, 8

mU = [ 0, 0, 0, 1, 0, 0, 0, 0, 0 ; ...

0, 0, 0, 0, 0, 0, 1, 0, 0 ; ...

0, 0, 0, 0, 0, 0, 0, 1, 0 ];

assert(isequal(mL * vX, mX(logical(tril(mX, -1)))));

assert(isequal(mU * vX, mX(logical(triu(mX, 1)))));

I am after sparse represenation of mU and mL in the most efficient way.

My current implementation is given by:

function [ mLU ] = GenerateTriangleExtractorMatrix( numRows, triangleFlag, diagFlag )

EXTRACT_LOWER_TRIANGLE = 1;

EXTRACT_UPPER_TRIANGLE = 2;

INCLUDE_DIAGONAL = 1;

EXCLUDE_DIAGONAL = 2;

switch(diagFlag)

case(INCLUDE_DIAGONAL)

numElements = 0.5 * numRows * (numRows + 1);

diagIdx = 0;

case(EXCLUDE_DIAGONAL)

numElements = 0.5 * (numRows - 1) * numRows;

diagIdx = 1;

end

vJ = zeros(numElements, 1);

if(triangleFlag == EXTRACT_LOWER_TRIANGLE)

elmntIdx = 0;

for jj = 1:numRows

for ii = (jj + diagIdx):numRows

elmntIdx = elmntIdx + 1;

vJ(elmntIdx) = ((jj - 1) * numRows) + ii;

end

end

elseif(triangleFlag == EXTRACT_UPPER_TRIANGLE)

elmntIdx = numElements + 1;

for jj = numRows:-1:1

for ii = (jj - diagIdx):-1:1

elmntIdx = elmntIdx - 1;

vJ(elmntIdx) = ((jj - 1) * numRows) + ii;

end

end

end

mLU = sparse(1:numElements, vJ, 1, numElements, numRows * numRows, numElements);

end

Is there a more efficient way to generate vJ without extensive allocation of memory (In order to allow generating really large matrices)?

Thank You.

James Tursa
on 22 Apr 2020

Edited: James Tursa
on 22 Apr 2020

Here is a mex routine that generates the sparse double matrices mL and mU directly, so no wasted memory in creating them. Seems to run about 3x-5x faster than m-code for somewhat large sizes.

/* S = GenerateTriangleExtractorMatrixMex(numRows,triangleFlag,diagFlag)

*

* S = double sparse matrix

* numRows = integer > 0

* triangleFlag = 1 , extract lower triangle

* 2 , extract upper triangle

* diagFlag = 1 , include diagonal

* 2 , exclude diagonal

* where

*

* M = an numRows X numRows matrix of non-zero terms

* assert(isequal(S * M(:), mX(logical(tril(M, -1))))); % for lower

* assert(isequal(S * M(:), mX(logical(triu(M, 1))))); % for upper

*

* Programmer: James Tursa

* Date: 2020-April-22

*/

#include "mex.h"

void mexFunction( int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[])

{

mwSize numRows, triangleFlag, diagFlag, numElements;

mwIndex *Ir, *Jc;

mwIndex i, j, k, m;

double *pr;

if( nrhs != 3 || !mxIsNumeric(prhs[0]) || !mxIsNumeric(prhs[1]) || !mxIsNumeric(prhs[2]) ||

mxGetNumberOfElements(prhs[0]) != 1 || mxGetNumberOfElements(prhs[1]) != 1 ||

mxGetNumberOfElements(prhs[2]) != 1 ) {

mexErrMsgTxt("Need three numeric scalar inputs");

}

if( nlhs > 1 ) {

mexErrMsgTxt("Too many outputs");

}

numRows = mxGetScalar(prhs[0]);

triangleFlag = mxGetScalar(prhs[1]);

diagFlag = mxGetScalar(prhs[2]);

if( numRows < 1 ) {

mexErrMsgTxt("Invalid numRows, should be > 0");

}

if( triangleFlag != 1 && triangleFlag != 2 ) {

mexErrMsgTxt("Invalid triangleFlag, should be 1 or 2");

}

if( diagFlag != 1 && diagFlag != 2 ) {

mexErrMsgTxt("Invalid diagFlag, should be 1 or 2");

}

if( diagFlag == 1 ) {

numElements = numRows * (numRows + 1) / 2; /* include diagonal */

} else {

numElements = (numRows - 1) * numRows / 2; /* exclude diagonal */

}

plhs[0] = mxCreateSparse(numElements, numRows*numRows, numElements, mxREAL);

pr = (double *) mxGetData(plhs[0]);

Ir = mxGetIr(plhs[0]);

Jc = mxGetJc(plhs[0]);

Jc[0] = 0;

diagFlag--;

k = 0;

m = 1;

if( triangleFlag == 1 ) { /* Lower */

for( j=0; j<numRows; j++ ) {

for( i=0; i<numRows; i++ ) {

if( i >= j+diagFlag ) {

*pr++ = 1.0;

*Ir++ = k++;

Jc[m] = Jc[m-1] + 1;

} else {

Jc[m] = Jc[m-1];

}

m++;

}

}

} else { /* Upper */

for( j=0; j<numRows; j++ ) {

for( i=0; i<numRows; i++ ) {

if( i+diagFlag <= j ) {

*pr++ = 1.0;

*Ir++ = k++;

Jc[m] = Jc[m-1] + 1;

} else {

Jc[m] = Jc[m-1];

}

m++;

}

}

}

}

You mex the routine as follows (you need a supported C compiler installed):

mex GenerateTriangleExtractorMatrixMex.c

And some test code:

% GenerateTriangleExtractorMatrix_test.m

n = 300;

disp('m-code timing')

tic

GenerateTriangleExtractorMatrix(10000,1,1);

toc

disp('mex code timing')

tic

GenerateTriangleExtractorMatrixMex(10000,1,1);

toc

for k=1:n

numRows = ceil(rand*5000+100);

numCols = numRows;

triangleFlag = (rand<0.5) + 1;

diagFlag = (rand<0.5) + 1;

Mm = GenerateTriangleExtractorMatrix(numRows,triangleFlag,diagFlag);

Mx = GenerateTriangleExtractorMatrixMex(numRows,triangleFlag,diagFlag);

if( ~isequal(Mm,Mx) )

error('Not equal');

end

end

disp('Random tests passed')

With a sample run:

>> GenerateTriangleExtractorMatrix_test

m-code timing

Elapsed time is 9.964882 seconds.

mex code timing

Elapsed time is 1.901741 seconds.

Random tests passed

Matt J
on 23 Apr 2020

Edited: Matt J
on 23 Apr 2020

Another approach to consider is to use my MatrixObj class

to construct an object that has the same effect as the operations mL*X and mL.'*Y, but doesn't require you to actually build the matrix,

N=5000;

tic;

mL0=GenerateTriangleExtractorMatrix( N, 1, 2);

toc

%Elapsed time is 0.678702 seconds.

tic;

B=tril(true(N),-1);

Bd=double(B(:));

mL=MatrixObj;

mL.Params.B=B;

mL.Params.Bd=Bd;

mL.Ops.mtimes=@(obj,z) z(obj.Params.B);

mL.Trans.mtimes=@mtimesT;

toc;

%Elapsed time is 0.086228 seconds.

function out=mtimesT(obj,z)

out=obj.Params.Bd;

out(obj.Params.B)=z;

end

In addition to requiring less time to construct, you can verify that it gives the same results as multiplications with mL and mL.',

>> X=rand(N^2,1); isequal(mL0.'*(mL0*X),mL.'*(mL*X))

ans =

logical

1

but with considerably less memory consumption:

>> whos mL mL0

Name Size Kilobytes Class Attributes

mL 1x1 219739 MatrixObj

mL0 12497500x25000000 390586 double sparse

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