# Most Efficient Way to Construct the Matrices to Extract the Lower and Upper Triangle from a Vectorized Matrix

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Royi Avital on 20 Apr 2020
Commented: Royi Avital on 24 Apr 2020
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.
Royi Avital on 23 Apr 2020
@Matt, I know that. Whenever I can use other features of the solver I user. I have cases I need those extractors in Matrix Form. I appericiate the dialogue. I think other who will read it will gain something. I still hope someone will bring a different point of view to the pattern of vJ. Though I guess @James' solution as practically as good as it gets (Also appericate if there is something to make it even faster).

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) || !mxIsNumeric(prhs) || !mxIsNumeric(prhs) ||
mxGetNumberOfElements(prhs) != 1 || mxGetNumberOfElements(prhs) != 1 ||
mxGetNumberOfElements(prhs) != 1 ) {
mexErrMsgTxt("Need three numeric scalar inputs");
}
if( nlhs > 1 ) {
mexErrMsgTxt("Too many outputs");
}
numRows = mxGetScalar(prhs);
triangleFlag = mxGetScalar(prhs);
diagFlag = mxGetScalar(prhs);
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 = mxCreateSparse(numElements, numRows*numRows, numElements, mxREAL);
pr = (double *) mxGetData(plhs);
Ir = mxGetIr(plhs);
Jc = mxGetJc(plhs);
Jc = 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
Royi Avital on 23 Apr 2020
Edited: Royi Avital on 23 Apr 2020
By the way, I tried optimizing the code:
if( triangleFlag == 1 ) { // Lower Triangle
for( jj = 1; jj < numRows + 1; jj++ ) {
for( ii = 1; ii < jj + diagFlag; ii++ ) {
ll++;
Jc[ll] = Jc[ll - 1];
}
for( ii = jj + diagFlag; ii < numRows + 1; ii++ ) {
ll++;
Jc[ll] = Jc[ll - 1] + 1;
vV[kk] = 1.0;
Ir[kk] = kk;
kk++;
}
}
} else { // Upper Triangle
for( jj = 1; jj < numRows + 1; jj++ ) {
for( ii = 1; ii < jj + 1 - diagFlag; ii++ ) {
ll++;
Jc[ll] = Jc[ll - 1] + 1;
vV[kk] = 1.0;
Ir[kk] = kk;
kk++;
}
for( ii = jj + 1 - diagFlag; ii < numRows + 1; ii++ ) {
ll++;
Jc[ll] = Jc[ll - 1];
}
}
}
But for some reason even removing the branching inside the loop didn't improve results.
Really Nice! If nothing comes up I will mark this as an answer. Thank You!

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
Royi Avital on 24 Apr 2020
This is really nice. Thank you for the effort!

Royi Avital on 21 Apr 2020
My current solution:
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
I like the memory allocation is kept to a minimum.
I wonder if there is a more efficient way to generate vJ. It is trivial to remove the inner loop and just count the number of elements yet in MATLAB it will mean each iteration will allocate memory (As we don't have iterators).
##### 2 CommentsShow 1 older commentHide 1 older comment
Royi Avital on 21 Apr 2020
@Tommy, Really liked your analysis. Yes, when dealing with sparse matrices the whole point it making sure allocation is kept to minimum. I agree if one wants both, it is better to do the trick you mentioned.
Let's see if someone can think on a different pattern to populate vJ which is more efficient.

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