TY - GEN

T1 - Generating a d-dimensional linear subspace efficiently

AU - Yuster, Raphael

PY - 2010

Y1 - 2010

N2 - We present an algorithm for computing a d-dimensional subspace of the row space of a matrix. For an n x n matrix A with m nonzero entries and with rank(A) ≥ d the algorithm generates a d x n matrix with full row rank and which is a subspace of Rows(A). If rank(A) < d the algorithm generates a rank(A) x n row-equivalent matrix. The running time of the algorithm is O(min{n 2-2/w m1/w dw-2+1/w, n2 d w-2}) where w < 2.376 is the matrix multiplication exponent. An immediate corollary of the algorithm is the construction of a row-reduced equivalent matrix of A, and hence the computation of rank(A), in time O(min{n2-2/w m1/w rank(A)w-2+1/w, n2 rank(A)w-2}). We note that the running time is sub-quadratic if d < (n2/m)0.528.

AB - We present an algorithm for computing a d-dimensional subspace of the row space of a matrix. For an n x n matrix A with m nonzero entries and with rank(A) ≥ d the algorithm generates a d x n matrix with full row rank and which is a subspace of Rows(A). If rank(A) < d the algorithm generates a rank(A) x n row-equivalent matrix. The running time of the algorithm is O(min{n 2-2/w m1/w dw-2+1/w, n2 d w-2}) where w < 2.376 is the matrix multiplication exponent. An immediate corollary of the algorithm is the construction of a row-reduced equivalent matrix of A, and hence the computation of rank(A), in time O(min{n2-2/w m1/w rank(A)w-2+1/w, n2 rank(A)w-2}). We note that the running time is sub-quadratic if d < (n2/m)0.528.

UR - http://www.scopus.com/inward/record.url?scp=77951691318&partnerID=8YFLogxK

U2 - 10.1137/1.9781611973075.39

DO - 10.1137/1.9781611973075.39

M3 - Conference contribution

AN - SCOPUS:77951691318

SN - 9780898717013

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 467

EP - 470

BT - Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms

PB - Association for Computing Machinery (ACM)

T2 - 21st Annual ACM-SIAM Symposium on Discrete Algorithms

Y2 - 17 January 2010 through 19 January 2010

ER -