[□=□]
[lemma=□]
Routines for the associated matrix factorizations are also provided , as are related computations such as reordering Schur factorizations , condition number estimation and forward and backward error estimation . sparse large sparse systems of linear equations using LU factorization . This ca n't be happening , for we chose N originally to be the smallest positive number with two different prime factorizations . Second , even if it is possible to track a large number of features , the computational cost of the SVD involved in the factorization of the rank 3 measurement matrix would be very high . The factorization of the measurement matrix , which is rank 3 in a noiseless situation , is computed by using a Singular Value Decomposition ( SVD ) expansion technique . Other topics that might be included , according to student interest , include : various other games , probability , notions of number and area , numerical topological invariants , infinite numbers , prime numbers , the mathematical basis of public key cryptography , irrational numbers , factorization of integers , the mathematical basis of algorithms for correcting errors in the transmission of digital data over noisy channels , factorization of polynomials . Replacing both parenthesized expressions above by factorizations of them into primes and eliminating the middle of the equation yields something like the following equation . Future releases of Java and the Java Compatibility Kit will provide improved factorization of user interface support so that vendors of Java server platforms can better address this issue .