Search results
Results from the WOW.Com Content Network
The Gram–Schmidt process can be stabilized by a small modification; this version is sometimes referred to as modified Gram-Schmidt or MGS. This approach gives the same result as the original formula in exact arithmetic and introduces smaller errors in finite-precision arithmetic.
More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×m unitary matrix Q and an m×n upper triangular matrix R.As the bottom (m−n) rows of an m×n upper triangular matrix consist entirely of zeroes, it is often useful to partition R, or both R and Q:
In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information theory , for example in entanglement characterization and in state purification , and plasticity .
Jørgen Pedersen Gram (27 June 1850 – 29 April 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark. Important papers of his include On series expansions determined by the methods of least squares , and Investigations of the number of primes less than a given number .
An early successful application of the LLL algorithm was its use by Andrew Odlyzko and Herman te Riele in disproving Mertens conjecture. [5]The LLL algorithm has found numerous other applications in MIMO detection algorithms [6] and cryptanalysis of public-key encryption schemes: knapsack cryptosystems, RSA with particular settings, NTRUEncrypt, and so forth.
In other words, the sequence is obtained from the sequence of monomials 1, x, x 2, … by the Gram–Schmidt process with respect to this inner product. Usually the sequence is required to be orthonormal , namely, P n , P n = 1 , {\displaystyle \langle P_{n},P_{n}\rangle =1,} however, other normalisations are sometimes used.
Main page; Contents; Current events; Random article; About Wikipedia; Contact us; Donate; Help; Learn to edit; Community portal; Recent changes; Upload file
$ tex Gram-Schmidt_process.tex && dvips -E Gram-Schmidt_process.dvi; Outline fonts $ eps2eps -dNOCACHE Gram-Schmidt_process.ps Gram-Schmidt_process2.eps; Fix bounding box $ ps2epsi Gram-Schmidt_process2.eps Gram-Schmidt_process.eps; Convert to Sketch $ pstoedit -f sk Gram-Schmidt_process.eps Gram-Schmidt_process.sk; Convert to SVG