Backslash checks whether the matrix is triangular (or a permutation of a triangular matrix), upper Hessenberg, symmetric, or symmetric positive definite, and applies an appropriate method. A notable example of such a meta-algorithm is the MATLAB backslash function x = A for solving.

Ideally, linear algebra software would detect structure in a matrix and call an algorithm that exploits that structure. Using this property one can solve in operations, rather than the operations required if the circulant structure is ignored. A second example is a circulant matrixĬirculant matrices have the important property that they are diagonalized by a unitary matrix called the discrete Fourier transform matrix. Much work has been done on developing numerical methods for solving that exploit the block structure and possible sparsity in and. Matrices from saddle point problems are symmetric indefinite and of the form Here are two examples of structures that can be exploited. Sparsity (a matrix having a large number of zeros) is particularly important to exploit, since algorithms intended for dense matrices may be impractical for sparse matrices because of extensive fill-in (zeros becoming nonzero). One of the fundamental tenets of numerical linear algebra is that one should try to exploit any matrix structure that might be present. This sin takes the top spot in Schmelzer and Hauser’s Seven Sins in Portfolio Optimization, because in portfolio optimization a negative eigenvalue in the covariance matrix can identify a portfolio with negative variance, promising an arbitrarily large investment with no risk! In the case of failure, the partially computed factor is returned in the first argument, and it can be used to compute a direction of negative curvature (as needed in optimization), for example. The MATLAB function chol returns an error message if the factorization fails, and a second output argument can be requested, which is set to the number of the stage on which the factorization failed, or to zero if the factorization succeeded. The best way to check definiteness is to compute a Cholesky factorization, which is often needed anyway. Missing or inconsistent data in forming a covariance matrix or a correlation matrix can cause a loss of definiteness, and rounding errors can cause a tiny positive eigenvalue to go negative.īut none of these conditions, or even all taken together, guarantees that the matrix has positive eigenvalues. However, a matrix that is supposed to be positive definite may fail to be so for a variety of reasons. Symmetric positive definite matrices (symmetric matrices with positive eigenvalues) are ubiquitous, not least because they arise in the solution of many minimization problems. Which is singular, and the information in has been lost.Īnother problem with the cross product matrix is that the -norm condition number of is the square of that of, and this leads to numerical instability in algorithms that work with when the condition number is large. Is positive definite but, since, in floating-point arithmetic rounds to and so Where is the unit roundoff of the floating point arithmetic, then

What is wrong with the cross-product matrix (also known as the Gram matrix)? It squares the data, which can cause a loss of information in floating-point arithmetic. By contrast, solving the least squares problem via QR factorization is always numerically stable. While fast, this method is numerically unstable when is ill conditioned. It is therefore natural to form the symmetric positive definite matrix and solve the normal equations by Cholesky factorization. The matrix analysis functions det, rcond, hess, and expm also show significant increase in speed on large double-precision arrays.The solution to the linear least squares problem, where is a full-rank matrix with, satisfies the normal equations. The matrix multiply (X*Y) and matrix power (X^p) operators show significant increase in speed on large double-precision arrays (on order of 10,000 elements). As a general rule, complicated functions speed up more than simple functions. The operation is not memory-bound processing time is not dominated by memory access time. For example, most functions speed up only when the array contains several thousand elements or more. The data size is large enough so that any advantages of concurrent execution outweigh the time required to partition the data and manage separate execution threads. They should require few sequential operations. These sections must be able to execute with little communication between processes.

The function performs operations that easily partition into sections that execute concurrently.

Matlab matrix to vector