Numerical Matrix Analysis: Linear Systems and Least Squares

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In particular, this includes the half-precision arithmetic that is becoming prevalent on GPUs and arbitrary precision arithmetic. His research is focused on developing effective strategies for scheduling large computational tasks on modern high-performance computing HPC systems, with a particular emphasis on the potential for application of reinforcement learning to the problem.

His work mainly focuses on nonlinear eigenvalue problems NLEVP , with a more specific interest on holomorphic functions and contour integrals. His main objective is developing a general nonlinear solver for medium-sized dense matrices.

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His work focuses on new rational Krylov techniques and machine learning techniques with applications in time series modelling. Prior to joining the group, he completed my MMath degree at the University of Manchester. Her research is focused on generalised eigenvalue problems arising from structural dynamics. Ramaseshan Kannan is a senior engineer at Arup, where he develops simulation software for structural analysis.

Find the least squares solution to the matrix equation or Pseudo-inverse

His work covers algorithm development, mathematical modelling, linear algebra solvers, software performance optimisation, data structures and parallel programming, and technology transfer. Research interests include matrix algorithms, structural dynamics, sparse linear algebra, and the applications of machine learning in engineering simulation. Ramaseshan is an alumnus of the NLA group and manages ongoing collaborations and sponsored research initiatives with the School of Maths.

He works on numerical and multithreaded software projects including writing linear algebra and nearest correlation matrix software for the NAG Libraries. Her research revolves around numerical optimization and machine learning. Bunch and L. Kaufman, Some stable methods for calculating inertia and solving symmetric linear systems , Mathematics of Computation, 31 , pp. Chu, J. George, J. Liu, and E.

ENGS 106: Numerical Linear Algebra

Coleman, A. Edenbrandt, and J. Gilbert, Predicting fill for sparse orthogonal factorization , J. ACM, 33 , pp. Dennis and R. Duff, Pivot selection and row orderings in Givens reduction on sparse matrices , Computing, 13 , pp. Duff, N. Gould, J. Reid, J. Scott, and K. Duff, R. Grimes, and J.


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Duff and J. Reid, A comparison of some methods for the solution of sparse overdetermined systems of linear equations , J. Software, 9 , pp. Eisenstat, M. Schultz, and A. Dooren, eds. Davis, Trans. Gentleman, Least squares computations by Givens transformations without square roots , J. Gentleman, Row elimination for solving sparse linear systems and least squares problems , in Proceedings the 6th Dundee Conference on Numerical Analysis, G.

Numerical Linear Algebra and Matrix Analysis | Nick Higham

Watson, ed. George and M. Heath, Solution of sparse linear least squares problems using Givens rotations , Linear Algebra Appl. George and J. It is a staple of statistics and is often considered a good introductory machine learning method.

It is also a method that can be reformulated using matrix notation and solved using matrix operations. In this tutorial, you will discover the matrix formulation of linear regression and how to solve it using direct and matrix factorization methods. Discover vectors, matrices, tensors, matrix types, matrix factorization, PCA, SVD and much more in my new book , with 19 step-by-step tutorials and full source code.

Linear regression is a method for modeling the relationship between two scalar values: the input variable x and the output variable y. The model can also be used to model an output variable given multiple input variables called multivariate linear regression below, brackets were added for readability. The objective of creating a linear regression model is to find the values for the coefficient values b that minimize the error in the prediction of the output variable y.

Where X is the input data and each column is a data feature, b is a vector of coefficients and y is a vector of output variables for each row in X. Reformulated, the problem becomes a system of linear equations where the b vector values are unknown. This type of system is referred to as overdetermined because there are more equations than there are unknowns, i.


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It is a challenging problem to solve analytically because there are multiple inconsistent solutions, e. Further, all solutions will have some error because there is no line that will pass nearly through all points, therefore the approach to solving the equations must be able to handle that. The way this is typically achieved is by finding a solution where the values for b in the model minimize the squared error.

This is called linear least squares. This formulation has a unique solution as long as the input columns are independent e.

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When the length of e is as small as possible, xhat is a least squares solution. This can be solved directly, although given the presence of the matrix inverse can be numerically challenging or unstable. We will use a simple 2D dataset where the data is easy to visualize as a scatter plot and models are easy to visualize as a line that attempts to fit the data points.

A scatter plot of the dataset is then created showing that a straight line cannot fit this data exactly.


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That is, given X, what are the set of coefficients b that when multiplied by X will give y. As we saw in a previous section, the normal equations define how to calculate b directly. This can be calculated directly in NumPy using the inv function for calculating the matrix inverse. Putting this together with the dataset defined in the previous section, the complete example is listed below. A scatter plot of the dataset is then created with a line plot for the model, showing a reasonable fit to the data.

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LeastSquares

A problem with this approach is the matrix inverse that is both computationally expensive and numerically unstable. An alternative approach is to use a matrix decomposition to avoid this operation. We will look at two examples in the following sections.

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