Doing Linear Algebra in Sage – Part 2 – Simple Matrix Calculations
... Setting the underlying field of numbers When you work with matrices (or many other things) in Sage you need to tell it what kinds of numbers you will be using. The basic choices are: QQ for rational numbers RR for real numbers CC for complex numbers. In this course we will always work with RR ...
... Setting the underlying field of numbers When you work with matrices (or many other things) in Sage you need to tell it what kinds of numbers you will be using. The basic choices are: QQ for rational numbers RR for real numbers CC for complex numbers. In this course we will always work with RR ...
Sketching as a Tool for Numerical Linear Algebra Lecture 1
... Solving least squares regression via the normal equations How to find the solution x to minx |Ax-b|2 ? Equivalent problem: minx |Ax-b |22 Write b = Ax’ + b’, where b’ orthogonal to columns of A Cost is |A(x-x’)|22 + |b’|22 by Pythagorean theorem Optimal solution x if and only if AT(Ax-b) = ...
... Solving least squares regression via the normal equations How to find the solution x to minx |Ax-b|2 ? Equivalent problem: minx |Ax-b |22 Write b = Ax’ + b’, where b’ orthogonal to columns of A Cost is |A(x-x’)|22 + |b’|22 by Pythagorean theorem Optimal solution x if and only if AT(Ax-b) = ...
Chapter 9 The Transitive Closure, All Pairs Shortest Paths
... R is computed in lg(n-1) + 1 matrix multiplications. Each multiplication requires O(n3) operations so R can be computed in O(n3 * lgn) 9.6.1 Kronrod's Algorithm It is used to multiply boolean matrices. C = A x B For example suppose: The A matrix row determines which rows of B are to be unioned to pr ...
... R is computed in lg(n-1) + 1 matrix multiplications. Each multiplication requires O(n3) operations so R can be computed in O(n3 * lgn) 9.6.1 Kronrod's Algorithm It is used to multiply boolean matrices. C = A x B For example suppose: The A matrix row determines which rows of B are to be unioned to pr ...
Sketching as a Tool for Numerical Linear Algebra
... Solving least squares regression via the normal equations How to find the solution x to minx |Ax-b|2 ? Equivalent problem: minx |Ax-b |22 Write b = Ax’ + b’, where b’ orthogonal to columns of A Cost is |A(x-x’)|22 + |b’|22 by Pythagorean theorem Optimal solution x if and only if AT(Ax-b) = ...
... Solving least squares regression via the normal equations How to find the solution x to minx |Ax-b|2 ? Equivalent problem: minx |Ax-b |22 Write b = Ax’ + b’, where b’ orthogonal to columns of A Cost is |A(x-x’)|22 + |b’|22 by Pythagorean theorem Optimal solution x if and only if AT(Ax-b) = ...
computer science 349b handout #36
... converge if A is defective, and even if the dominant eigenvalue λ1 has multiplicity > 1. In this latter case, the method will converge with the same order of convergence as in the nondefective case if λ1 has a full set of eigenvectors (but very slowly if not). The critical limitation of the Power Me ...
... converge if A is defective, and even if the dominant eigenvalue λ1 has multiplicity > 1. In this latter case, the method will converge with the same order of convergence as in the nondefective case if λ1 has a full set of eigenvectors (but very slowly if not). The critical limitation of the Power Me ...