1 Box Muller - NYU Courant
... It may seem odd that X and Y in (13) are independent given that they use the same R and Θ. Not only does our algebra shows that this is true, but we can test the independence computationally, and it will be confirmed. Part of this method was generating a point “at random” on the unit circle. We sugg ...
... It may seem odd that X and Y in (13) are independent given that they use the same R and Θ. Not only does our algebra shows that this is true, but we can test the independence computationally, and it will be confirmed. Part of this method was generating a point “at random” on the unit circle. We sugg ...
Lecture 3 Linear Equations and Matrices
... sparse matrix techniques (studied in numerical linear algebra) it’s not uncommon to solve for hundreds of thousands of variables, with hundreds of thousands of (sparse) equations, even on a small computer . . . which is truly amazing (and the basis for many engineering and scientific programs, like ...
... sparse matrix techniques (studied in numerical linear algebra) it’s not uncommon to solve for hundreds of thousands of variables, with hundreds of thousands of (sparse) equations, even on a small computer . . . which is truly amazing (and the basis for many engineering and scientific programs, like ...
3 The positive semidefinite cone
... and B are both a multiple of xxT . Let u be any vector orthogonal to x, i.e., uT x = 0. Then 0 ≤ uT Au ≤ uT (A + B)u = uT (λxxT )u = 0. Thus for any u ∈ {x}⊥ we have uT Au = 0 which implies, since A 0, u ∈ ker(A) (see Exercise 3.2). Since im(A) = ker(A)⊥ for any symmetric matrix A we get im(A) = k ...
... and B are both a multiple of xxT . Let u be any vector orthogonal to x, i.e., uT x = 0. Then 0 ≤ uT Au ≤ uT (A + B)u = uT (λxxT )u = 0. Thus for any u ∈ {x}⊥ we have uT Au = 0 which implies, since A 0, u ∈ ker(A) (see Exercise 3.2). Since im(A) = ker(A)⊥ for any symmetric matrix A we get im(A) = k ...
Polar Decomposition of a Matrix
... The matrix representation of systems reveals many useful and fascinating properties of linear transformations. One such representation is the polar decomposition. This paper will investigate the polar decomposition of matrices. The polar decomposition is analogous to the polar form of coordinates. W ...
... The matrix representation of systems reveals many useful and fascinating properties of linear transformations. One such representation is the polar decomposition. This paper will investigate the polar decomposition of matrices. The polar decomposition is analogous to the polar form of coordinates. W ...
Lecture 4 Two_level_minmization
... Let Mmxn be a Boolean matrix (like the constraint matrix in Q-M), the UCP is to find a minimum number of columns to cover M in the sense that any row with a 1-entry has at least one of its 1entries covered ...
... Let Mmxn be a Boolean matrix (like the constraint matrix in Q-M), the UCP is to find a minimum number of columns to cover M in the sense that any row with a 1-entry has at least one of its 1entries covered ...
