Lab 3: Using MATLAB for Differential Equations 1
... where t0 is the initial time, tf is the final time, and y0 is the initial condition, y(t0 ) = y0 . The same syntax (1) works for equations and systems alike. Example 1. y 0 = y 2 − t, y(0) = 0, for 0 ≤ t ≤ 4. 1. Creating the M-file. Start up MATLAB; the Command Window appears with the prompt >> awai ...
... where t0 is the initial time, tf is the final time, and y0 is the initial condition, y(t0 ) = y0 . The same syntax (1) works for equations and systems alike. Example 1. y 0 = y 2 − t, y(0) = 0, for 0 ≤ t ≤ 4. 1. Creating the M-file. Start up MATLAB; the Command Window appears with the prompt >> awai ...
An ergodic theorem for permanents of oblong matrices
... permanent of the truncated m ˆ n matrix is asymptotically equal to nÓm λ, where λ is the product of the expectations of the entries of X1 . Asymptotic results for the permanents of similar sequences of matrices have been obtained under much stronger assumptions on the distribution of the entries. Se ...
... permanent of the truncated m ˆ n matrix is asymptotically equal to nÓm λ, where λ is the product of the expectations of the entries of X1 . Asymptotic results for the permanents of similar sequences of matrices have been obtained under much stronger assumptions on the distribution of the entries. Se ...
2016 SN P1 ALGEBRA - WebCampus
... 1. Introduction to algebra (Simplification) For expressions in the form ax2 + bx + c with a6=1 → Have to find 2 numbers which multiply together give c but also have to find 2 other numbers for coefficients of the 2 terms in x which multiplied together equal a and allow the coefficient b to be deriv ...
... 1. Introduction to algebra (Simplification) For expressions in the form ax2 + bx + c with a6=1 → Have to find 2 numbers which multiply together give c but also have to find 2 other numbers for coefficients of the 2 terms in x which multiplied together equal a and allow the coefficient b to be deriv ...
MATH 310, REVIEW SHEET 1 These notes are a very short
... translate it to a linear system: give a variable name xi to the coefficient in front of each of the reactants and products. For each of the elements that appears in the reaction, you get a linear equation: the number of atoms of the element appearing on the left is equal to the number of atoms on th ...
... translate it to a linear system: give a variable name xi to the coefficient in front of each of the reactants and products. For each of the elements that appears in the reaction, you get a linear equation: the number of atoms of the element appearing on the left is equal to the number of atoms on th ...
Section 1.6: Invertible Matrices One can show (exercise) that the
... definition, f = e1 ◦ . . . ◦ ek , where e1 , . . . , ek are elementary row operations on Fm×n . Since each elementary row operation is an invertible function, the theorem follows. In fact, the result of this theorem is an important part of the reason for using admissible row operations on the augmen ...
... definition, f = e1 ◦ . . . ◦ ek , where e1 , . . . , ek are elementary row operations on Fm×n . Since each elementary row operation is an invertible function, the theorem follows. In fact, the result of this theorem is an important part of the reason for using admissible row operations on the augmen ...
Error in dot products; forward and backward error
... true even under blocked rearrangements of the algorithm (though this error bound does not necessarily hold for Strassen’s algorithm). Algorithms whose computed results in floating point correspond to a small relative backward error, such as the standard dot-product and matrix-vector multiplication a ...
... true even under blocked rearrangements of the algorithm (though this error bound does not necessarily hold for Strassen’s algorithm). Algorithms whose computed results in floating point correspond to a small relative backward error, such as the standard dot-product and matrix-vector multiplication a ...
Matrix Decomposition and its Application in Statistics
... Theorem: If A is a n×n real, symmetric and positive definite matrix then there exists a unique lower triangular matrix G with positive diagonal element such that A GG T . Proof: Since A is a n×n real and positive definite so it has a LU decomposition, A=LU. Also let the lower triangular matrix L t ...
... Theorem: If A is a n×n real, symmetric and positive definite matrix then there exists a unique lower triangular matrix G with positive diagonal element such that A GG T . Proof: Since A is a n×n real and positive definite so it has a LU decomposition, A=LU. Also let the lower triangular matrix L t ...
MATRICES part 2 3. Linear equations
... A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is a n by n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n by n identity matrix is augmented to the right of ...
... A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is a n by n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n by n identity matrix is augmented to the right of ...
Combining systems: the tensor product and partial trace
... The state of a quantum system is a vector in a complex vector space. (Technically, if the dimension of the vector space is infinite, then it is a separable Hilbert space). Here we will always assume that our systems are finite dimensional. We do this because everything we will discuss transfers with ...
... The state of a quantum system is a vector in a complex vector space. (Technically, if the dimension of the vector space is infinite, then it is a separable Hilbert space). Here we will always assume that our systems are finite dimensional. We do this because everything we will discuss transfers with ...
Structured Multi—Matrix Variate, Matrix Polynomial Equations
... Suppose all the matrices , , … , . have common set of right eigenvectors corresponding to all the eigenvalues , , … , . ( not necessarily same ). Then, we necessarily have that all the eigenvalues of unknown matrices , , … , . are zeroes ( solutions ) of the determinent ...
... Suppose all the matrices , , … , . have common set of right eigenvectors corresponding to all the eigenvalues , , … , . ( not necessarily same ). Then, we necessarily have that all the eigenvalues of unknown matrices , , … , . are zeroes ( solutions ) of the determinent ...
Multilinear spectral theory
... For matrices (order-2), only one way to take transpose (ie. swapping row and column indices) since Σ2 has only one non-trivial element. For an order-k tensor, there are k! − 1 different ‘transposes’ — one for each non-trivial element of Σk . An order-k tensor A = Jai1···ik K ∈ Rn×···×n is called sym ...
... For matrices (order-2), only one way to take transpose (ie. swapping row and column indices) since Σ2 has only one non-trivial element. For an order-k tensor, there are k! − 1 different ‘transposes’ — one for each non-trivial element of Σk . An order-k tensor A = Jai1···ik K ∈ Rn×···×n is called sym ...
Non-negative matrix factorization
NMF redirects here. For the bridge convention, see new minor forcing.Non-negative matrix factorization (NMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically.NMF finds applications in such fields as computer vision, document clustering, chemometrics, audio signal processing and recommender systems.