# MTH603 Quiz 3 Solution and Discussion

• Euler’s method is only useful for a few steps and small step sizes; however Euler’s method together with Richardson extrapolation may be used to increase the ____________.

• Given that dydt=y−ty+tdydt=y−ty+t with the initial condition y=1.01 at t=0.01. Using Euler’s method, y at t= 0.04, h=0.05, the value of y(0.05) is

• Generally, Adams methods are superior of output at many points is needed.

The Adams methods are useful to reduce the number of function calls, but they usually require more CPU time than the Runge-Kutta methods.

• Generally, Adams methods are superior of output at many points is needed.

• Given that dydt=t+y√ with the initial condition y0=1att0=0 find the 3rd term in Taylor series when t=1, y/ =0.2, y// =2, and h=0.1.

• The Power method can be used only to find the eigenvalue of A that is largest in absolute value—we call this eigenvalue the dominant eigenvalue of A.

True
False

As presented here, the method can be used only to find the eigenvalue of A that is largest in absolute value—this eigenvalue is called the dominant … The eigenvectors corresponding to are called dominant eigenvectors of A. 1 i. 2, . . . , n.

• Central difference method seems to be giving a better approximation, however it requires more computations.

True
False

Numerical method. In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.

Reff

• If x is an eigen value corresponding to eigen value of V of a matrix A. If a is any constant, then x – a is an eigen value corresponding to eigen vector V is an of the matrix A - a I.

True
False

If an eigenvalue l of A is known, the corresponding eigenvector(s) may be obtained by … l of a matrix A is the maximum number of linearly independent eigen vectors x of A … If v1, v2, …, vn are the eigenvectors associated with the respective … the eigenvalues of A and then if some of them are multiple, to check if there exist …

• If n x n matrices A and B are similar, then they have the different eigenvalues (with the same multiplicities).

True
False

If A and B are positive definite, is A + B positive definite? We don’t know … to A. If two matrices have the same n distinct eigenvalues, they’ll be similar to the same diagonal

• For differences methods we require the set of values.

True
False

Method of Differences. The method of finite differences gives us a way to calculate a polynomial using its values at several consecutive points. This is often a good approach to finding the general term in a pattern, if we suspect that it follows a polynomial form.

• The characteristics polynomial of a 3x 3 identity matrix is __________, if x is the eigen values of the given 3 x 3 identity matrix. where symbol ^ shows power.

• (x-1)^3
• (x+1)^3
• x^3-1
• x^3+1

• The Jacobi’s method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal.

True
False

Jacobi Method.
The Jacobi method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal (Bronshtein and Semendyayev 1997, p. … Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges.

• The Power method can be used only to find the eigenvalue of A that is largest in absolute value—we call this eigenvalue the dominant eigenvalue of A

True
False

For large values of n, polynomial equations like this one are difficult and time-consuming to solve. Moreover, numerical techniques for approximating roots of polynomial equations of high degree are sensitive to rounding errors. In this section we look at an alternative method for approximating eigenvalues. As presented here, the method can be used only to find the eigenvalue of A that is largest in absolute value—we call this eigenvalue the dominant eigenvalue of A. Although this restriction may seem severe, dominant eigenval- ues are of primary interest in many physical applications.

• Eigenvalues of a symmetric matrix are all _________.

• real
• zero
• positive
• negative

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, … geometry, for each tangent space to a manifold may be endowed with an inner product, giving rise to what is called a Riemannian manifold. … Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real.

• The determinant of a _______ matrix is the product of the diagonal elements.

• diagonal
• upper triangular
• lower triangular
• scalar

By linearity then, the determinant of AB is the product of the diagonal elements of A times the determinant of B, that is, it is the product of the determinant of A and that of B, as we have claimed. and we will have det A = det A’ , and det AB = det A’B.

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