Computational Methods Final

The following is true about the matrix condition number, EXCEPT. A. A measure of the singularity of a matrix B. It's greater than or equal to 1. C. A measure of the size of a matrix. D. Can be used for predicting rounding errors.

C. A measure of the size of a matrix.

The following is true about the Lagrange interpolating polynomials, EXCEPT A. It does not provide a polynomial in conventional form (i.e. axn+bxn-1+....) B. It's simply a reformulation of the Newton's polynomial that avoids the computation of divided differences. C. It applies lower-order polynomials to subsets of data points. D. It has similar estimated error as the Newton's method.

C. It applies lower-order polynomials to subsets of data points.

The following statement is correct for the Simpson's 1/3 rule, EXCEPT. A. It is limited to equi-spaced segaments in multiple-application version. B. It's a Newton-Cotes closed integration formula. C. It can only be applied to even number of points in multiple-application version. D. It is more accurate than the trapezoidal rule

C. It can only be applied to even number of points in multiple-application version.

The following is correct for the Hessian of a function, EXCEPT A. It provides a way to discern whether a multidimensional function has reached an optimum. B. It only involves second-order partial derivatives. C. It indicates the steepest ascent direction. D. It is a matrix.

C. It indicates the steepest ascent direction.

What will lead to no-solution for a linear system (n=2) containing two simultaneous equations using graphical depiction? A. Intersecting but getting too close to each other B. Overlapping each other C. Parallel to each other D. Perpendicular to each other

C. Parallel to each other

In Gauss elimination, after forward elimination, the main matrix will be changed to a _____. A. Lower triangular matrix B. Diagonal matrix C. Upper triangular matrix D. Tridiagonal matrix

C. Upper triangular matrix

What is the best description for a multidimensional multimodal problem? A. Single dependent variable and multiple global maximums B. Single dependent variable and single global maximum C. multiple dependent variables and multiple local maximum D. multiple dependent variables and multiple global maximums

C. multiple dependent variables and multiple local maximum

A second order ordinary differential equation includes ______________. A. A first derivative as its highest derivative and one independent variable. B. A first derivative as its highest derivative and two independent variable. C. A second derivative as its highest derivative and one independent variable. D. A second derivative as its highest derivative and two independent variable.

C. A second derivative as its highest derivative and one independent variable

The following is true for numerical approaches, EXCEPT A.Are always discretized solution B.Can sometimes equal to analytical solutions C.Contain errors D.Are always approximate solutions

C. Contain errors

When applying finite different methods for solving a BVP, the finite differences are substituted for the _____ in the original differential equation. A. Initial values B. Boundary values C. Derivatives D. Initial guesses

C. Derivatives

The following statement is correct for Runge-Kutta (RK) methods, EXCEPT. A. They are derived from Taylor series expansion. B. They don't require the calculation of higher derivatives. C. Huen's method is a first-order RK method. D. Raltson's method is a second-order RK method.

C. Heun's method is a first-order RK method

Eigen value problems are originated from ______. A. Sparse linear systems B. Tridiagonal linear systems C. Homogeneous linear systems D. Nonhomogeneous linear systems

C. Homogenous Linear Systems

The following statement is correct for the multistep methods using variable step size in solving ODEs, EXCEPT. A. It uses doubling and halving the step sizes based on the needs. B. The step size is chosen so that the corrector converges within two iterations C. It improves the order of accuracy of the method. D. If the corrector error is greater than some specified error, the step size is decreased.

C. It improves the order of accuracy of the method

The following is correct about the Non-Self-Starting Heun's Method, EXCEPT. A. It has a local error of O(h3). B. Its corrector can be applied iteratively. C. It uses the Euler's method as the predictor. D. It uses two steps.

C. It uses the Euler's method as the predictor

For searching out optima, the second derivatives of one dimensional functions can tell us about __________ . A. if we have reached maximum. B. the conjugate direction of the search. C. whether the optima is a maximum or minimum. D. the steepest trajectory of the function.

C. Whether the optima is a maximum or minimum

Symmetric matrix is one where _____ for all i and j. A. aii=ajj B. aii=-ajj C. aij=aji D. aij=-aji

C. aij=aji

How many initial guesses do we need for the Muller method? A.1 B.2 C.3 D.Order of polynomials

C.3

A tridiagonal matrix is a kind of ______. A.Identity matrix B.Upper triangular matrix C.Banded matrix D.Lower triangular matrix

C.Banded matrix

The following test can be used to check if the system is ill-conditioned, EXCEPT A.Multiply the inverse by the original matrix B.Invert the scaled matrix C.Calculate the matrix norm D.Invert the inverted matrix

C.Calculate the matrix norm

Which one of the following will result in decrease of round-off errors? A.Increase in the number of computations B.Decrease in step size C.Increase in number of significant figures D.Decrease in the number of terms in series

C.Increase in number of significant figures

The follow statement is correct about "Norm", EXCEPT A.It is used to check the singularity of the matrix. B.It is a real-valued function. C.It is a measure of the size of the matrix. D.It can be used to check the rounding errors of the solution.

C.It is a measure of the size of the matrix.

Any numerical algorithm can be composed of using the following fundamental logical representations, EXCEPT A.Sequence B.Repetition C.Iteration D.Selection

C.Iteration

In the following equation, the term c/m is so called v(t) = (gm/c)(1-e^(-(c/m)t)) A.Independent variables B.Forcing functions C.Parameters D.Dependent variables

C.Parameters

Newton-Raphson method is derived from ___. A.Fixed-point iteration method B.Graphical method C.Taylor series D.Bracketing method

C.Taylor series

Müller's method obtains a root estimate by _______. A.bracketing the root using two points B.projecting a straight line to the x axis using two points C.projecting a parabola to the x axis D.projecting a straight line to the x axis using the slope

C.projecting a parabola to the x axis

The Golden Ratio means _______. A. half B. 0.6 C. (5^(1/2))/2 D. 0.618

D. 0.618

A tridiagonal matrix is a kind of ______. A. Lower triangular matrix B. Identity matrix C. Upper triangular matrix D. Banded matrix

D. Banded matrix

The following test can be used to check if the system is ill-conditioned, EXCEPT A. Multiply the inverse by the original matrix B. Invert the scaled matrix C. Invert the inverted matrix D. Calculate the matrix norm

D. Calculate the matrix norm

The following statement is correct for the Newton's optimization method, EXCEPT. A. It is faster than golden-section method if converges. B. It's an open method. C. It can be derived from a second-order Taylor series for f(x). D. It is to find f''(x) = 0.

D. It is to find f''(x) = 0.

The follow statement is correct about "Norm", EXCEPT A. It is a measure of the size of the matrix. B. It is a real-valued function. C. It can be used to check the rounding errors of the solution. D. It is used to check the singularity of the matrix.

D. It is used to check the singularity of the matrix.

The following statement is correct for the optimal steepest ascent method, EXCEPT. A. It is faster than Pattern search method. B. It's a gradient-based method. C. It uses the maximum step in the gradient direction. D. It uses second order derivative to search for the optimum.

D. It uses second order derivative to search for the optimum.

Gauss-Seidel method is the most commonly used _____. A. Bracketing method B. LU decomposition method C. Guass elimination method D. Iterative method

D. Iterative method

Optimization and ______ are related, both involve guessing and searching for a point on a function. A. Interpolation B. Curve fitting C. Integration D. Root finding

D. Root finding

Which one of the following methods is NOT a direct optimization method? A. Powell's search method B. Univariate search C. Random search D. Steepest ascent method

D. Steepest ascent method

In the method of quadratic splines, if we have a total of N points (including interior knots and end points), how many equations do we need to find all coefficients in splines? A. 3N-1 B. 3N C. 3N+1 D. 3(N-1)

D. 3(N-1)

Partial differential equations (PDEs) are different from ordinary differential equations by ______________. A. Having nonlinear coefficients. B. Having more dependent variables . C. Having higher-order derivatives. D. Having more independent variables .

D. Having more independent variables

If all conditions are specified at the same value of the independent variable, then we have a (an) ___________ problem. A. boundary value B. constant value C. single value D. initial value

D. Initial Value

The following is correct for the Power method, EXCEPT. A. It can be used to find both the largest eigen value and the smallest eigen value. B. It results in the corresponding eigen vector. C. It requires normalization of the resulting vector in each iteration. D. It requires setting determinant of the system to Zero in each iteration.

D. It requires setting determinant of the system to Zero in each iteration

The following methods are 2nd-order methods, EXCEPT A. The Heun's method B. The midpoint method. C. The Ralston's method. D. The implicit Euler's method.

D. The implicit Euler's method

Methods for stiff ODEs face challenges on maintaining stability. This can be solved by ______. A. Using multistep methods. B. Using higher order methods. C. Using larger step size. D. Using implicit methods.

D. Using implicit methods

The following are used for the Polynomial method, EXCEPT A. Solving the roots of a polynomial equation derived from the determinant of the system. B. Substituting a central finite-divided difference approximation for the derivatives. C. Discretizing the problem with a series of nodes. D. Using iterations to find the largest eigenvalue.

D. Using iterations to find the largest eigenvalue

How many significant digits does 0.0001753 have? A.7 B.6 C.5 D.4

D.4

Which one of the following is a bracketing method? A.Fixed-point iteration B.Secant method C.Newton-Raphson method D.False-position method

D.False-position method

Which one of the following is a bracketing method? A.Secant method B.Fixed-point iteration C.Newton-Raphson method D.False-position method

D.False-position method

Following methods use two initial guesses for finding root, EXCEPT A.Secant method B.False-position method C.Bisection method D.Fixed-point iteration

D.Fixed-point iteration

What method is the most appropriate for solving larger (n > 3) sets of simultaneous equations? A.graphical method B.elimination of unknowns C.Cramer's rule D.Gauss elimination

D.Gauss elimination

Which one of the following will result in decrease of round-off errors? A.Increase in the number of computations B.Decrease in the number of terms in series C.Decrease in step size D.Increase in number of significant figures

D.Increase in number of significant figures

The following statements are True about the Secant method, EXCEPT A.It's an open method. B.It needs two initial guesses. C.Its convergence is not guaranteed. D.It's faster than New-Raphson method.

D.It's faster than New-Raphson method.

Bairstow's method is an iterative approach loosely related to ________________. A.Müller method and Secant method B.Bracketing method and Secant method C.Taylor series and Newton Raphson method D.Müller method and Newton Raphson method

D.Müller method and Newton Raphson method

What are the initial guesses used for the Bairstow's method? A.cn and bn B.Δr and Δs C.x0 and x1 D.r and s

D.r and s

Shooting methods use ______ in updating the guess to improve the efficiency for solving the BVP. A. Golden Section B. Interpolation C. Integration D. Finite Difference

Interpolation

The following statement is correct for Heun's method, EXCEPT. A. It can be applied in an iterative fashion. B. It is a first-order RK method. C. It is to improve the estimate of the slope . D. It is a predictor-corrector approach.

It is a first-order RK method

Which one of the following is correct for a stiff ODE? A. Its solution only involves fast changing components. B. Its solution only involves slow changing components. C. Its solution involves both fast changing and slow changing components, but the fast ones die out in a short period. D. Its solution involves both fast changing and slow changing components, but the slow ones die out in a short period.

Its solution involves both fast changing and slow changing components, but the fast ones die out in a short period

To perform polynomial interpolation that fits all N points, __________ polynomial equation is required. A. Quadratic B. N-1th order C. Nth order D. N+1th order

B. N-1th order

What will lead to infinite-solutions for a linear system (n=2) containing two simultaneous equations using graphical depiction? A. Parallel to each other B. Overlapping each other C. Perpendicular to each other D. Intersecting but getting too close to each other

B. Overlapping each other

Among computation integral methods, _____ has the best accuracy. A. Second order polynomial Newton-Cotes formula B. Simpson's 3/8 rule C. Trapezoidal rule D. Simpson's 1/3 rule

B. Simpson's 3/8 rule

Sr is the sum of the squares of residuals around the regression line and r^2 is the coefficient of determination. For a perfect fit, ____________. A. Sr=1 and r=r^2=1 B. Sr=0 and r=r^2=1 C. Sr=St, r=r^2=0 D. Sr=St, r=r^2=1

B. Sr=0 and r=r^2=1

What is the fast method for solving a banded matrix? A. Gauss elimination B. Thomas algorithm C. Cholesky decomposition D. LU decomposition

B. Thomas algorithm

The Golden-section optimization method uses ______ to determine the interval containing the maximum. A. Two initial guesses and one interior points B. Two initial guesses and two interior points C. One initial guess and zero interior point D. One initial guess and one interior point

B. Two initial guesses and two interior points

Total sum of the squares around the mean for the dependent variable, y, is St. Sum of the squares of residuals around the regression line is Sr. So St-Sr quantifies __________. A. the variance of describing data using a straight line than an average value. B. the improvement of describing data using a straight line than an average value. C. the residuals of describing data using a straight line than an average value. D. the standard deviation of describing data using a straight line than an average value.

B. the improvement of describing data using a straight line than an average value.

Best strategy for Least Squares Regression is to minimize the sum of ________. A. the residuals for all available data B. the squares of the residuals between the measured and the calculated data. C. the absolute values of the residuals D. the square root of the residuals.

B. the squares of the residuals between the measured and the calculated data.

For searching out optima, the second derivatives of one dimensional functions can tell us about __________ . A. the conjugate direction of the search. B. whether the optima is a maximum or minimum. C. if we have reached maximum. D. the steepest trajectory of the function.

B. whether the optima is a maximum or minimum.

If the band width of a banded system is 5, then the half-bandwidth of this system is ______. A. 1 B. 2 C. 3 D. 4

B. 2

Every version of RK-2 methods would yield exactly the same results if the solution to ODE were as follows, EXCEPT: A. Linear B. High-order polynomial C. Constant D. Quadratic

B. High-order polynomial

The following is correct for the shoot method, EXCEPT. A. It uses the method used in solving initial value problems. B. It uses finite difference equation for all the derivatives in the equation. C. It uses a trial-and-error approach. D. It reaches very good approximation for linear equations when converges .

B. It uses finite difference equation for all the derivatives in the equation

The Euler's method for solving ODEs is derived from _________. A. Interpolation methods. B. Taylor series. C. Root finding methods. D. Integration methods.

B. Taylor series

The following is correct for elliptic PDEs, EXCEPT A. They are used to characterize steady-state problems. B. They are solved by using shooting methods. C. They are used to characterize boundary value problems. D. They are solved by transforming into algebraic difference equation.

B. They are solved by using shooting methods

Numerical solutions of ODEs involve two types of error, ________ A. Local truncation error and round-off error B. Truncation error and round-off error. C. Local truncation error and propagated truncation error. D. True truncation error and approximate truncation error

B. Truncation error and round-off error

Boundary-value problems are solved by specifying the conditions ______________. A. at different value of the dependent variable. B. at different value of the independent variable. C. at the same value of the independent variable. D. at the same value of the dependent variable.

B. at a different value of the independent variable

If one reads the odometer to 87324.452, how many significant digits for this reading? Take careful note of the number of digits that the odometer dial can show in the image. A.5 B.6 C.7 D.8

B.6

Which one of the following will reduce truncation errors? A.Increase in number of significant figures B.Decrease in step size C.Increase in the number of computations D.Decrease in the number of terms in series

B.Decrease in step size

Which one of the following will reduce truncation errors? A.Increase in the number of computations B.Decrease in step size C.Increase in number of significant figures D.Decrease in the number of terms in series

B.Decrease in step size

Which one in the following is NOT the advantage of using bisection method? A.# of iteration predictable B.Fast C.Always find root D.Easy

B.Fast

Gauss-Seidel method is the most commonly used _____. A.Guass elimination method B.Iterative method C.Bracketing method D.LU decomposition method

B.Iterative method

In general, which one of the following method has the fastest convergence? A.False-position method B.Newton-Raphson method C.Secant method D.Fixed-point iteration

B.Newton-Raphson method

What will lead to infinite-solutions for a linear system (n=2) containing two simultaneous equations using graphical depiction? A.Perpendicular to each other B.Overlapping each other C.Parallel to each other D.Intersecting but getting too close to each other

B.Overlapping each other

What will lead to no-solution for a linear system (n=2) containing two simultaneous equations using graphical depiction? A.Intersecting but getting too close to each other B.Parallel to each other C.Perpendicular to each other D.Overlapping each other

B.Parallel to each other

In Gauss elimination, after forward elimination, the main matrix will be changed to a _____. A.Diagonal matrix B.Upper triangular matrix C.Lower triangular matrix D.Tridiagonal matrix

B.Upper triangular matrix

Taylor series provides a means to predict the value of a function at one point in terms of the function value and its __ at another point. A.n-th power B.derivatives C.factorial D.absolute values

B.derivatives

Simpson's 1/3 rule employs _______ interpolation. A. A second order Lagrange polynomial B. A second order Newton polynomial C. A first order Newton polynomial D. A second order Spline

A. A second order Lagrange polynomial

In equal-area graphical differentiation, ______ is used to estimate the derivative for each interval. A. Centered finite difference B. Polynomial integration C. Linear regression curve fitting D. Simpson's rule

A. Centered finite difference

What method is the most appropriate for solving larger (n > 3) sets of simultaneous equations? A. Gauss elimination B. elimination of unknowns C. graphical method D. Cramer's rule

A. Gauss elimination

The following is correct for the gradient vector, EXCEPT A. It is the conjugate direction. B. It tells us the steepest ascent direction. C. It's usually changed in every iteration. D. It tells us how much we will gain in search by taking the direction.

A. It is the conjugate direction.

What in the following will improve the convergence of the Gauss-Seidel method? A. Pivoting to get diagonally dominant matrix B. Converting matrix to a banded matrix C. Scaling to ensure elements no greater than 1 D. LU decomposition

A. Pivoting to get diagonally dominant matrix

_______ is the representation of spread of data by the square of the standard deviation. A. Variance B. Arithmetic average C. Arithmetic mean D. Coefficient of variation

A. Variance

The strip graphical method employs _____ polynomials to approximate integrals. A. Zeroth order B. First order C. Second order D. Third order

A. Zeroth order

In the method of spline interpolation, ______ are called spline functions. A. connecting polynomials B. linear polynomials C. cubic polynomials D. quadratic polynomials

A. connecting polynomials

The residual in linear regression represents the ________ between a data point and the straight line. A. vertical distance B. normal distance C. tangential distance D. horizontal distance

A. vertical distance

Which one in the following is the simplest form of interpolation? A. Newton's linear interpolation B. Newton's quadratic interpolation C. Linear spline interpolation D. Linear Lagrange interpolation

A. Newton's linear interpolation

The following is a type of partial differential equations, EXCEPT. A. Polynomial equation. B. Elliptic equation. C. Hyperbolic equation. D. Parabolic equation.

A. Polynomial Equation

The following can be used to reduce the error for the Euler's method, EXCEPT A. Using different initial conditions. B. Modification of the estimate of the slope. C. Using more significant digits for computation. D. Reducing the step size.

A. Using different initial conditions

LU decomposition provides efficient means to compute the ______. A. matrix inverse B. root finding C. matrix transpose D. determinant of a matrix

A. matrix inverse

In bisection method, an estimate of the root xr is determined by __________. A.(xl+xu)/2 B.(xl-xu)/2 C.xl D.xu

A.(xl+xu)/2

The following is true about the matrix condition number, EXCEPT. A.A measure of the singularity of a matrix B.It's greater than or equal to 1. C.A measure of the size of a matrix. D.Can be used for predicting rounding errors.

A.A measure of the singularity of a matrix

Which method in the following is developed for finding complex roots of polynomials? A.Bairstow B.Fixed-point iteration C.False-position method D.Newton-Raphson method

A.Bairstow

What does a parallelogram represent in drawing a flow chart? A. Input/output B. Decision C.Process D.Terminal

A.Input/output

Stable algorithms refer to those for which the cumulative effect of the __ are limited. A.Round-off errors B.Relative true errors C.Absolute true errors D.Truncation errors

A.Round-off errors

Following descriptions are correct about the bisection method, EXCEPT A.The approximate errors have "ragged" nature. B.The true and approximate errors are far apart when the interval happens to be centered on the true root. C.The root must lie within ±∆x/2 of the estimate. D.Approximate errors are always greater than true errors.

A.The approximate errors have "ragged" nature.

What is the fast method for solving a banded matrix? A.Thomas algorithm B.LU decomposition C.Gauss elimination D.Cholesky decomposition

A.Thomas algorithm

LU decomposition provides efficient means to compute the _____. A. Root Finding B. Determinant of a Matrix C. Matix Inverse D. Matrix Transpose

C . Matrix Inverse