[Numeth] Bracketing, Open, Direct, and Indirect Methods

Question 1

False-position method is also called ___

Answer 1

linear interpolation method

Question 2

a procedure for converting a matrix to reduced row echelon form using elementary row operations.

Answer 2

Gauss-Jordan Elimination

Question 3

Unlike direct methods, these use iterations to find the approximate solution.

Answer 3

Indirect Methods

Question 4

the most familiar method for solving a system of linear equations.

Answer 4

Gauss Elimination

Question 5

Gaussian elimination is also known as ____

Answer 5

row reduction

Question 6

It is done by rearranging the function so that x is on the left-hand side of the equation

Answer 6

Simple fixed-point iteration

Question 7

a modification on the bisection method.

Answer 7

False-position method

Question 8

It works by narrowing the gap between the positive and negative intervals until it closes in on the correct answer.

Answer 8

Bisection Method

Question 9

False-position in Latin

Answer 9

regula falsi

Question 10

an algorithm in linear algebra for solving a system of linear equations. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients.

Answer 10

Gaussian Elimination

Question 11

2 categories for solving linear systems of equations

Answer 11

direct and indirect/iterative methods

Question 12

Types of Direct Methods

Answer 12

Gauss Elimination and Gauss-Jordan Ellimination

Question 13

In this method, a tangent can be extended from the point [xi,f(xi)]. The point where this tangent cross the x-axis usually represents an improved estimate of the root.

Answer 13

NRM

Question 14

A matrix in reduced-row echelon form is also known as

Answer 14

row canonical form

Question 15

In general, odd multiple roots ______, whereas even ones do not.

Answer 15

cross the x-axis

Question 16

It separates the interval and subdivides the interval in which the root of the equation lies.

Answer 16

Bisection Method

Question 17

a modification of the Jacobi method is called the Gauss-Seidel method, named after ____ and ____

Answer 17

Carl Friedrich Gauss and Philipp L. Seidel

Question 18

Types of Indirect Methods

Answer 18

Jacobi Iteration and Gauss-Seidel Method

Question 19

These methods transform the original system into an equivalent system in which the coefficient matrix is upper-triangular, lower-triangular, or diagonal, making the system much easier to solve.

Answer 19

Direct Methods

Question 20

2 phases of the gauss elimination

Answer 20

Elimination and solution phase

Question 21

This method involves a fractional perturbation ( of the independent variable to estimate f’(x)

Answer 21

Modified Secant method

Question 22

The lack of refinements in Gaussian Elimination can be a problem because Gaussian elimination is sensitive to _____

Answer 22

rounding error.

Question 23

The principle behind this method is the intermediate theorem for continuous functions.

Answer 23

Bisection Method

Question 24

Steps in Bisection Method

Answer 24

1. Choose Xl and Xu
2. Xr = (Xl + Xu) /2
3. Find f(xl) and f(xu)
   f(xl)f(xu) is (-) = lower subinterval, xu = xr
   f(xl)f(xu) is (+) = upper subinterval, xl = xr

Question 25

These techniques are called ____ because two initial guesses for the root are required.

Answer 25

bracketing methods

Question 26

Methods are said _____ when they move closer to the truth as the computation progresses.

Answer 26

convergent

Question 27

they sometimes diverge or move away from the true root as the computation progresses. When these method converges, they usually do so much more quickly.

Answer 27

open methods

Question 28

based on formulas that they require only a single starting value of x or two starting values that do not necessarily bracket the root.

Answer 28

open methods

Question 29

εs stands for?

Answer 29

pre-specified stopping criterion

Question 30

This is an alternative method that exploits the graphical insight to join f(XL) and f(XU) by a straight line.

Answer 30

false-position method

Question 31

deal with the fact that a function typically changes sign in the vicinity of a root.

Answer 31

Bracketing Methods

Question 32

Types of Open Methods:

Answer 32

Simple fixed-point iteration, newton-raphson method (NRM), secant method, modified secant method, and modified NRM

Question 33

A simple method for obtaining an estimate of the root of the equation f(x) = 0 is by making a plot of the function and observe where it crosses the x-axis.

Answer 33

Graphical Method

Question 34

In this method, a system of equations that is given in a general form is manipulated to be in the upper triangular form, which is then be solved by using the back substitution.

Answer 34

Gauss Elimination

Question 35

Jacobi Method is named after _____ (1804-1851)

Answer 35

Carl Gustav Jacob Jacobi

Question 36

The bisection method is also known as ____

Answer 36

interval halving, root-finding, binary search, dichotomy, binary chopping, or Bolzano's Method

Question 37

the most widely used of all root-locating formulas

Answer 37

Newton-raphson method

Question 38

the refinement of the Gauss elimination method.

Answer 38

Gauss-Jordan Method

Question 39

Types of Bracketing Methods?

Answer 39

Bisection and False-Position Method