Numerical Methods Exam 1

What are the value(s) of α for which the following system has infinite solutions? | 2 -6α | |x| = | 3 | | 3α -1 | |y| | 1.5 |

1/3

During the solution of A x = b by Gauss Jordan Elimination method, the diagonal system after elimination is | 2 -2 5 | |x| = | 13 | | 2 3 4 | |y| | 20 | | 3 -1 3 | |z| | 10 |

| 2 0 0 | |x| = | 2 | | 0 5 0 | |y| | 10 | | 0 0 4.1 | |z| | -12.3 |

What are the value(s) of α for which the following system has no solution? | 2 -6α | |x| = | 3 | | 3α -1 | |y| | 1.5 |

-1/3

In finding root of a transcendental function f(x) = exp(x) - x - 2 by the Bisection Method, a unique root is in the interval (0, 2). In the next iteration of the Bisection method, the search interval should be a. (1, 2) b. (0, 2) c. (0, 0.5) d. (0, 1)

a. (1, 2)

A computer engineer is designing a microprocessor based on octal (base = 8) bits. It uses thirty two octal bits to represent a floating point value with one bit for mantissa sign, 4 bits for the exponent, and the remaining 27 bits for mantissa. The magnitude of its underflow limit in decimal representation will be a. -1850 b. -1852 c. None of the choices shown d. -1851

a. -1850

If A = 0.6455 ± 0.0001 and B = 0.6454 ± 0.0002 then using calculus principles, the result C = A / B - B / A2 will have at least the following # of correct decimal digits a. 2 b. 0 c. 3 d. 1

a. 2

If A = 0.6455 ± 0.0001 and B = 0.6454 ± 0.0002 then using calculus principles, the result C = A / B - B / A2 will have at least the following # of significant digits a. 2 b. 0 c. 3 d. 1

a. 2

The Newton-Raphson Method for finding root of a transcendental function f(x) belongs to the following category: a. Open or Non-bracketing b. Random c. Bracketing d. Graphical

a. Open or Non-bracketing

In finding the root of a transcendental function f(x) by the Newton Raphson Method, the convergence rate is a. Quadratic b. Super-linear c. Linear d. none of the choices shown

a. Quadratic

In finding the root of a transcendental function f(x) by the Secant Method, the convergence rate is a. Super-linear b. Linear c. Quadratic d. none of the choices shown

a. Super-linear

Given a smooth transcendental function f(x), if f(a) * f(b) is greater than zero then the open x-interval (a, b) a. all of the choices shown are possible b. contains no root of f(x) c. contains two unique roots of the function f(x) d. contains an even number of roots of the function f(x)

a. all of the choices shown are possible

When performing the matrix multiplication: FT * F a. matrix F can be of any size b. matrix F must be a column vector c. matrix F must be a square matrix d. matrix F must be a row vector

a. matrix F can be of any size

The quadratic equation: 2x2 + 20x + 50 = 0 has a. two equal real roots b. none of the choices shown c. no real roots d. two unique real roots

a. two equal real roots

Given | 0.25 0.10 | |x| = | 0.10 | | 0.12 0.41 | |y| | 0.20 | a. x = 0.2320, y = 0.4199 b. x = 0.9285, y = 0.3714 c. x = 0.4199, y = 0.2320 d. x = 0.2809, y = 0.9597

a. x = 0.2320, y = 0.4199

In finding the root of a transcendental function f(x) by the Bisection Method, if the search interval at the beginning of an iteration is x = (L, R) then the relative error bound at the end of the iteration is a. | (L - R) / (L + R) | b. | (L) / (L + R) | c. | (R) / (L + R) | d. | (L + R) / (L - R) |

a. | (L - R) / (L + R) |

For all real vectors x(nx1), which of the following is true? a. √n ∥x∥∞ ≥ ∥x∥2 b. ∥x∥∞ > ∥x∥2 c. √n∥x∥∞ ≤ ∥x∥2 d. ∥x∥∞ ≥ ∥x∥2

a. √n ∥x∥∞ ≥ ∥x∥2

In finding the root of a transcendental function f(x) by the Bisection Method, if the search interval at the beginning of an iteration is x = (L, R) then the approximate root value at the end of the iteration is a. |L - R| / 2 b. (L + R) / 2 c. (L - R) / 2 d. |L + R| / 2

b. (L + R) / 2

In searching for the root of a transcendental function f(x) by Secant Method, we have the following table of values: x1 x2 f1 f2 Dx 1.0 1.1 -0.2817 -0.09583 ? The change Dx is a. -0.03291 b. -0.05156 c. +0.1 d. +0.02219

b. -0.05156

In finding the root of the transcendental function f(x) = x - 2-x by the Secant Method with the initial guesses x1 = 0.5 and x2 = 0.6, the approximate root value after ONE iteration will be a. 0.6213 b. 0.6406 c. 0.6250 d. 0.5595

b. 0.6406

The determinant of the matrix A after normalization is A = | 0.25 0.10 | | 0.12 0.41 | a. 0.09 b. 0.79 c. 0.93 d. 0.49

b. 0.79

We want to determine the root of a transcendental function in the interval (0.02, 0.03) to FOUR significant digits. How many iterations will the Bisection Method require to find the root with such a precision? a. 7 b. 11 c. 10 d. 8

b. 11

A computer engineer is designing a microprocessor based on octal (base = 8) bits. It uses thirty two octal bits to represent a floating point value with one bit for mantissa sign, 4 bits for the exponent, and the remaining 27 bits for mantissa. Its machine precision in decimal representation will be a. 22 significant digits b. 23 significant digits c. 24 significant digits d. 21 significant digits

b. 23 significant digits

The Secant Method for finding root of a transcendental function f(x) belongs to the following category: a. Bracketing b. Open or Non-bracketing c. Graphical d. Random

b. Open or Non-bracketing

In finding the fixed point of a transcendental function g(x) by the Fixed Point Iteration Method, the convergence rate is a. Linear b. Super-linear c. Quadratic d. none of the choices shown

b. Super-linear

In finding root of a transcendental function f(x) by the Bisection method, if the relative error bound is 0.0004 then the approximate root value has a. at least three correct decimal digits b. at least three significant digits c. at least four significant digits d. at least four correct decimal digits

b. at least three significant digits

Given a smooth transcendental function f(x), if f(a) * f(b) is less than zero then the open x-interval (a, b) a. all of the choices shown are possible b. contains an odd number of roots of function f(x) c. contains no root of f(x) d. contains an even number of roots of the function f(x)

b. contains an odd number of roots of function f(x)

If matrix F = FT then a. matrix F must be a skew-symmetric matrix b. matrix F must be a symmetric matrix c. matrix F must be a square matrix d. matrix F must be a diagonal matrix

b. matrix F must be a symmetric matrix

The quadratic equation: 2x2 - 3x + 5 = 0 has a. none of the choices shown b. no real roots c. two equal real roots d. two unique real roots

b. no real roots

The following matrix is | 7 2 0 | | 3 5 -1 | | 0 5 -6 | a. singular b. strictly diagonally dominant c. not strictly diagonally dominant d. triangular

b. strictly diagonally dominant

Unit or Identity matrix I is a a. skew-symmetric matrix b. symmetric matrix c. square matrix d. all of the choices shown

b. symmetric matrix

A root of a smooth transcendental function f(x) is a. the value of x where the function f(x) crosses the x-axis b. the value of x where the function f(x) evaluates to zero c. all of the choices shown d. the value of x where the function f(x) evaluates to x

b. the value of x where the function f(x) evaluates to zero

The function f(x) = 2 * e-x - sin(x) has a root {please note that x is in radians} a. x = 3.5 ± 0.2 b. x = 1 ± 0.2 c. x = -1 ± 0.2 d. all of the choices shown

b. x = 1 ± 0.2

In class, we found square root of "a" by finding the root of the transcendental function f(x) = x2 - a by Newton-Raphson Method. The iteration scheme simplifies to a. xnew = x2 / (2 * x) b. xnew = (x2 + a) / (2 * x) c. xnew = (x2 - a) / (2 * x) d. xnew = (x2 - a) / x

b. xnew = (x2 + a) / (2 * x)

A computer engineer is designing a microprocessor based on octal (base = 8) bits. It uses eight octal bits to represent an integer. Its overflow limit in decimal representation will be a. +16,277,216 b. +8,388,608 c. +8,388,607 d. +16,277,215

c. +8,388,607

Bragg's Law states that when x-rays of wavelength "λ" are incident on parallel crystal planes spaced "d" apart at angle "θ" such that λ = 2 * d * sinθ then diffraction occurs. If the wavelength λ = 15.4 nm = constant, and we want to measure d ≅ 23.4 nm to FIVE SIGNIFICANT DIGITS, then we must measure "θ" with an absolute precision equal to or better than a. 0.0000007 deg b. 0.004 deg c. 0.0001 deg d. 0.00001 deg

c. 0.0001 deg

The determinant of matrix A is A = | 0.25 0.10 | | 0.12 0.41 | a. 0.1145 b. 0.0225 c. 0.0905 d. 0.0243

c. 0.0905

The Maximum Norm ∥x∥∞ of vector x = [+0.12 +0.14 -0.15 -0.10]T is a. 0.0665 b. 0.2579 c. 0.1500 d. 0.1400

c. 0.1500

The Euclidean Norm ∥x∥2 of vector x = [+0.12 +0.14 -0.15 -0.10]T is a. 0.1400 b. 0.0665 c. 0.2579 d. 0.1500

c. 0.2579

The function f(x) = ex - 3x2 has a root a. 3.20 ± 0.02 b. 0.50 ± 0.02 c. 0.90 ± 0.02 d. 2.65 ± 0.02

c. 0.90 ± 0.02

In searching for the root of a transcendental function f(x) by Newton-Raphson Method with the initial guess x0 = 1, we have the following table of values: x f(x) f'(x) Xnew 1 -0.2817 1.7183 ? The next guess xnew is a. 0 b. 0.8361 c. 1.1639 d. 0.5

c. 1.1639

We want to determine the root of a transcendental function in the interval (5, 6) to FOUR significant digits. How many iterations will the Bisection Method require to find the root with such a precision? a. 14 b. 10 c. 11 d. 15

c. 11

We want to determine the root of a transcendental function in the interval (5, 6) to FOUR correct decimal digits. How many iterations will the Bisection Method require to find the root with such a precision? a. 10 b. 11 c. 15 d. 14

c. 15

A computer engineer is designing a microprocessor based on octal (base = 8) bits. It uses thirty two octal bits to represent a floating point value with one bit for mantissa sign, 4 bits for the exponent, and the remaining 27 bits for mantissa. The magnitude of its overflow limit in decimal representation will be a. 1850 b. 1848 c. 1849 d. None of the choices shown

c. 1849

If A = 0.6455 ± 0.0001 and B = 0.6454 ± 0.0002 then using calculus principles, the result C = A2 / B - B / A2 will have at least the following # of significant digits a. 3 b. 1 c. 2 d. 0

c. 2

We want to determine the root of a transcendental function in the interval (0.02, 0.03) to FOUR correct decimal digits. How many iterations will the Bisection Method require to find the root with such a precision? a. 7 b. 10 c. 8 d. 11

c. 8

If x = [-1 2 3 -2]T, and y = [4 1 5 -3]T, then 5 * x - 3 * y is a. [7 13 30 -19]T b. [-5 1 -2 4]T c. [-17 7 0 -1]T d. [3 3 8 -5]T

c. [-17 7 0 -1]T

A= | 1 -2 2 | B= | -1 -2 4 | | 3 1 1 | | 1 3 -5 | | 2 0 1 | | 2 4 -7 | a. A * B = B * A b. A * B = I (unit matrix) c. all of the answers shown are correct d. B * A = I (unit matrix)

c. all of the answers shown are correct

A quadratic algebraic function: f(x) = a * x2 + b * x + c a. may have two unique real roots b. may have two equal real roots c. all of the choices shown are possible d. may have two complex roots that are complex conjugates

c. all of the choices shown are possible

In finding root of a transcendental function f(x) by the Bisection method, if the relative error bound is 0.00006 then the approximate root value has a. at least four significant digits b. at least three correct decimal digits c. at least three significant digits d. at least four correct decimal digits

c. at least three significant digits

If matrix F = - FT then a. matrix F must be a square matrix b. matrix F must be a symmetric matrix c. matrix F must be a skew-symmetric matrix d. matrix F must be a diagonal matrix

c. matrix F must be a skew-symmetric matrix

If the matrix products: F * FT = FT * F then a. matrix F must be a diagonal matrix b. matrix F must be a square matrix c. matrix F must be a symmetric or skew-symmetric matrix d. matrix F must be a scalar matrix

c. matrix F must be a symmetric or skew-symmetric matrix

In finding the fixed point of the function g(x) = exp(x) - 2 by the Fixed Point Iteration Method, the method is guaranteed to convergefor any initial guess in the following range: a. (0, 2) b. (0, 2.718) c. none of the choices shown d. (0.693, 2.718)

c. none of the choices shown

The following matrix is | 6 4 -3 | | 4 -2 0 | | -3 0 1 | a. triangular b. strictly diagonally dominant c. not strictly diagonally dominant d. singular

c. not strictly diagonally dominant

The quadratic equation: 2x2 - 3x - 5 = 0 has a. two equal real roots b. no real roots c. two unique real roots d. none of the choices shown

c. two unique real roots

In finding the root of a transcendental function f(x) by the Bisection Method, if the search interval at the beginning of an iteration is x = (L, R) then the absolute error bound at the end of the iteration is a. (L + R) / 2 b. (L - R) / 2 c. |L - R| / 2 d. |L + R| / 2

c. |L - R| / 2

Given any two real vectors x and y, which of the following is false? a. ∥x + y∥2 ≤ ∥x∥2 + ∥y∥2 b. ∥x + y∥ ≤ ∥x∥ + ∥y∥ c. ∥x + y∥ ≥ ∥x∥ + ∥y∥ d. ∥x∥ ≥ 0 and ∥y∥ ≥ 0

c. ∥x + y∥ ≥ ∥x∥ + ∥y∥

We want to explore the interval (0, 2) that may contain a root of the transcendental function f(x) = ex - x - 2. Which of the following statement is correct? a. f(x) has no root in the interval (0, 2) b. f(x) has more than one root in the interval (0, 2) c.f(x) has a unique root in the interval (0, 2) d. none of the choices shown

c.f(x) has a unique root in the interval (0, 2)

A computer engineer is designing a microprocessor based on octal (base = 8) bits. It uses eight octal bits to represent an integer. Its underflow limit in decimal representation will be a. -8,388,607 b. -16,277,216 c. -16,277,215 d. -8,388,608

d. -8,388,608

Bragg's Law states that when x-rays of wavelength "λ" are incident on parallel crystal planes spaced "d" apart at angle "θ" such that λ = 2 * d * sinθ then diffraction occurs. If the wavelength λ = 15.4 nm = constant, and we want to measure d ≅ 23.4 nm to FOUR CORRECT DECIMAL DIGITS, then we must measure "θ" with an absolute precision equal to or better than a. 0.00001 deg b. 0.004 deg c. 0.0001 deg d. 0.00004 deg

d. 0.00004 deg

In finding the fixed point of the function g(x) = exp(x) - 2 by the Fixed Point Iteration Method, if the initial guess x0 = 1 the the next guess xnew will be a. 0.3891 b. -1.6321 c. -0.2817 d. 0.7183

d. 0.7183

In searching for the root of a transcendental function f(x) by Secant Method, we have the following table of values: x1 x2 f1 f2 xnew 1.0 1.1 -0.5634 -0.09167 ? The next guess xnew is a. 1.081 b. 1.133 c. 1.152 d. 1.119

d. 1.119

In searching for the root of a transcendental function f(x) by Secant Method, we have the following table of values: x1 x2 f1 f2 Dx 1.0 1.1 -0.2817 -0.09583 ? The next guess xnew is a. 1.048 b. 1.133 c. 1.200 d. 1.152

d. 1.152

If A = 0.6455 ± 0.0001 and B = 0.6454 ± 0.0002 then using calculus principles, the result C = A2 / B - B / A2 will have at least the following # of correct decimal digits a. 0 b. 1 c. 3 d. 2

d. 2

In finding root of a transcendental function f(x) by Newton-Raphson Method, the successive iterative values for xnew are shown below: iteration = 0, x = 1.6333 iteration = 1, x = 1.5911 iteration = 2, x = 1.5874 iteration = 3, x = 1.5873 iteration = 4, x = 1.5873 Using the convergence criteria taught in the class, the iteration # after which we have TWO correct decimal digits is a. 1 b. 3 c. 4 d. 2

d. 2

In finding the root of the transcendental function f(x) = x * exp(-x) - 0.3 in the search interval x = (1, 5) by the Bisection Method, the approximate root value after TWO iterations will be a. 1.5 b. 3.0 c. 0.0 d. 2.0

d. 2.0

In finding root of a transcendental function f(x) by Newton-Raphson Method, the successive iterative values for xnew are shown below: iteration = 0, x = 2.0000 iteration = 1, x = 1.6667 iteration = 2, x = 1.5911 iteration = 3, x = 1.5874 iteration = 4, x = 1.5873 Using the convergence criteria taught in the class, the iteration # after which we have TWO significant digits is a. 4 b. 2 c. 1 d. 3

d. 3

The Bisection Method for finding root of a transcendental function f(x) belongs to the following category: a. Random b. Open or Non-bracketing c. Graphical d. Bracketing

d. Bracketing

The following gas stations were cited for irregular dispensation by the Department of Agriculture. Which one cheated the most? Station Actual Gasoline Dispensed Pump Reading Houston 9.90 10.00 Dallas 19.90 20.00 Austin 29.80 30.00 Laredo 29.95 30.00 a. Austin b. Laredo c. Dallas d. Houston

d. Houston

In finding the root of a transcendental function f(x) by the Bisection Method, the convergence rate is a. Super-linear b. none of the choices shown c. Quadratic d. Linear

d. Linear

The Fixed Point Iteration Method for finding fixed point of a transcendental function g(x) belongs to the following category: a. Random b. Bracketing c. Graphical d. Open or Non-bracketing

d. Open or Non-bracketing

In finding root of a transcendental function f(x) by the Bisection method, if the absolute error bound is 0.0004 then the approximate root value has a. at least four correct decimal digits b. at least three significant digits c. at least four significant digits d. at least three correct decimal digits

d. at least three correct decimal digits

We want to explore the open interval (0, 2) that may contain a root of the transcendental function f(x) = ex - x - 2. Which of the following statement is correct? a. f(x) has a maxima in the open interval (0, 2) b. f(x) has a minima in the open interval (0, 2) c. the derivative f'(x) changes sign in the open interval (0, 2) d. f(x) has no maxima or minima in the open interval (0, 2)

d. f(x) has no maxima or minima in the open interval (0, 2)

The following matrix is a | 2 0 0 | | 0 -4 0. | | 6 0 4 | a. diagonal matrix b. upper triangular matrix c. symmetric matrix d. lower triangular matrix

d. lower triangular matrix

Given matrix A A = | 0.25 0.10 | | 0.12 0.41 | a. matrix A is likely ill-conditioned b. matrix A is singular c. matrix A is diagonal d. matrix A is likely well-conditioned

d. matrix A is likely well-conditioned

When performing the matrix multiplication: F * FT a. matrix F must be a column vector b. matrix F must be a row vector c. matrix F must be a square matrix d. matrix F can be of any size

d. matrix F can be of any size

In solving an engineering problem, the sequence of the four major steps is: a. model, validate, formulate, solve b. model, solve, formulate, validate c. formulate, model, solve, validate d. model, formulate, solve, validate

d. model, formulate, solve, validate

The following matrix is a | 1 -0.5 -0.5 | | 0 1 -0.5 | | 0 0 1 | a. diagonal matrix b. lower triangular matrix c. symmetric matrix d. upper triangular matrix

d. upper triangular matrix

Assuming that α ≠ ± 1/3, the solution of the following system is | 2 -6α | |x| = | 3 | | 3α -1 | |y| | 1.5 | a. x1 = -1.5 / (1 + 3α), x2 = +1.5 / (1 + 3α) b. x1 = -1.5 / (1 - 3α), x2 = +1.5 / (1 + 3α) c. x1 = +1.5 / (1 - 3α), x2 = -1.5 / (1 - 3α) d. x1 = +1.5 / (1 + 3α), x2 = -1.5 / (1 + 3α)

d. x1 = +1.5 / (1 + 3α), x2 = -1.5 / (1 + 3α)

For all real vectors x, which of the following is true? a. √n ∥x∥∞ ≤ ∥x∥2 b. ∥x∥∞ < ∥x∥2 c. ∥x∥∞ ≥ ∥x∥2 d. ∥x∥∞ ≤ ∥x∥2

d. ∥x∥∞ ≤ ∥x∥2

After transformation by Gauss-Jordan Elimination method, we have the following equation. The solution is | 2 0 0 | |x| = | 2 | | 0 5 0 | |y| | 10 | | 0 0 4 | |z| | 12 |

x = 1, y = 2, z = 3

Given the following matrix A. Its inverse A-1 is A =⎢2 -2 5 | | 2 3 4 | | 3 −1 3⎥

| -0.317 -0.024 0.561 | | -0.146 0.220 -0.049 | | 0.268 0.098 -0.244 |

The product A * B of the following two matrices is A = | 2 0 0 | | 0 -4 0 | | 6 0 4 | B= | 1 -0.5 -0.5 | | 0 1 -0.5 | | 0 0 1 |

| 2 -1 -1 | | 0 -4 2 | | 6 -3 1 |

The inverse A-1 of matrix A is A = | 0.25 0.10 | | 0.12 0.41 |

| 4.53 -1.10 | | -1.33 2.76 |