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Chapter 1: Problem 11

All of these equations are mathematically true. Nonetheless, floating point error causes some of them to be false according to Octave. Which ones? HINT: Use the boolean operator wo to check. For example, to check if \(\sin (0)=0,\) type \(\sin (0)==0\) into Octave. ans=1 means true (the two sides are equal according to Octave - no round-off error) and ans=0 means false (the two sides are not equal according to Octave round-off error). (a) \((2)(12)=9^{2}-4(9)-21\) (b) \(e^{3 \ln (2)}=8\) (c) \(\ln (10)=\ln (5)+\ln (2)\) (d) \(g\left(\frac{1+\sqrt{5}}{2}\right)=\frac{1+\sqrt{5}}{2}\) where \(g(x)=\sqrt[3]{x^{2}+x}\) (e) \(\lfloor 153465 / 3\rfloor=153465 / 3\) (f) \(3 \pi^{3}+7 \pi^{2}-2 \pi+8=((3 \pi+7) \pi-2) \pi+8\)

Short Answer

Expert verified
Equations (c) and (f) are likely false due to floating-point errors in Octave.

Step by step solution

01

Checking Equation (a) with Octave

Write the equation from part **(a)** in Octave: `2*12 == 9^2 - 4*9 - 21`. Run this command. If `ans=1`, it implies the equation holds true in Octave. If `ans=0`, it suggests a floating-point error.
02

Checking Equation (b) with Octave

Write the equation from part **(b)** as `exp(3*log(2)) == 8` and enter this into Octave. Utilize Octave's `exp` for the exponent and `log` for the natural logarithm. Observe the output; `ans=1` indicates truth, whereas `ans=0` shows a lack of precision agreement.
03

Checking Equation (c) with Octave

Enter the equation from part **(c)** using the command `log(10) == log(5) + log(2)`. Logarithms are typically sensitive to floating-point representation, so verify whether Octave returns `ans=1` for truth or `ans=0` for a floating-point discrepancy.
04

Checking Equation (d) with Octave

For part **(d)**, calculate `g((1+sqrt(5))/2)` by defining `g(x) = nthroot(x^2+x, 3)`. Evaluate the equation using `g( (1+sqrt(5))/2 ) == (1+sqrt(5))/2`. Check if Octave's response is `ans=1` (true) or `ans=0` (false).
05

Checking Equation (e) with Octave

Type `floor(153465/3) == 153465/3` to evaluate part **(e)**. The `floor` function rounds down to the nearest integer. Determine whether `ans=1` implies correctness or `ans=0` shows a discrepancy due to floating-point division.
06

Checking Equation (f) with Octave

For part **(f)**, input `(3*pi^3 + 7*pi^2 - 2*pi + 8) == ((3*pi + 7)*pi - 2)*pi + 8` to check the identity. Compute both sides to identify if Octave considers them equal (`ans=1`) or not (`ans=0`) due to floating-point arithmetical errors.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Octave Programming
Octave is a high-level programming language primarily intended for numerical computations. It's especially beneficial for students and professionals who are dealing with mathematical problems.

One of Octave's strengths lies in its ability to quickly evaluate, test, and verify mathematical equations. With straightforward commands such as `sin`, `log`, and `exp`, you can efficiently work out complex mathematical challenges.
  • Typing commands: You can enter mathematical expressions directly into the Octave command line.
  • Using operators: Compare different mathematical statements with relational operators like `==` for equality checks.
Overall, Octave simplifies the process of solving and verifying mathematical algorithms, making it a convenient tool for anyone learning or working in various fields of mathematics and engineering.
Mathematical Equations
Mathematical equations are statements that assert the equality of two expressions. They are foundational in many fields, including physics, engineering, and computer science.

Equations can be simple, like linear equations, or they can be more complex, involving exponential, logarithmic, or trigonometric functions.
  • Linear Equations: These have the standard form of ax + b = 0 and are fundamental in algebra.
  • Logarithmic and Exponential Equations: Equations involving log functions or exponential growth and decay are commonly used in science.
  • Trigonometric Equations: Often used in geometry and physics, involving functions such as sine, cosine, and tangent.
Understanding equations is crucial for verifying their correctness or determining solutions under various conditions, as shown when using Octave for numerical check-ups.
Numerical Analysis
Numerical analysis is a branch of mathematics focusing on algorithms for solving problems with continuous variables. It involves approximating solutions for mathematical problems that cannot be solved exactly.

In Octave, you can use numerical analysis techniques to evaluate and verify mathematical equations effectively.
  • Error Analysis: It's crucial to understand potential errors from the representation of numbers and operations.
  • Algorithm Efficiency: Assessing how quickly and efficiently an algorithm can solve a problem is part of numerical analysis.
  • Approximations: Techniques such as interpolation and numerical integration help in estimating solutions.
By applying these principles, you can identify and minimize floating-point errors, allowing for more accurate and reliable results in computational mathematics.
Floating Point Representation
Floating point representation is a way to encode real numbers that supports a wide range of values. It is vital in computing and numerical analysis, especially when dealing with very large or very small numbers.

However, this representation is not without its faults, often causing small errors due to its limited precision.
  • Precision Limits: Floating point numbers have finite precision which can lead to rounding errors.
  • Underflow and Overflow: When numbers are too small or too large to be represented within the available precision.
  • Comparing Values: Checking equality with floating-point numbers can often result in inaccuracies.
Understanding these limitations is crucial in Octave programming, where slight errors might cause a mathematical equation to evaluate incorrectly. This highlights the importance of numerical analysis to detect and correct such floating-point errors.

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Most popular questions from this chapter

Compute the absolute and relative errors in the approximation of \(\pi\) by \(3 .\)

(The number in question 7 is an approximation of \(1 / \pi .\) Using Octave, find the absolute and relative errors in the approximation.

The Fibonacci sequence, \(\left\langle F_{n}\right\rangle,\) is recursively defined by $$ \begin{aligned} F_{n+1} &=F_{n}+F_{n-1}, \quad n \geq 1 \\ F_{0} &=1 \\ F_{1} &=1 \end{aligned} $$ so the first few terms are 1,1,2,3,5,8 . (a) Write a recursive function that calculates the \(n^{t h}\) Fibonacci number. Your function should have one argument, \(n\). (b) Write a function that uses a for loop to calculate the \(n^{\text {th }}\) Fibonacci number. Your function should have one argument, \(n\). (c) Write a program that calls the function from \(5 \mathrm{a}\) to calculate \(F_{30}\) (d) Write a program that calls the function from \(5 \mathrm{~b}\) to calculate \(F_{30}\) (e) Which code is simpler (recursive or nonrecursive)? (f) Which code is faster? (g) Which code is more accurate?

The Tower of Hanoi, part 1. The Tower of Hanoi is a game played with a number of different sized disks stacked on a pole in decreasing size, the largest on the bottom and the smallest on top. There are two other poles, initially with no disks on them. The goal is to move the entire stack of disks to one of the initially empty poles following two rules. You are allowed to move only one disk at a time from one pole to another. You may never place a disk upon a smaller one. [3] (a) Starting with a stack of three disks, what is the minimum number of moves it takes to complete the game? Answer this question with a number. (b) Starting with a stack of four disks, what is the minimum number of moves it takes to complete the game? i. Answer this question recursively. ii. Answer this question with a number based on your recursive answer.

Tet \(f(x)=x^{3}\) (a) Find the second Taylor polynomial, \(P_{2}(x),\) about \(x_{0}=0\) (b) Find the remainder term, \(R_{2}(0.5),\) and the actual error in using \(P_{2}(0.5)\) to approximate \(f(0.5)\). (c) Repeat part (a) using \(x_{0}=1\). (d) Repeat part (b) using the polynomial from part (c).
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