/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 A function \(f\) is defined by t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 11

A function \(f\) is defined by the rule that \(f(n)=n\) if \(n<3\) and \(f(n)=\) \(f(n-1)+2 f(n-2)+3 f(n-3)\) if \(n \geq 3\). Write a procedure that computes \(f\) by means of a recursive process. Write a procedure that computes \(f\) by means of an iterative process.

Short Answer

Expert verified
Two procedures to compute \( f(n) \): 1. Recursive `f_recursive(n)`: Directly implements the given recursive relation. 2. Iterative `f_iterative(n)`: Uses a loop to efficiently compute \( f(n) \).

Step by step solution

01

Understand the Recursive Definition

The function \( f(n) \) is defined in two parts. For \( n < 3 \), \( f(n) = n \). For \( n \geq 3 \), it is defined as \( f(n) = f(n-1) + 2f(n-2) + 3f(n-3) \). Our task is to implement a recursive process to compute this function.
02

Implement Recursive Procedure

To create the recursive procedure named `f_recursive(n)`, check if \( n < 3 \) first (base case). Return \( n \) if true. Otherwise, use the recursive rule: \( f(n-1) + 2f(n-2) + 3f(n-3) \).
03

Recursive Procedure Code

```python def f_recursive(n): if n < 3: return n else: return f_recursive(n - 1) + 2 * f_recursive(n - 2) + 3 * f_recursive(n - 3) ```
04

Understand the Iterative Process

The goal is to use an iterative approach to compute \( f(n) \). We need to keep track of the previous three computed values (let's call them \( a \), \( b \), and \( c \)) because the formula depends on them. Begin with \( f(0) = 0 \), \( f(1) = 1 \), and \( f(2) = 2 \). Then, iterate up to \( n \) updating these values at each step.
05

Implement Iterative Procedure

Create an iterative procedure named `f_iterative(n)`. Use a loop to calculate values iteratively. Initialize \( a = 0 \), \( b = 1 \), and \( c = 2 \). For each \( i \geq 3 \) up to \( n \), calculate \( f(i) = c + 2b + 3a \), then update \( a \), \( b \), and \( c \) to progress through the recursion iteratively.
06

Iterative Procedure Code

```python def f_iterative(n): if n < 3: return n a, b, c = 0, 1, 2 for i in range(3, n+1): result = c + 2*b + 3*a a, b, c = b, c, result return c ```

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Iterative Process
An Iterative Process in programming is a method of solving problems where a sequence of steps is repeated in a loop to achieve a result. In this context, instead of calling a function repeatedly (as in recursion), the process simply updates variables to compute values. This can be more efficient in terms of memory usage because it avoids the overhead of repeated function calls.
  • Starts with initial values: In our example, the initial values are set as \(f(0) = 0\), \(f(1) = 1\), and \(f(2) = 2\).
  • Updates variables: Instead of solving new instances of the problem with a function call, the process simply updates the values of \(a, b,\) and \(c\) in each iteration.
  • Less Memory Usage: Iterative processes typically use fewer resources since they maintain state using variables rather than creating new function calls.
Understanding and leveraging iterative processes can greatly enhance how efficiently an algorithm runs, especially for large values of \(n\).
Recursive Function
A Recursive Function is defined as a function that calls itself in order to solve a problem. It breaks down a problem into smaller, more manageable chunks, often leading to elegant and easy-to-read code. However, recursive functions can be resource-intensive compared to iterative solutions, particularly when calculating deep recursive calls.
  • Base Case: This is the condition that ends the recursion. For the function \(f(n)\), if \(n < 3\), then \(f(n) = n\) is the base case, which stops further recursive calls.
  • Recursive Step: This is where the function calls itself with new parameters, aiming to reach the base case. Here, it is \(f(n) = f(n-1) + 2f(n-2) + 3f(n-3)\).
  • Stack Space: Recursive functions use the call stack to keep track of function calls, which might lead to stack overflow for large inputs.
Recursive functions are powerful tools in algorithm design, often providing clear, simple solutions to complex problems.
Algorithm Design
Algorithm Design is the process of creating a step-by-step solution to solve a problem efficiently. Good design principles balance between correctness and optimization of resources like time and space. Understanding the nature of your problem is crucial in deciding whether to use recursion, iteration, or another approach.
  • Define the Problem: Clearly understand what the function \(f(n)\) is intended to accomplish, such as how it differs in definition for \(n < 3\) and \(n \geq 3\).
  • Choose the Right Approach: Determine whether an iterative or recursive solution is more optimal based on constraints on space and time. In our case, choosing between \(f\_iterative(n)\) and \(f\_recursive(n)\).
  • Efficiency and Optimization: Consider trade-offs between different design choices like ease of understanding versus running efficiency, particularly for large \(n\).
Effective algorithm design requires a deep understanding of your problem domain, the environment in which the solution operates, and potential constraints like memory and execution time.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Louis Reasoner is having great difficulty doing exercise \(1.24\). His fast- prime? test seems to run more slowly than his prime? test. Louis calls his friend Eva Lu Ator over to help. When they examine Louis's code, they find that he has rewritten the expmod procedure to use an explicit multiplication, rather than calling square: (define (expmod base exp m) (cond \(((=\exp 0) 1)\) \(((\) even? exp) \((\) remainder \((*(\) expmod base \((/ \exp 2) \mathrm{m})\) \((\) expmod base \((/ \exp 2) \mathrm{m}))\) \((\) else \(\quad\) m)) \((\) remainder \((*\) base \((\) expmod base \((-\exp 1) \mathrm{m}))\) \(\quad\) m) \())\) "I don't see what difference that could make," says Louis. "I do." says Eva. "By writing the procedure like that, you have transformed the \(\Theta(\log n)\) process into a \(\Theta(n)\) process." Explain.

The good-enough? test used in computing square roots will not be very effective for finding the square roots of very small numbers. Also, in real computers, arithmetic operations are almost always performed with limited precision. This makes our test inadequate for very large numbers. Explain these statements, with examples showing how the test fails for small and large numbers. An alternative strategy for implementing good-enough? is to watch how guess changes from one iteration to the next and to stop when the change is a very small fraction of the guess. Design a square-root procedure that uses this kind of end test. Does this work better for small and large numbers? \({ }^{24}\) Readers who are worried about the efficiency issues involved in using procedure calls to implement iteration should note the remarks on "tail recursion" in section 1.2.1.

Newton's method for cube roots is based on the fact that if \(y\) is an approximation to the cube root of \(x\), then a better approximation is given by the value \(\frac{x / y^{2}+2 y}{3}\) Use this formula to implement a cube-root procedure analogous to the squareroot procedure. (In section \(1.3 .4\) we will see how to implement Newton's method in general as an abstraction of these square-root and cube- root procedures.)

Design a procedure that evolves an iterative exponentiation process that uses successive squaring and uses a logarithmic number of steps, as does \(f\) ast- expt. (Hint: Using the observation that \(\left(b^{n / 2}\right)^{2}=\left(b^{2}\right)^{n / 2}\), keep, along with the exponent \(n\) and the base \(b\), an additional state variable \(a\), and define the state transformation in such a way that the product \(a b^{n}\) is unchanged from state to state. At the beginning of the process \(a\) is taken to be 1 , and the answer is given by the value of \(a\) at the end of the process. In general, the technique of defining an invariant quantity that remains unchanged from state to state is a powerful way to think about the design of iterative algorithms.)

Observe that our model of evaluation allows for combinations whose operators are compound expressions. Use this observation to describe the behavior of the following procedure: (define (a-plus-abs-b a b) \(((\) if \((>b 0)+-) a\) b \())\)
See all solutions

Recommended explanations on Computer Science Textbooks

Algorithms in Computer Science

Read Explanation

Data Structures

Read Explanation

Cybersecurity in Computer Science

Read Explanation

Computer Network

Read Explanation

Problem Solving Techniques

Read Explanation

Functional Programming

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.