/*! 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 8 Suppose \(a \in \mathbf{R}^{n}\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 8

Suppose \(a \in \mathbf{R}^{n}\) is given and that \(A \in \mathbf{R}^{n \times n}\) has the property that \(\left.a_{i j}=a_{((i+j-1)} \bmod n\right)+1\), Give an algorithm that overwrites \(y\) with \(y+A x\) where \(x, y \in \mathbf{R}^{n}\) are given.

Short Answer

Expert verified
Calculate \( z = Ax \) and update each \( y_i \) by adding \( z_i \).

Step by step solution

01

Understand the Matrix A

Matrix \( A \) is an \( n \times n \) matrix where each element \( a_{ij} \) is defined by \( a_{((i+j-1) \mod n) + 1} \). This means the elements are determined by a cyclic pattern based on the vector \( a \).
02

Compute Matrix-Vector Product Ax

To compute \( Ax \), for each \( i \) from 1 to \( n \), calculate \( z_i = \sum_{j=1}^{n} a_{ij} x_j \). Here, \( a_{ij} = a_{((i+j-1) \mod n) + 1} \) as per the definition.
03

Update the Vector y

Overwrite the vector \( y \) by adding the result from Step 2 so that \( y \gets y + z \), where \( z = Ax \). For each \( i \), update \( y_i \) with \( y_i + z_i \) using the values calculated in Step 2.

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.

Matrix-Vector Product
When it comes to matrices and vectors, one crucial operation is the *matrix-vector product*. This operation is a way to transform a vector using a matrix. Here's a simple breakdown:
  • Consider a matrix \( A \) with dimensions \( n \times n \), meaning it has \( n \) rows and \( n \) columns.
  • Also have a vector \( x \) with \( n \) elements.
  • The product \( Ax \) results in a new vector, say \( z \), also with \( n \) elements.
To find each component \( z_i \) of the resulting vector \( z \), you need to calculate a sum of products from the corresponding row of \( A \) and the vector \( x \): \[ z_i = \sum_{j=1}^{n} a_{ij} x_j \]This basically scales each element in \( x \) by the corresponding values in a row of \( A \) and then sums them up. This operation is central to many algorithms and is used extensively in fields such as computer graphics, physics simulations, and machine learning.
Cyclic Pattern
In mathematics, a cyclic pattern involves a repeating sequence, often seen in operations like matrix fills or during programming loops. Here鈥檚 how this concept applies to our problem:
  • The matrix \( A \) in the problem is constructed using a cyclic pattern based on the vector \( a \).
  • Each element \( a_{ij} \) of the matrix \( A \) is determined by repeatedly cycling through the vector \( a \).
  • The formula \( a_{ij} = a_{((i+j-1) \mod n) + 1} \) utilizes modular arithmetic to "wrap around" the index as it exceeds \( n \), ensuring that the indices never fall outside the bounds of vector \( a \).
This cyclic use of indices helps to either create matrices with specific symmetrical properties or efficiently manage datasets in programming scenarios. It's akin to having a wheel of numbers that keeps rotating, never stopping even when reaching the end.
Update Vector Algorithm
After calculating the matrix-vector product in the exercise, the next step is to update a given vector using an *update vector algorithm*. Let's understand this step clearly:
  • First, you calculate the product \( Ax = z \) using the established matrix-vector product method.
  • Next, you need to "add" this result to another vector \( y \), essentially updating it with the changes implied by this product.
  • For each component, you perform the addition: \( y_i \leftarrow y_i + z_i \).
This operation, \( y \gets y + Ax \), can be visualized as a shift or move from one set of values (the original \( y \)) to another by incorporating the transformation \( Ax \) has brought. It鈥檚 like adding a new layer or update over existing data, a process fundamental in areas such as iterative algorithms, simulations, and differential equation solvers.

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

Specify an algorithm that computes the first column of the matrix \(M=\left(A-\lambda_{1} I\right) \cdots\left(A-\lambda_{r} I\right)\) where \(A \in \mathbf{R}^{n \times n}\) is upper Hessenberg and \(\lambda_{1}, \ldots, \lambda_{r}\) are given scalars. How many flops are required assuming that \(r \ll n ?\)

Show that if \(B\) and \(C\) are each permutation matrices, then \(B \otimes C\) is also a permutation matrix.

Suppose \(D=A B C\) where \(A \in \mathbf{R}^{m \times n}, B \in \mathbf{R}^{n \times p}\), and \(C \in \mathbf{R}^{p \times s}\), Compare the flop count of an algorithm that computes \(D\) via the formula \(D=(A B) C\) versus the flop count for an algorithm that computes \(D\) using \(D=A(B C)\). Under what conditions is the former procedure more flop-efficient than the latter?

Suppose \(a \in \mathbf{R}^{n}\) is given and that \(A \in \mathbf{R}^{n \times n}\) has the property that \(a_{i j}=a_{|i-j|+1}\). Give an algorithm that overwrites \(y\) with \(y+A x\) where \(x, y \in \mathbf{R}^{n}\) are given.

Adapt strass so that it can handle nonsquare products, e.g., \(C=A B\) where \(A \in \mathbf{R}^{m \times r}\) and \(B \in \mathbf{R}^{r \times n}\). Is it better to augment \(A\) and \(B\) with zeros so that they become square and equal in size, or to "tile" \(A\) and \(B\) with square submatrices?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.