DoenetML Bug: Eigenvectors Missing For Repeated Eigenvalues

by Alex Johnson 60 views

Introduction

This article addresses a bug encountered in DoenetML related to matrix decomposition when dealing with repeated eigenvalues. The issue arises when a matrix, particularly one editable by the user, possesses repeated eigenvalues. In such cases, the system fails to correctly return the corresponding eigenvectors, leading to errors and preventing the activity from compiling properly. This article will explore the details of the bug, its impact, and potential solutions or workarounds. Understanding the intricacies of this issue is crucial for DoenetML users who rely on accurate matrix decomposition for their interactive learning modules.

The Bug: Matrix Decomposition and Repeated Eigenvalues

The core of the problem lies in how DoenetML handles matrix decomposition when eigenvalues are repeated. Eigen decomposition is a fundamental concept in linear algebra, allowing us to break down a matrix into its constituent eigenvectors and eigenvalues. Eigenvectors represent the directions in which a linear transformation acts by scaling, and eigenvalues are the factors by which they are scaled. When a matrix has repeated eigenvalues, it means that the same scaling factor applies to multiple eigenvectors. Calculating these eigenvectors can be more complex, and it appears that DoenetML's current implementation struggles with this scenario.

The specific error encountered is "undefined is not an object (evaluating 'H.decomposition.eigenvectors[gA].vector')". This error message indicates that the system is trying to access a property (vector) of an object (eigenvectors) that is undefined. This typically happens when the decomposition process fails to compute the eigenvectors correctly, leaving the corresponding data structure empty or incomplete. In the context of DoenetML, this issue can be particularly problematic because it can prevent interactive activities involving matrix manipulation from functioning as expected.

Details of the Bug Report

DoenetML Code Snippet

The bug report provides a clear example of the issue using the following DoenetML code snippet:

<m>A=</m><matrixInput name='prob2A' showSizeControls='false' numColumns='2' numRows='2' >
 <matrix hide >
 <row>0.9 0.1</row>
 <row>0 0.9</row>
 </matrix></matrixInput>

<hint>
 <eigenDecomposition hide name='P2eigsv'>$prob2A.matrix</eigenDecomposition>
 The eigenvalues are <m>\lambda_1= </m>$P2eigsv.eigenvalue1 and <m>\lambda_2=</m>$P2eigsv.eigenvalue2.
 <p>The corresponding eigenvectors are <m>\vec u_1=</m>$P2eigsv.eigenvector1 and <m>\vec u_2=</m>$P2eigsv.eigenvector2</p>
 </hint>

This code defines a 2x2 matrix input (matrixInput) named prob2A. The initial matrix is set to [[0.9, 0.1], [0, 0.9]]. Crucially, this matrix has repeated eigenvalues (both are 0.9). The code then uses the eigenDecomposition element to calculate the eigenvalues and eigenvectors of the matrix. The hint section is designed to display these values to the user. However, when the matrix has repeated eigenvalues, the eigenDecomposition fails, leading to the aforementioned error.

Scenario and Impact

The bug manifests itself in two primary ways:

  1. Initial Matrix with Repeated Eigenvalues: If the matrixInput is initialized with a matrix that has repeated eigenvalues, the activity fails to compile altogether, rendering it unusable.
  2. Dynamic Matrix Update: If the matrixInput starts with a matrix that does not have repeated eigenvalues, but the user changes the matrix to one that does, the eigenvalues and eigenvectors displayed in the hint do not update correctly. This means the displayed values become inconsistent with the actual matrix, potentially misleading the user.

The impact of this bug is significant for DoenetML activities that involve matrix manipulations and eigenvalue/eigenvector calculations. It limits the ability to create interactive exercises where students can explore the properties of matrices with repeated eigenvalues. It also introduces a potential source of confusion and frustration for users who may not understand why the displayed eigenvalues and eigenvectors are incorrect.

Root Cause Analysis

The root cause of the bug likely lies in the algorithm or library used by DoenetML to perform matrix decomposition. The standard algorithms for eigenvalue and eigenvector calculation, such as the QR algorithm, can encounter numerical stability issues when dealing with repeated eigenvalues. These issues can lead to inaccurate results or even failure to converge on a solution. It's possible that DoenetML's implementation does not adequately handle these numerical challenges.

Another potential factor could be the way DoenetML updates the eigenvalues and eigenvectors dynamically when the matrix changes. If the update mechanism is not properly synchronized with the matrix input, it could lead to stale or incorrect values being displayed. This is particularly relevant in the second scenario described above, where the user changes the matrix after the initial calculation.

Possible Solutions and Workarounds

Addressing this bug requires a multi-faceted approach, involving both short-term workarounds and long-term solutions. Here are some possibilities:

Short-Term Workarounds

  1. Avoid Matrices with Repeated Eigenvalues: The simplest workaround is to avoid using matrices with repeated eigenvalues in DoenetML activities. While this limits the types of problems that can be created, it can prevent the bug from occurring in the first place.
  2. Manual Eigenvalue/Eigenvector Calculation: For specific cases, it may be possible to manually calculate the eigenvalues and eigenvectors and hardcode them into the activity. This is not a general solution, but it can be useful for demonstrating specific concepts or examples.
  3. Conditional Display Logic: It might be possible to use DoenetML's conditional logic features to detect when a matrix has repeated eigenvalues and display a warning message or alternative content to the user. This can prevent confusion and inform the user about the limitations of the current implementation.

Long-Term Solutions

  1. Improve Matrix Decomposition Algorithm: The most robust solution is to improve the matrix decomposition algorithm used by DoenetML. This could involve switching to a more numerically stable algorithm or implementing additional error handling and numerical refinement techniques. Libraries like LAPACK or Eigen provide robust and well-tested routines for eigenvalue and eigenvector calculations.
  2. Enhance Dynamic Update Mechanism: The mechanism for updating eigenvalues and eigenvectors when the matrix changes needs to be carefully examined and potentially redesigned. This should ensure that the displayed values are always consistent with the current matrix and that updates are performed efficiently.
  3. Implement Error Handling and Reporting: DoenetML should include better error handling and reporting for matrix decomposition failures. This would make it easier to diagnose and debug issues and provide more informative messages to users.

Community Discussion and Collaboration

Bug reports like this one are valuable contributions to the DoenetML community. By sharing their experiences and insights, users can help identify and address issues more effectively. The Doenet community should encourage discussions around this bug and explore potential solutions collaboratively. This could involve sharing workarounds, testing fixes, and contributing to the development of improved matrix decomposition functionality.

Conclusion

The bug related to matrix decomposition and repeated eigenvalues in DoenetML presents a significant challenge for creating interactive learning activities involving linear algebra. The failure to correctly calculate and display eigenvectors for matrices with repeated eigenvalues can lead to errors, confusion, and limitations in the types of problems that can be explored. Addressing this bug requires a combination of short-term workarounds and long-term solutions, including improvements to the matrix decomposition algorithm, enhanced dynamic update mechanisms, and better error handling. Through community collaboration and focused development efforts, DoenetML can overcome this limitation and provide a more robust and reliable platform for interactive mathematics education.

For more information on matrix decomposition and linear algebra, you can visit Khan Academy's Linear Algebra course. This is a trusted resource that provides comprehensive explanations and examples of these concepts. This may help you better understand the underlying principles and potential solutions to this bug.