1. What is the QR Algorithm?

The QR algorithm is an eigenvalue algorithm that computes the eigenvalues and eigenvectors of a matrix through iterative QR decompositions. It's one of the most important algorithms in numerical linear algebra.

2. How Does the Calculator Work?

The calculator implements the basic QR algorithm:

Where:

• \( A \) — Input square matrix
• \( Q \) — Orthogonal matrix from QR decomposition
• \( R \) — Upper triangular matrix from QR decomposition
• \( k \) — Iteration count

Explanation: The algorithm iteratively decomposes the matrix into Q and R factors, then multiplies them in reverse order until convergence to an upper triangular matrix with eigenvalues on the diagonal.

3. Importance of QR Algorithm

Details: The QR algorithm is fundamental for eigenvalue computations in scientific computing, structural analysis, quantum mechanics, and many other fields requiring matrix analysis.

4. Using the Calculator

Tips: Enter a square matrix using comma-separated values within rows and semicolon-separated rows. Specify the number of iterations (typically 10-50 for reasonable convergence).

5. Frequently Asked Questions (FAQ)

Q1: What's the convergence rate of QR algorithm?
A: The basic QR algorithm has linear convergence. With shifts, it can achieve cubic convergence.

Q2: When does QR algorithm fail?
A: It may fail for non-square matrices or matrices with complex eigenvalues if real arithmetic is used exclusively.

Q3: How to improve convergence?
A: Use shifts (Wilkinson shifts) or transform to Hessenberg form first to reduce computational cost.

Q4: What's the computational complexity?
A: O(n³) per iteration for a dense n×n matrix, though practical implementations are O(n²) per iteration for Hessenberg matrices.

Q5: Are there alternatives to QR algorithm?
A: Yes, including power iteration, Jacobi method, and divide-and-conquer approaches, but QR is generally preferred for most applications.