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Related Concept Videos

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
Design Example: Underdamped Parallel RLC Circuit01:17

Design Example: Underdamped Parallel RLC Circuit

Consider designing an oscillator circuit, a crucial component in various electronic devices and systems. The objective is to create an oscillator circuit with specific characteristics: a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. To accomplish this, a parallel RLC circuit is employed, known for its ability to sustain oscillations at a resonant frequency. In this case, the damping factor is pivotal in achieving the desired performance.
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RLC Circuit as a Damped Oscillator01:30

RLC Circuit as a Damped Oscillator

An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
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Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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Standing Waves in a Cavity01:28

Standing Waves in a Cavity

A household microwave and lasers are examples of standing electromagnetic waves in a cavity. When two conducting metal plates are placed parallel at the nodal planes, it creates a cavity where standing waves are formed. The cavity between the two planes is analogous to a stretched string held at the points x = 0 and x = L. Here, the distance 'L' between the two planes must be an integer multiple of half of the wavelength. The wavelengths that satisfy this condition are given by:
Gaussian Elimination: Problem Solving01:30

Gaussian Elimination: Problem Solving

Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...

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Fabrication and Characterization of High-Q Silicon Nitride Membrane Resonators
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Published on: August 8, 2025

Numerical procedures for solving nonsymmetric eigenvalue problems associated with optical resonators.

W D Murphy, M L Bernabe

    Applied Optics
    |March 6, 2010
    PubMed
    Summary
    This summary is machine-generated.

    The Prony method now efficiently solves nonsymmetric eigenvalue problems and automatically identifies dominant eigenvalues. This advancement enables accurate computation of eigenvectors for complex optical resonator designs.

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    The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry

    Published on: August 12, 2013

    Area of Science:

    • Numerical analysis
    • Computational physics
    • Optical engineering

    Background:

    • The algebraic eigenvalue problem is fundamental in many scientific disciplines.
    • Existing methods for eigenvalue problems may have limitations in handling nonsymmetric matrices or determining the number of dominant eigenvalues.
    • Accurate computation of eigenvalues and eigenvectors is crucial for analyzing physical systems, such as optical resonators.

    Purpose of the Study:

    • To extend the Prony method for solving nonsymmetric algebraic eigenvalue problems.
    • To enhance the Prony method for automatic detection of the number of dominant eigenvalues.
    • To develop an iterative algorithm for computing associated eigenvectors and assess the accuracy of matrix approximations.

    Main Methods:

    • Extension of the Prony method to address nonsymmetric eigenvalue problems.
    • Development of an iterative algorithm for eigenvector computation.
    • Utilization of the QR method for resolution studies and accuracy assessment of matrix approximations.

    Main Results:

    • The extended Prony method successfully handles nonsymmetric eigenvalue problems.
    • The improved method automatically determines the number of dominant eigenvalues.
    • Numerical results demonstrate the accuracy of the method for both simple and complex optical resonator designs, including those with multiple propagation geometries and misaligned mirrors.

    Conclusions:

    • The enhanced Prony method provides an effective and accurate approach for solving nonsymmetric eigenvalue problems.
    • The automatic search for dominant eigenvalues simplifies the analysis of complex systems.
    • The method's applicability to advanced optical resonator designs highlights its practical significance in optical engineering and computational physics.