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Related Concept Videos

Lagrange Multipliers: Problem Solving01:30

Lagrange Multipliers: Problem Solving

A silo with a cylindrical base, flat bottom, and hemispherical roof is a common design in agricultural and industrial storage due to its structural efficiency and ease of construction. Optimizing its dimensions to maximize storage capacity for a given amount of material—i.e., a fixed surface area—is a classic problem in applied calculus and engineering design. The key parameters are the radius r of the base and the height h of the cylindrical section.The total volume of the silo is obtained 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...
Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...
Construction of Root Locus01:15

Construction of Root Locus

The construction of a root locus involves several key steps to analyze and visualize the behavior of a system's poles with varying gain. The number of branches in the root locus equals the number of closed-loop poles and is symmetrical about the real axis.
For positive gain values, the root locus exists on the real axis to the left of an odd number of finite open-loop poles or zeros. The root locus starts at the open-loop poles and traces the paths of the closed-loop poles as the gain increases.
Quadratic Equations01:29

Quadratic Equations

A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...
Lagrange Multipliers: One Constraint01:29

Lagrange Multipliers: One Constraint

In constrained optimization, the objective is to maximize or minimize a quantity while satisfying a fixed condition. A standard example is a rectangular pen built against a barn wall using 100 meters of fencing. Because the wall provides one side of the enclosure, only the other three sides require fencing. The problem is to find the dimensions that produce the greatest possible area.Let L represent the length parallel to the wall and W the width perpendicular to it. The area of the pen is A =...

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Related Experiment Video

Updated: Jun 23, 2026

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

Implicit restart Lanczos as an eigensolver.

Reza Rajaie Khorasani1, Randall S Dumont

  • 1Department of Chemistry, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4M1, Canada.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 28, 2009
PubMed
Summary
This summary is machine-generated.

Implicit restart Lanczos algorithms are more efficient for large-scale molecular physics computations than simple Lanczos. A modified version offers memory savings and broader applicability for finding eigenvalues.

Related Experiment Videos

Last Updated: Jun 23, 2026

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

Area of Science:

  • Computational Physics
  • Quantum Chemistry
  • Molecular Physics

Background:

  • Large-scale computations in molecular and chemical physics require efficient eigensolvers.
  • Traditional simple Lanczos algorithms face limitations in efficiency and memory usage.

Purpose of the Study:

  • To investigate and compare the efficiency of implicit restart Lanczos and simple Lanczos algorithms for large-scale computations.
  • To present a modified implicit restart Lanczos algorithm with enhanced capabilities.

Main Methods:

  • Utilizing the cardioid billiard and the hydrogen cyanide/hydrogen isocyanide (HCN/HNC) molecule as model systems.
  • Implementing and comparing implicit restart Lanczos and simple (without reorthogonalization) Lanczos algorithms.
  • Developing and testing a modified implicit restart Lanczos algorithm.

Main Results:

  • Implicit restart Lanczos demonstrates superior efficiency compared to the simple Lanczos algorithm.
  • The modified implicit restart Lanczos algorithm reduces memory requirements by using a smaller Krylov space.
  • The modified algorithm can handle larger basis sets and facilitates the retrieval of all eigenpairs or eigenvalues below a threshold.

Conclusions:

  • Implicit restart Lanczos algorithms are highly efficient for molecular and chemical physics eigensolving.
  • The modified implicit restart Lanczos algorithm offers significant advantages in memory efficiency and flexibility for large-scale quantum calculations.