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Numerical methods for matrix eigenvalue calculation
is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Given an
Eigenvalue_algorithm
Numerical linear algebra algorithm
numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric
Jacobi_eigenvalue_algorithm
Algorithm to calculate eigenvalues
linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix
QR_algorithm
Numerical eigenvalue calculation
In 1988, Ojalvo produced a more detailed history of this algorithm and an efficient eigenvalue error test. Input a Hermitian matrix A {\displaystyle A}
Lanczos_algorithm
Algorithm on Hermitian matrices
Divide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa 1990s)
Divide-and-conquer eigenvalue algorithm
Divide-and-conquer_eigenvalue_algorithm
Iterative method for approximating eigenvectors
iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors
Arnoldi_iteration
Eigenvalue algorithm
known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A {\displaystyle A} , the algorithm will produce a number λ {\displaystyle
Power_iteration
Matrix decomposition
In linear algebra, eigendecomposition (also known as eigenvalue decomposition or EVD) is a factorization of a matrix A {\displaystyle A} into a canonical
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Quantum algorithm for eigenvalue estimation
estimation algorithm is a quantum algorithm to estimate the phase corresponding to an eigenvalue of a given unitary operator. Because the eigenvalues of a unitary
Quantum phase estimation algorithm
Quantum_phase_estimation_algorithm
Matrix decomposition
{\displaystyle M} . Two-sided Jacobi SVD algorithm—a generalization of the Jacobi eigenvalue algorithm—is an iterative algorithm where a square matrix is iteratively
Singular_value_decomposition
but not exactly, equal eigenvalues Convergent matrix — square matrix whose successive powers approach the zero matrix Algorithms for matrix multiplication:
List of numerical analysis topics
List_of_numerical_analysis_topics
Eigenvalue algorithm
an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates
Rayleigh_quotient_iteration
Concepts from linear algebra
Eigenmoments Eigenvalue algorithm Quantum states Jordan normal form List of numerical-analysis software Nonlinear eigenproblem Normal eigenvalue Quadratic
Eigenvalues_and_eigenvectors
Trigonometric interpolation Eigenvalue algorithms Arnoldi iteration Inverse iteration Jacobi method Lanczos iteration Power iteration QR algorithm Rayleigh quotient
List_of_algorithms
Method for approximating eigenvalues
enabling the use of a numerical eigenvalue algorithm. It is used in all applications that involve approximating eigenvalues and eigenvectors, often under
Rayleigh–Ritz_method
Matrix decomposition
squares (LLS) problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as A = Q R
QR_decomposition
Mathematical algorithm
an iterative eigenvalue algorithm. It allows one to find an approximate eigenvector when an approximation to a corresponding eigenvalue is already known
Inverse_iteration
Quantum algorithm for integer factorization
part of the algorithm. The gate thus defined satisfies U r = I {\displaystyle U^{r}=I} , which immediately implies that its eigenvalues are the r {\displaystyle
Shor's_algorithm
real symmetric matrix, A. It is the core operation in the Jacobi eigenvalue algorithm, which is numerically stable and well-suited to implementation on
Jacobi_rotation
Overview of and topical guide to algorithms
Karatsuba algorithm Schönhage–Strassen algorithm Gaussian elimination LU decomposition QR decomposition Singular value decomposition Eigenvalue algorithm Strassen
Outline_of_algorithms
Quantum search algorithm
In quantum computing, Grover's algorithm, also known as the quantum search algorithm, is a quantum algorithm for unstructured search that finds with high
Grover's_algorithm
Algorithm to be run on quantum computers
the ground-state eigenvector and eigenvalue of a Hermitian operator. The quantum approximate optimization algorithm takes inspiration from quantum annealing
Quantum_algorithm
Matrix equal to its conjugate-transpose
theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation
Hermitian_matrix
eigenvalue of matrices. The standard method for finding all roots of a polynomial in MATLAB uses the Francis QR algorithm to compute the eigenvalues of
Polynomial_root-finding
Clustering methods
A mathematically equivalent algorithm takes the eigenvector u {\displaystyle u} corresponding to the largest eigenvalue of the random walk normalized
Spectral_clustering
Number, approximately 3.14
form of the Dirichlet eigenvalue problem in one dimension, the Poincaré inequality is the variational form of the Neumann eigenvalue problem, in any dimension
Pi
German mathematician (1804–1851)
(PhD, Dr. habil.) Known for Popularizing the character ∂ Jacobi eigenvalue algorithm Jacobi ellipsoid Jacobi elliptic functions Jacobi identity Jacobi
Carl_Gustav_Jacob_Jacobi
Algorithm used by Google Search to rank web pages
by Karl T. Muth, the Hummingbird algorithm, and the SALSA algorithm. The eigenvalue problem behind PageRank's algorithm was independently rediscovered and
PageRank
American mathematician
an American engineer mainly known for the Arnoldi iteration, an eigenvalue algorithm used in numerical linear algebra. His main research interests included
Walter_Edwin_Arnoldi
Matrix with nonzero elements on the main diagonal and the diagonals above and below it
003. Dhillon, Inderjit Singh (1997). A New O(n2) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem (PDF) (PhD). University of California
Tridiagonal_matrix
Numerical simulations of physical problems via computers
difference method and relaxation method) matrix eigenvalue problem (using e.g. Jacobi eigenvalue algorithm and power iteration) All these methods (and several
Computational_physics
Topics referred to by the same term
avoirdupois QR decomposition, a decomposition of a matrix QR algorithm, an eigenvalue algorithm to perform QR decomposition Quadratic reciprocity, a theorem
QR
Branch of mathematics
of V such that f(v) = av for some scalar a in F. This scalar a is an eigenvalue of f. If the dimension of V is finite, and a basis has been chosen, f
Linear_algebra
Construct for Hermitian matrices
exact values of all eigenvalues. It is also used in eigenvalue algorithms (such as Rayleigh quotient iteration) to obtain an eigenvalue approximation from
Rayleigh_quotient
Quantum algorithm for solving systems of linear equations
The Harrow–Hassidim–Lloyd (HHL) algorithm is a quantum algorithm for obtaining certain limited information about the solution to a system of linear equations
HHL_algorithm
Directed graph with no directed cycles
proved, that the same numbers count the (0,1) matrices for which all eigenvalues are positive real numbers. The proof is bijective: a matrix A is an adjacency
Directed_acyclic_graph
Optimization algorithms using quantum computing
Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. Mathematical optimization deals with finding the
Quantum optimization algorithms
Quantum_optimization_algorithms
points). QR algorithm In numerical linear algebra, the QR algorithm is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors
List of inventions and discoveries by women
List_of_inventions_and_discoveries_by_women
Method of data analysis
eigenvalues of C. This step will typically involve the use of a computer-based algorithm for computing eigenvectors and eigenvalues. These algorithms
Principal_component_analysis
Degree of connectedness within a graph
the total number of vertices n. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector. Furthermore
Centrality
Algorithm to solve Wahba's problem
to efficiently solve the eigenvalue problem and construct a numerically stable representation of the solution. The algorithm was introduced by Malcolm
Quaternion estimator algorithm
Quaternion_estimator_algorithm
Mechanism in quantum computing
the eigenvalue of U {\displaystyle U} . Phase kickback allows a quantum setup to estimate eigenvalues exponentially quicker than classical algorithms. This
Phase_kickback
Topics referred to by the same term
the U.S. state of Washington MRRR algorithm (multiple relatively robust representations), an eigenvalue algorithm MRR (disambiguation) This disambiguation
MRRR
Methods for numerical approximations
phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm is based on the singular
Numerical_analysis
In mathematics, invariant of square matrices
the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry, the signed n-dimensional volume of a n-dimensional parallelepiped
Determinant
Function in discrete mathematics
linear combination of eigenvectors for the same eigenvalue is also an eigenvector for that eigenvalue. Various researchers have proposed different choices
Discrete_Fourier_transform
Topics referred to by the same term
Divide-and-conquer algorithm, in computer science Divide-and-conquer eigenvalue algorithm, in mathematics Divide and conquer algorithm for matrix multiplication
Divide and conquer (disambiguation)
Divide_and_conquer_(disambiguation)
Field of mathematics
used to solve linear least-squares problems, and eigenvalue problems (by way of the iterative QR algorithm). An LU factorization of a matrix A consists of
Numerical_linear_algebra
Statistical algorithm
Least mean squares (LMS) algorithms are a class of adaptive filter used to mimic a desired filter by finding the filter coefficients that relate to producing
Least_mean_squares_filter
Type of equation involving matrix-valued functions
nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically
Nonlinear_eigenproblem
Method used in statistics, pattern recognition, and other fields
where the larger the eigenvalue, the better the function differentiates. This however, should be interpreted with caution, as eigenvalues have no upper limit
Linear_discriminant_analysis
Kind of square matrix in linear algebra
triangular matrix, often economizes the arithmetic involved in the QR algorithm for eigenvalue problems. Any n × n {\displaystyle n\times n} matrix can be transformed
Hessenberg_matrix
Numerical analysis concept
an eigenvalue problem which is solved by the QR algorithm. This algorithm was popular, but significantly more efficient algorithms exist. Algorithms based
Gauss–Legendre_quadrature
Algorithm used for frequency estimation and radio direction finding
MUSIC (MUltiple SIgnal Classification) is an algorithm used for frequency estimation and radio direction finding. In many practical signal processing
MUSIC_(algorithm)
Square matrix used to represent a graph or network
adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph
Adjacency_matrix
Algorithm for finding zeros of functions
method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes)
Newton's_method
3.265. Kublanovskaya, Vera N. (1961). "On some algorithms for the solution of the complete eigenvalue problem". USSR Computational Mathematics and Mathematical
Timeline_of_algorithms
Concept in linear algebra
generalized eigenvalues of a pencil is called the generalized eigenvalue problem. The most popular algorithm for this task is the QZ algorithm, an implicit
Matrix_pencil
Quantum algorithm for counting solutions to search problems
with the two eigenvalues e ± i θ {\displaystyle e^{\pm i\theta }} . From here onwards, we follow the quantum phase estimation algorithm scheme: we apply
Quantum_counting_algorithm
Mathematical optimization algorithm
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose
Conjugate_gradient_method
Dimensionality reduction algorithm
accurate eigenvalues on both synthetic and experimental data sets. Exact DMD: The Exact DMD algorithm generalizes the original DMD algorithm in two ways
Dynamic_mode_decomposition
Measure in graph theory
the total number of vertices n. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector. Furthermore
Eigenvector_centrality
Algorithm in numerical linear algebra
{R} ^{m\times n}} , and assume that the eigenvalues of A {\displaystyle A} are distinct from the eigenvalues of B {\displaystyle B} . Then, the matrix
Bartels–Stewart_algorithm
Adaptive filter algorithm for digital signal processing
over conventional LMS algorithms such as faster convergence rates, modular structure, and insensitivity to variations in eigenvalue spread of the input
Recursive least squares filter
Recursive_least_squares_filter
Quantum algorithm framework
transformation". A variant of this algorithm can also be performed when A is Hermitian, corresponding to an "eigenvalue transformation". That is, given a
Quantum singular value transformation
Quantum_singular_value_transformation
Machine learning framework for portfolio construction
eigenvalues must be strictly positive. When the matrix is numerically ill-conditioned—that is, when the ratio of its largest to smallest eigenvalue (its
Hierarchical_Risk_Parity
Specialist field of computer science
methods Numerical linear algebra, including decompositions and eigenvalue algorithms Linear programming Branch and cut Branch and bound Molecular dynamics
Computational_science
Feature detection algorithm in computer vision
The scale-invariant feature transform (SIFT) is a computer vision algorithm to detect, describe, and match local features in images, invented by David
Scale-invariant feature transform
Scale-invariant_feature_transform
Eigenvalue problem for the Laplace operator
In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation:
Helmholtz_equation
Iterative method used to solve a linear system of equations
the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system
Jacobi_method
Algorithmic runtime requirements for matrix multiplication
Unsolved problem in computer science What is the fastest algorithm for matrix multiplication? More unsolved problems in computer science In theoretical
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
Mathematical algorithm
In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial
Faddeev–LeVerrier_algorithm
Grouping a set of objects by similarity
analysis refers to a family of algorithms and tasks rather than one specific algorithm. It can be achieved by various algorithms that differ significantly
Cluster_analysis
Solving an optimization problem with a quadratic objective function
negative-definite (has n negative eigenvalues); Pardalos and Vavasis proved (strong) NP-hardness whenever Q has at least one negative eigenvalue. They do this by showing
Quadratic_programming
Matrix factorisation in mathematics
similar to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix. The complex Schur decomposition reads as follows:
Schur_decomposition
Array of numbers
matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of its characteristic polynomial, det
Matrix_(mathematics)
Quantum algorithm
eigensolver (VQE) is a quantum algorithm for quantum chemistry, quantum simulations and optimization problems. It is a hybrid algorithm that uses both classical
Variational quantum eigensolver
Variational_quantum_eigensolver
Topics referred to by the same term
dominant system of linear equations Jacobi eigenvalue algorithm, a method for calculating the eigenvalues and eigenvectors of a real symmetric matrix
Jacobi
Methodic assignment of colors to elements of a graph
\lambda _{\max }(W),\lambda _{\min }(W)} are the largest and smallest eigenvalues of W {\displaystyle W} . Define χ H ( G ) = max W χ W ( G ) {\textstyle
Graph_coloring
strong resemblance to, the Lanczos algorithm for finding eigenvalues of large sparse real matrices. The algorithm is essentially not parallel: it is of
Block_Lanczos_algorithm
Polynomial whose roots are the eigenvalues of a matrix
a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its
Characteristic_polynomial
Search for an atomic arrangement with the lowest inter-atomic force
pre-requisites, a local optimization algorithm can then move "uphill" along the eigenvector with the most negative eigenvalue and "downhill" along all other
Energy_minimization
Statistical analysis technique
k-sparse largest eigenvalue. If one takes k=p, the problem reduces to the ordinary PCA, and the optimal value becomes the largest eigenvalue of covariance
Sparse_PCA
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
short for coordinate rotation digital computer, is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots
CORDIC
Method for finding largest (or smallest) eigenvalues
finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric generalized eigenvalue problem A x = λ B x , {\displaystyle
LOBPCG
Diagnostic plot in multivariate statistics
In multivariate statistics, a scree plot is a line plot of the eigenvalues of factors or principal components in an analysis. The scree plot is used to
Scree_plot
for solving large scale eigenvalue problems in the matrix-free fashion. The package is designed to compute a few eigenvalues and corresponding eigenvectors
ARPACK
expansion Jacobi–Perron algorithm Jacobi−Trudi identities Jacobi conformal projections Jacobi coordinates Jacobi eigenvalue algorithm Jacobi ellipsoid Jacobi
List of things named after Carl Gustav Jacob Jacobi
List_of_things_named_after_Carl_Gustav_Jacob_Jacobi
Decomposition in multilinear algebra
nonlinear conjugate gradient (NCG) limited memory BFGS (L-BFGS) Eigenvalue algorithms: Power iteration Factorization Machines: Support Vector Machines
Tensor_rank_decomposition
Optimization algorithm
unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate function. The idea is to
Gradient_descent
Method in linear algebra
characteristic polynomial Δ(t). Step 2: Find the eigenvalues of A, which are the roots of Δ(t). Step 3: For each eigenvalue λ of A from step 2, find an orthogonal
Orthogonal_diagonalization
Method for satisfying the Newtonian motion of a rigid body which consists of mass points
This approximation only works for matrices with eigenvalues smaller than 1, making the LINCS algorithm suitable only for molecules with low connectivity
Constraint (computational chemistry)
Constraint_(computational_chemistry)
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
develop a robust numerical algorithm for the Jordan normal form, as the result depends critically on whether two eigenvalues are deemed to be equal. For
Jordan_normal_form
Change of basis applied in quantum computing
quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator, and algorithms for the hidden subgroup problem. The quantum Fourier
Quantum_Fourier_transform
dx} Rayleigh's quotient Rayleigh quotient iteration Eigenvalue algorithm Generalized eigenvalue problem Meirovitch, Leonard (2003). Fundamentals of Vibration
Rayleigh's quotient in vibrations analysis
Rayleigh's_quotient_in_vibrations_analysis
Root-finding algorithm for polynomials
connection with the shifted QR algorithm for computing matrix eigenvalues. See Dekker and Traub The shifted QR algorithm for Hermitian matrices. Again
Jenkins–Traub_algorithm
Idempotent linear transformation from a vector space to itself
decomposition Reduction to Hessenberg form (the first step in many eigenvalue algorithms) Linear regression Projective elements of matrix algebras are used
Projection_(linear_algebra)
Algorithms for matrix decomposition
factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized
Non-negative matrix factorization
Non-negative_matrix_factorization
Matrix decomposition method
(2010-05-01). "Toward a parallel solver for generalized complex symmetric eigenvalue problems". Procedia Computer Science. ICCS 2010. 1 (1): 437–445. doi:10
Cholesky_decomposition
Form of radar used to create images of landscapes
whitens or equalizes, the clutter eigenvalues. Resolution loss due to the averaging operation. Backprojection Algorithm has two methods: Time-domain Backprojection
Synthetic-aperture_radar
EIGENVALUE ALGORITHM
EIGENVALUE ALGORITHM
EIGENVALUE ALGORITHM
EIGENVALUE ALGORITHM
Boy/Male
American, British, English, French
Little Famous One; Deserving
Boy/Male
Tamil
Picture
Boy/Male
Indian, Marathi
One of the Ved of Hindu Dharma
Boy/Male
Australian, French, German, Polish
Priceless; Inestimable
Boy/Male
Hindu
Name of Lord Shiva
Surname or Lastname
English
English : variant of Lombard.
Girl/Female
Indian, Telugu
Queen of Snakes
Boy/Male
Muslim/Islamic
Servant of the Subtle One
Girl/Female
Hebrew
Plant.
Boy/Male
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
That which does Not Disappear
EIGENVALUE ALGORITHM
EIGENVALUE ALGORITHM
EIGENVALUE ALGORITHM
EIGENVALUE ALGORITHM
EIGENVALUE ALGORITHM
n.
The art of calculating by nine figures and zero.
n.
The art of calculating with any species of notation; as, the algorithms of fractions, proportions, surds, etc.
n.
Alt. of Algorithm