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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
Concept in mathematics
In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate gradient method to nonlinear optimization. For a quadratic
Nonlinear conjugate gradient method
Nonlinear_conjugate_gradient_method
Algorithm for solving matrix-vector equations
In numerical linear algebra, the conjugate gradient squared method (CGS) is an iterative algorithm for solving systems of linear equations of the form
Conjugate gradient squared method
Conjugate_gradient_squared_method
Derivation of the conjugate gradient method Nonlinear conjugate gradient method Biconjugate gradient method Biconjugate gradient stabilized method Elijah Polak
Gradient_method
In numerical linear algebra, the conjugate gradient method is an iterative method for numerically solving the linear system A x = b {\displaystyle {\boldsymbol
Derivation of the conjugate gradient method
Derivation_of_the_conjugate_gradient_method
Optimization algorithm
Gradient descent is a method for unconstrained mathematical optimization. It is a first-order iterative algorithm for minimizing a differentiable multivariate
Gradient_descent
Mathematical optimization method
iterates. This method, and modifications, are globally convergent under mild conditions, and perform competitively with conjugate gradient methods for many
Barzilai–Borwein_method
Form of projection
steepest descent method and the conjugate gradient method, but proximal gradient methods can be used instead. Proximal gradient methods starts by a splitting
Proximal_gradient_method
Algorithm for solving systems of linear equations
biconjugate gradient method is an algorithm to solve systems of linear equations A x = b . {\displaystyle Ax=b.\,} Unlike the conjugate gradient method, this
Biconjugate_gradient_method
similar to the much more popular conjugate gradient method, with similar construction and convergence properties. This method is used to solve linear equations
Conjugate_residual_method
Class of reinforcement learning algorithms
Policy gradient methods are a class of reinforcement learning algorithms and a sub-class of policy optimization methods. Unlike value-based methods which
Policy_gradient_method
Numerical approximation algorithm
method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of an iterative method or a method of
Iterative_method
Concept in mathematics
biconjugate gradient method (BiCG) and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugate gradient squared
Biconjugate gradient stabilized method
Biconjugate_gradient_stabilized_method
Mathematical term
Nonlinear conjugate gradient method, generalizes the conjugate gradient method to nonlinear optimization Stochastic gradient descent, iterative method for optimizing
Slope
Preconditioned Conjugate Gradient Method (LOBPCG), Wiedemann's coordinate recurrence algorithm, the conjugate gradient method, Krylov subspace methods. Distributed
Matrix-free_methods
iteration Conjugate gradient method (CG) — assumes that the matrix is positive definite Derivation of the conjugate gradient method Nonlinear conjugate gradient
List of numerical analysis topics
List_of_numerical_analysis_topics
Transforms equations for numerical solution
preconditioned iterative methods for linear systems include the preconditioned conjugate gradient method, the biconjugate gradient method, and generalized minimal
Preconditioner
Mathematical method for solving large eigenvalue problems
\mathbf {1} } the Identity matrix. In contrast to the Conjugate gradient method, here the gradient calculates by twice multiplying matrix H : G ∼ H → G
Folded_spectrum_method
Benchmark in high-performance computing
The High Performance Conjugate Gradients Benchmark (HPCG benchmark) is a supercomputing benchmark test proposed by Michael Heroux from Sandia National
HPCG_benchmark
American mathematician (1906–1991)
optimal control. As a pioneer in computer science, he devised the conjugate gradient method, published jointly with Eduard Stiefel. Born in Bricelyn, Minnesota
Magnus_Hestenes
Topics referred to by the same term
conjugate, in geometry Conjugate gradient method, an algorithm for the numerical solution of particular systems of linear equations Conjugate points, in differential
Conjugation
Mathematical algorithm
\mathbf {J_{r}} } . For large systems, an iterative method, such as the conjugate gradient method, may be more efficient. If there is a linear dependence
Gauss–Newton_algorithm
Methods for numerical approximations
usually used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact solution
Numerical_analysis
Study of mathematical algorithms for optimization problems
Polyak, subgradient–projection methods are similar to conjugate–gradient methods. Bundle method of descent: An iterative method for small–medium-sized problems
Mathematical_optimization
unknowns associated with subdomain interfaces is solved by the conjugate gradient method. Suppose we want to solve the Poisson equation − Δ u = f , u |
Schur_complement_method
Type of numerical method
iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG. In overlapping domain decomposition methods, the subdomains overlap by more
Domain_decomposition_methods
Numerical optimization algorithm
NEWUOA LINCOA Nonlinear conjugate gradient method Levenberg–Marquardt algorithm Broyden–Fletcher–Goldfarb–Shanno or BFGS method Differential evolution
Nelder–Mead_method
Numerical method for solving physical or engineering problems
is symmetric and positive definite, so a technique such as the conjugate gradient method is favored. For problems that are not too large, sparse LU decompositions
Finite_element_method
Topics referred to by the same term
art Conceptual graph, a formalism for knowledge representation Conjugate gradient method, an algorithm for the numerical solution of particular systems
CG
Topics referred to by the same term
forms on request. Preconditioned conjugate gradient square method, a variant of the preconditioned conjugate gradient method – an algorithm for the numerical
PCGS
Optimization algorithm
necessarily approximate the optimum. One example of the former is conjugate gradient method. The latter is called inexact line search and may be performed
Line_search
Swiss mathematician (1909–1978)
Together with Cornelius Lanczos and Magnus Hestenes, he invented the conjugate gradient method, and gave what is now understood to be a partial construction
Eduard_Stiefel
Solving an optimization problem with a quadratic objective function
problems a variety of methods are commonly used, including interior point, active set, augmented Lagrangian, conjugate gradient, gradient projection, extensions
Quadratic_programming
Method for finding largest (or smallest) eigenvalues
Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding
LOBPCG
learning) Winnow algorithm Backpropagation Conjugate gradient method Generalized Hebbian algorithm Gradient descent Levenberg–Marquardt algorithm PagedAttention
List of artificial intelligence algorithms
List_of_artificial_intelligence_algorithms
over-relaxation Conjugate gradient method Generalized minimal residual method Biconjugate gradient method IML++ "Chebyshev iteration method", Encyclopedia
Chebyshev_iteration
Concept in numerical linear algebra
preconditioner in another iterative solution algorithm such as the conjugate gradient method or GMRES. For a given matrix A ∈ R n × n {\displaystyle A\in \mathbb
Incomplete_LU_factorization
Topics referred to by the same term
shipping line Computer Games Magazine Computer Graphics Metafile Conjugate gradient method, an algorithm for the numerical solution of particular systems
CGM
Algorithm for finding a local minimum of a function
Powell's method, strictly Powell's conjugate direction method, is an algorithm proposed by Michael J. D. Powell for finding a local minimum of a function
Powell's_method
Topics referred to by the same term
PCGM may refer to: Political correctness gone mad Preconditioned conjugate gradient method Pacific Coast Gravity Meeting This disambiguation page lists articles
PCGM
Signal processing system
Sample Matrix Inversion Algorithm Recursive Least Square Algorithm Conjugate gradient method Constant Modulus Algorithm Beamforming is spatial signal processing
Adaptive_beamformer
Method of solving differential equations
using multigrid preconditioners in the locally optimal block conjugate gradient method. Electronic Transactions on Numerical Analysis, 15, 38–55, 2003
Multigrid_method
Array of numbers
solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method. An algorithm is, roughly speaking, numerically stable if little
Matrix_(mathematics)
Image processing technique
combining the filter with an iterative method, e.g., the Chebyshev iteration and the conjugate gradient method are proposed in for graph-based image denoising
Edge-preserving_smoothing
Approximation of a matrix's Cholesky factorization
factorization is often used as a preconditioner for algorithms like the conjugate gradient method. The Cholesky factorization of a positive definite matrix A of
Incomplete Cholesky factorization
Incomplete_Cholesky_factorization
Optimization algorithm
gradient descent methods can move in any direction that the ridge or alley may ascend or descend. Hence, gradient descent or the conjugate gradient method
Hill_climbing
Method for finding stationary points of a function
iterative methods. Many of these methods are only applicable to certain types of equations, for example the Cholesky factorization and conjugate gradient will
Newton's method in optimization
Newton's_method_in_optimization
Branch of numerical analysis
space iterative methods, such as the conjugate gradient method or GMRES. In overlapping domain decomposition methods, the subdomains overlap by more than
Numerical methods for partial differential equations
Numerical_methods_for_partial_differential_equations
Topics referred to by the same term
which remodel chromatin to silence other genes Preconditioned conjugate gradient method, an algorithm for the numerical solution of particular systems
PCG
Overview of and topical guide to deep learning
Backpropagation Conjugate gradient method Generalized Hebbian algorithm Gradient descent Levenberg–Marquardt algorithm Perceptron Quasi-Newton method Wake-sleep
Outline_of_deep_learning
Numerous methods exist to compute descent directions, all with differing merits, such as gradient descent or the conjugate gradient method. More generally
Descent_direction
Analysis tool used to find the approximate error in a result
the residual of the PDE. Shewchuk, Jonathan Richard (1994). "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" (PDF). p. 6.
Residual_(numerical_analysis)
expanded in a plane wave basis set and using a self-consistent conjugate gradient method to determine the energy minimum. Computational efficiency is achieved
ABINIT
Family of optimization algorithms
averaging methods, full-gradient snapshot methods, recursive estimator methods (e.g., SARAH), and dual methods. Each category contains methods designed
Stochastic_variance_reduction
Inequalities for inexact line search
9} for Newton or quasi-Newton methods and c 2 = 0.1 {\displaystyle c_{2}=0.1} for the nonlinear conjugate gradient method. Inequality i) is known as the
Wolfe_conditions
Type of mathematical function
2014-07-14. Michael J. D. Powell (1977). "Restart procedures for the conjugate gradient method". Mathematical Programming. 12 (1): 241–254. doi:10.1007/bf01593790
Radial_basis_function
Iterative method used to solve a linear system of equations
end end end Conjugate gradient method Gaussian belief propagation Iterative method: Linear systems Kaczmarz method (a "row-oriented" method, whereas Gauss-Seidel
Gauss–Seidel_method
Algorithm
cost than other iterative methods, such as the conjugate gradient method. In 2009, a randomized version of the Kaczmarz method for overdetermined linear
Kaczmarz_method
Numerical software
for solving the Newton system iteratively by a preconditioned conjugate gradient method, rather than directly, via an LDL* decomposition. The interior
HiGHS_optimization_solver
Method of estimating the parameters of a statistical model
This is the case when conjugate priors are used. Via numerical optimization such as the conjugate gradient method or Newton's method. This usually requires
Maximum a posteriori estimation
Maximum_a_posteriori_estimation
Dutch mathematician (born 1944)
contributions include preconditioned iterative methods, in particular the ICCG (incomplete Cholesky conjugate gradient) method (developed together with Koos Meijerink)
Henk_van_der_Vorst
positive-definite, we can apply standard iterative methods like the gradient descent method or the conjugate gradient method to solve S x 2 = B ∗ A − 1 b 1 − b 2 {\displaystyle
Uzawa_iteration
Software packages using DDA
typically use regular grids (cubical or rectangular cuboid), conjugate gradient method to solve large systems of linear equations and FFT-acceleration
Discrete dipole approximation codes
Discrete_dipole_approximation_codes
Krylov methods such as Krylov-Schur, Arnoldi and Lanczos. Davidson methods such as Generalized Davidson and Jacobi-Davidson. Conjugate gradient methods such
SLEPc
Method In numerical analysis
finite element method. BDDC is used as a preconditioner to the conjugate gradient method. A specific version of BDDC is characterized by the choice of
BDDC
Topics referred to by the same term
International Dose-Response Society, published by SAGE Nonlinear conjugate gradient method, an algorithm for numerically finding the minimum of a nonlinear
Nonlinearity_(disambiguation)
Machine learning and applied statistics
the method of conjugate gradients, Nordsieck methods, Gaussian quadrature rules, and quasi-Newton methods. In all these cases, the classic method is based
Probabilistic_numerics
Technique in computational electromagnetism
Large size problems can be solved using iterative techniques like Conjugate gradient method. For both generalized and normal eigenvalue problems, just a few
Plane_wave_expansion_method
The contemporary conjugate convective heat transfer model was developed after computers came into wide use in order to substitute the empirical relation
Conjugate convective heat transfer
Conjugate_convective_heat_transfer
Mixture of several programming languages in the same program
Conjugate gradient method Ford-Fulkerson algorithm Gauss–Seidel method Generalized minimal residual method Jacobi eigenvalue algorithm Jacobi method Karmarkar's
Pidgin_code
Mathematical algorithm
descent algorithm Conjugate gradient – Mathematical optimization algorithmPages displaying short descriptions of redirect targets Gradient descent – Optimization
Coordinate_descent
Matrix in which most of the elements are zero
Both iterative and direct methods exist for sparse matrix solving. Iterative methods, such as conjugate gradient method and GMRES utilize fast computations
Sparse_matrix
published 2001 – LOBPCG Locally Optimal Block Preconditioned Conjugate Gradient method finding extreme eigenvalues of symmetric eigenvalue problems by
Timeline_of_algorithms
Jewish American mathematician
Gertrude Blanch, and William Karush. Magnus Hestenes's work on the conjugate gradient method was a direct outgrowth of this group's work together over the
Marvin Stein (computer scientist)
Marvin_Stein_(computer_scientist)
Method for computing radiation
as the use of the conjugate gradient (CG) algorithm of Petravic and Kuo-Petravic. Subsequently, various conjugate gradient methods have been explored
Discrete_dipole_approximation
Discontinued online library
solutions methods are: Richardson Iteration Chebyshev Iteration Conjugate Gradient (CG) Conjugate Gradient Squared (CGS) BiConjugate Gradient (BiCG) BiConjugate
IML++
British mathematician (1939–2016)
in 2025. BFGS method Biconjugate gradient method Davidon–Fletcher–Powell formula Nonlinear conjugate gradient method Practical methods of optimization
Roger Fletcher (mathematician)
Roger_Fletcher_(mathematician)
Optimization algorithm
Pytlak, Radoslaw (2009). "Limited Memory Quasi-Newton Algorithms". Conjugate Gradient Algorithms in Nonconvex Optimization. Springer. pp. 159–190. ISBN 978-3-540-85633-7
Limited-memory_BFGS
Algorithm for statistical inference on graphical models
shown to be immune to numerical problems of the preconditioned conjugate gradient method. The previous description of BP algorithm is called the codeword-based
Belief_propagation
Algorithms for solving convex optimization problems
Interior-point methods (also referred to as barrier methods or IPMs) are algorithms for solving linear and non-linear convex optimization problems. IPMs
Interior-point_method
Numerical method in computational electromagnetics
large matrices with a large number of unknowns, iterative methods such as conjugate gradient method can be used for acceleration. The actual field distributions
Method of moments (electromagnetics)
Method_of_moments_(electromagnetics)
Parallel software library for linear algebra
mathematical libraries Conjugate gradient method Biconjugate gradient stabilized method (BiCGSTAB) Generalized minimal residual method (GMRES) Eigenvalue
Lis_(linear_algebra_library)
Overview of and topical guide to algorithms
Newton's method Gradient descent Conjugate gradient method Simulated annealing Expectation–maximization algorithm Numerical integration Monte Carlo method Linear
Outline_of_algorithms
Estimation method that minimizes the mean square error
decomposition, while for large sparse systems conjugate gradient method is more effective. Levinson recursion is a fast method when C Y {\displaystyle C_{Y}} is also
Minimum mean square error estimator
Minimum_mean_square_error_estimator
Computer optimization methods
Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies
Proximal gradient methods for learning
Proximal_gradient_methods_for_learning
benchmarks should feature new parallel-aware algorithmic and software methods, genericness and architecture neutrality, easy verifiability of correctness
NAS_Parallel_Benchmarks
Model-free reinforcement learning algorithm
algorithm for training an intelligent agent. Specifically, it is a policy gradient method, often used for deep RL when the policy network is very large. The
Proximal_policy_optimization
Gamma ray Imaging method
O. (June 1985). "Constrained Iterative Reconstruction by the Conjugate Gradient Method". IEEE Transactions on Medical Imaging. 4 (2): 65–71. doi:10.1109/TMI
Gamma_ray_tomography
Aspect of quantum chemistry
been developed, from simple conjugate gradient methods with exact line searches, to Newton–Raphson and trust-region methods. The Foster–Boys localization
Localized_molecular_orbitals
Algorithm for finding zeros of functions
Bisection method Euler method Fast inverse square root Fisher scoring Gradient descent Integer square root Kantorovich theorem Laguerre's method Methods of computing
Newton's_method
Optimization algorithm
Also known as the conditional gradient method, reduced gradient algorithm and the convex combination algorithm, the method was originally proposed by Marguerite
Frank–Wolfe_algorithm
Field of mathematics
linear problem Ax = b, the classical iterative approach is the conjugate gradient method. If A is not symmetric, then examples of iterative solutions to
Numerical_linear_algebra
Algorithm for searching a problem space
point methods, conjugate gradient method, line search, and other local heuristics. Note that most of the common individual learning methods are deterministic
Memetic_algorithm
Optimization algorithm
Quasi-Newton methods for optimization are based on Newton's method to find the stationary points of a function, points where the gradient is 0. Newton's method assumes
Quasi-Newton_method
Iterative method for solving the Sylvester matrix equations
example use of the conjugate gradient method preconditioned with incomplete Cholesky factorization). The idea behind the ADI method is to split the finite
Alternating-direction implicit method
Alternating-direction_implicit_method
Computer graphics simulation of deformable objects
the conjugate gradient method), which itself may also be difficult to achieve at interactive frame rates. An alternative is to use an explicit method with
Soft-body_dynamics
Mathematical optimization algorithms
truncated Newton methods to work, the inner solver needs to produce a good approximation in a finite number of iterations; conjugate gradient has been suggested
Truncated_Newton_method
systems of linear equations Biconjugate gradient method: solves systems of linear equations Conjugate gradient: an algorithm for the numerical solution
List_of_algorithms
Quantum algorithm for solving systems of linear equations
O(N^{3})} time. If A is s-sparse and positive semi-definite, then the Conjugate Gradient method can be used to find the solution vector x → {\displaystyle {\vec
HHL_algorithm
Mathematical optimization function
continuously differentiable. Indeed, many proximal gradient methods can be interpreted as a gradient descent method over M f {\displaystyle M_{f}} . The Moreau
Moreau_envelope
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CONJUGATE GRADIENT-METHOD