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In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues λ {\displaystyle \lambda } , left eigenvectors y {\displaystyle
Quadratic_eigenvalue_problem
Concepts from linear algebra
{\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=0} leads to a so-called quadratic eigenvalue problem, ( ω 2 m + ω c + k ) x = 0. {\displaystyle \left(\omega ^{2}m+\omega
Eigenvalues_and_eigenvectors
Solving an optimization problem with a quadratic objective function
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks
Quadratic_programming
Concept in linear algebra
matrix. Generalized eigenvalue problem Generalized pencil-of-function method Nonlinear eigenproblem Quadratic eigenvalue problem Generalized Rayleigh
Matrix_pencil
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
Process of calculating the causal factors that produced a set of observations
notion of eigenvalue does not make sense any longer. A mathematical analysis is required to make it a bounded operator and design a well-posed problem: an illustration
Inverse_problem
Polynomial with all terms of degree two
whether all other non-zero eigenvalues are of the same sign: If they are, then it is elliptic; otherwise, it is hyperbolic. Quadratic forms over the ring of
Quadratic_form
Computational analysis of vibrations
{\displaystyle [F]} is the force vector. The general problem, with nonzero damping, is a quadratic eigenvalue problem. However, for vibrational modal analysis, the
Modal_analysis_using_FEM
PEP is intended for polynomial eigenproblems, including the quadratic eigenvalue problem. Solvers based on explicit linearization, that rely on EPS solvers
SLEPc
scissors-congruent? Babai's problem: which groups are Babai invariant groups? Brouwer's conjecture on upper bounds for sums of eigenvalues of Laplacians of graphs
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Quantum search algorithm
algorithms for NP-complete problems require exponentially many steps, and Grover's algorithm provides at most a quadratic speedup over the classical solution
Grover's_algorithm
Numerical methods for matrix eigenvalue calculation
the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also
Eigenvalue_algorithm
Topics referred to by the same term
plan or query execution plan, in a database software system Quadratic eigenvalue problem, a special case of nonlinear eigenproblem in mathematics QEP
QEP
Conjecture on zeros of the zeta function
this is important open problem itself and part of Langlands program. Artin (1924) introduced global zeta functions of (quadratic) function fields and conjectured
Riemann_hypothesis
Branch of mathematics
electric power. Linear algebraic concepts such as matrix operations and eigenvalue problems are employed to enhance the efficiency, reliability, and economic
Linear_algebra
Algorithm for finding zeros of functions
^{2}-22.12728346\lambda +34.49868830=0} . Applying the quadratic formula gives the two eigenvalues as λ 1 = 22.12728346 + 351.62192 2 ≈ 20.43943 {\displaystyle
Newton's_method
Numerical linear algebra algorithm
following result of Schönhage yields locally quadratic convergence. To this end let S have m distinct eigenvalues λ 1 , . . . , λ m {\displaystyle \lambda
Jacobi_eigenvalue_algorithm
Subfield of mathematical optimization
can be seen as reducing a general unconstrained convex problem, to a sequence of quadratic problems.Newton's method can be combined with line search for
Convex_optimization
Method in linear algebra
3000 Solved Problems in Linear Algebra. Maxime Bôcher (with E.P.R. DuVal) (1907) Introduction to Higher Algebra, § 45 Reduction of a quadratic form to a
Orthogonal_diagonalization
Algorithm to solve Wahba's problem
Wahba's problem as a quadratic form, using the Cayley–Hamilton theorem and the Newton–Raphson method to efficiently solve the eigenvalue problem and construct
Quaternion estimator algorithm
Quaternion_estimator_algorithm
Type of homogeneous polynomial of degree 2
indefiniteness, of this quadratic form is equivalent to the same property of A, which can be checked by considering all eigenvalues of A or by checking the
Definite_quadratic_form
Mathematical function
approximation and in linear inverse problems, and as apodization tapers or window functions in quadratic problems of spectral density estimation. Slepian
Slepian_function
Spectral line splitting in electrical field
effect is either linear (proportional to the applied electric field) or quadratic with a high accuracy. The Stark effect can be observed both for emission
Stark_effect
Australian and American mathematician (born 1975)
sets into the setting of restriction to quadratic hypersurfaces.[T03] The multilinear setting for these problems was further developed by Tao in collaboration
Terence_Tao
Topics referred to by the same term
function in terms of eigenvalues. Sylvester's law of inertia, also called Sylvester's rigidity theorem, about the signature of a quadratic form. Sylvester's
Sylvester's_theorem
Differential calculus on function spaces
multi-dimensional eigenvalue problems can be formulated as variational problems. The Sturm–Liouville eigenvalue problem involves a general quadratic form Q [ y
Calculus_of_variations
American mathematician
David; Elman, Howard; Osborn, John E. (2009), "A non-self-adjoint quadratic eigenvalue problem describing a fluid-solid interaction. {II}. {A}nalysis of convergence"
John E. Osborn (mathematician)
John_E._Osborn_(mathematician)
Problem in physics and astronomy
the eigenvalues (energies) have been obtained: these are a generalization of the Lambert W function. Various generalizations of Euler's problem are known;
Euler's_three-body_problem
Algorithm to calculate eigenvalues
the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm
QR_algorithm
Method for approximating eigenvalues
numerical method of approximating eigenvalues, which originated in the context of solving physical boundary-value problems. It is named after Lord Rayleigh
Rayleigh–Ritz_method
Computer science problem
longest increasing subsequence problem is closely related to the longest common subsequence problem, which has a quadratic time dynamic programming solution:
Longest increasing subsequence
Longest_increasing_subsequence
maximized — i.e., the length of the quadratic form q(x) is maximized — this is the eigenvector, and its length is the eigenvalue. All other eigenvectors will
Courant_minimax_principle
Approximations used in machine learning
is at least quadratic in the number of training data points, but most kernel methods include computation of matrix inversion or eigenvalue decomposition
Low-rank matrix approximations
Low-rank_matrix_approximations
Transforms equations for numerical solution
the shift, the original eigenvalue problem A x = λ x {\displaystyle Ax=\lambda x} is replaced with the shift-and-invert problem ( A − α I ) − 1 x = μ x
Preconditioner
Algorithm to be run on quantum computers
general number field sieve. Likewise, Grover's algorithm would run quadratically faster than the best possible classical algorithm for the same task
Quantum_algorithm
Locus of the zeros of a polynomial of degree two
{radius}^{2}={\frac {C}{eigenvalue\;of{\mathbf {A}}}}} where the radial axes and along the eigenvectors of the corresponding eigenvalues, and C = p T A p −
Quadric
roots exist only when the degree of the polynomial is less than 5. The quadratic formula has been known since antiquity, and the cubic and quartic formulas
Polynomial_root-finding
Quantum algorithm for integer factorization
factoring problem to the problem of order-finding. This reduction is similar to that used for other factoring algorithms, such as the quadratic sieve. A
Shor's_algorithm
Method used in statistics, pattern recognition, and other fields
covariance matrix. These projections can be found by solving a generalized eigenvalue problem, where the numerator is the covariance matrix formed by treating the
Linear_discriminant_analysis
Statistical algorithm
the error. The mean-square error as a function of filter weights is a quadratic function which means it has only one extremum, that minimizes the mean-square
Least_mean_squares_filter
Types of problems: Linear-quadratic regulator — system dynamics is a linear differential equation, objective is quadratic Linear-quadratic-Gaussian control
List of numerical analysis topics
List_of_numerical_analysis_topics
Machine learning framework for portfolio construction
portfolios have been proposed as a robust alternative to traditional quadratic optimization methods, including the Critical Line Algorithm (CLA) of Markowitz
Hierarchical_Risk_Parity
Optimization algorithm
\mathbf {A} \mathbf {x} -\mathbf {b} =0} reformulated as a quadratic minimization problem. If the system matrix A {\displaystyle \mathbf {A} } is real
Gradient_descent
Curve from a cone intersecting a plane
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola
Conic_section
Subfield of convex optimization
of edges crossing from one side to the other. This problem can be expressed as an integer quadratic program: Maximize ∑ ( i , j ) ∈ E 1 − v i v j 2 , {\displaystyle
Semidefinite_programming
Method for finding stationary points of a function
iterate x k + 1 {\displaystyle x_{k+1}} is defined so as to minimize this quadratic approximation in t {\displaystyle t} , and setting x k + 1 = x k + t {\displaystyle
Newton's method in optimization
Newton's_method_in_optimization
Array of numbers
sprang from several sources. Number-theoretical problems led Gauss to relate coefficients of quadratic forms, that is, expressions such as x2 + xy − 2y2
Matrix_(mathematics)
Measure of variation in statistics
\mathbf {1} )} is the multivariate standard normal. The eigenvectors and eigenvalues of S {\displaystyle \mathbf {S} } correspond to the axes of the 1 sd
Standard_deviation
Mathematical measure of a function's variability
is a measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space H1. The Dirichlet energy is intimately
Dirichlet_energy
Visual representation used in non-linear control system analysis
q=AD-BC=\det(\mathbf {A} )\,.} The explicit solution of the eigenvalues are then given by the quadratic formula: λ = 1 2 ( p ± Δ ) {\displaystyle \lambda ={\frac
Phase_plane
Number, approximately 2.41421
is the positive solution of quadratic equation σ 2 − 2 σ − 1 = 0. {\displaystyle \sigma ^{2}-2\sigma -1=0.} The quadratic formula gives the two solutions
Silver_ratio
Property of a mathematical matrix
min-max theorem, the kth largest eigenvalue of M {\displaystyle M} is greater than or equal to the kth largest eigenvalue of N . {\displaystyle N.} If M
Definite_matrix
Matrix in mathematics
complementarity problem. Linear complementarity problems arise in linear and quadratic programming, computational mechanics, and in the problem of finding
M-matrix
Matrix equal to its conjugate-transpose
Ritz and Lord Rayleigh. Parlett, Beresford N. (1998). The symmetric eigenvalue problem. Classics in applied mathematics. Philadelphia, Pa: Society for Industrial
Hermitian_matrix
Nonlinear equation which arises on linear optimal control problems
time-invariant Linear-Quadratic Regulator problem (LQR) as well as that of the infinite horizon time-invariant Linear-Quadratic-Gaussian control problem (LQG). These
Algebraic_Riccati_equation
Principle in geometry and linear algebra
{1}{\sqrt {2}}}\end{bmatrix}}.} This applies to the present problem of "diagonalizing" the quadratic form through the observation that 5 x 2 + 8 x y + 5 y 2
Principal_axis_theorem
Method of improving artificial neural network
iteration III: A short and sharp convergence estimate for generalized eigenvalue problems". Linear Algebra and Its Applications. 358 (1–3): 95–114. doi:10
Batch_normalization
Quantum mechanical model
eigenstate. Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function ⟨ x | ψ ⟩ = ψ ( x )
Quantum_harmonic_oscillator
Mathematical problem
Ad_{L}} -unipotent elements, i.e. elements g for which 1 is the only eigenvalue of A d L ( g ) {\displaystyle Ad_{L}(g)} . Borel showed in 1909 that the
Littlewood_conjecture
Topics referred to by the same term
value or values; see Periodic points of complex quadratic mappings Characteristic multiplier, an eigenvalue of a monodromy matrix Multiplier algebra, a construction
Multiplier
Mathematical optimization algorithm
optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. Despite differences in their approaches, these derivations share
Conjugate_gradient_method
Concept in computational chemistry
some eigenvalues E j {\displaystyle \mathbf {E} ^{j}} and their corresponding eigenvectors c I j {\displaystyle \mathbf {c} _{I}^{j}} . The eigenvalues are
Configuration_interaction
Concept in graph theory
get a quadratic: p 2 + ( μ − λ ) p − ( k − μ ) = 0 {\displaystyle p^{2}+(\mu -\lambda )p-(k-\mu )=0} This gives the two additional eigenvalues 1 2 [ (
Strongly_regular_graph
Matrix-valued random variable
in the state equation and the problem is known as one of stochastic control. A key result in the case of linear-quadratic control with stochastic matrices
Random_matrix
Methods of mathematical approximation
finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is
Perturbation_theory
Polynomial equation of degree 3
arithmetic operations, square roots, and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree
Cubic_equation
Description of a quantum-mechanical system
the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate
Schrödinger_equation
Matrix representing a Euclidean rotation
odd, there will be a "dangling" eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the
Rotation_matrix
Matrix equal to its transpose
symmetric matrix A {\displaystyle A} is equal to the number of non-zero eigenvalues of A {\displaystyle A} . Any square matrix can uniquely be written as
Symmetric_matrix
Concept in mathematics
A&B/2\\B/2&C\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}},} is the quadratic form associated with the equation, and the matrix A 33 = ( A B / 2 B /
Matrix representation of conic sections
Matrix_representation_of_conic_sections
Branch of numerical optimization
usually eigenvalue bounds derived from interval Hessian matrices. One of the most general second-order methodologies for handling problems of general
Deterministic global optimization
Deterministic_global_optimization
Regularization technique for ill-posed problems
the inverse-problem, the inverse mapping operates as a high-pass filter that has the undesirable tendency of amplifying noise (eigenvalues / singular values
Ridge_regression
Linear operators with a common spectrum
spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity. The theory of isospectral
Isospectral
Field of engineering
developed a Rayleigh quotient approximation to improve the accuracy of eigenvalue approximations. Barthelemy and Haftka published a comprehensive review
Multidisciplinary design optimization
Multidisciplinary_design_optimization
Relation of a matrix of variables between two points in time
are stable if and only if all eigenvalues of the matrix L are smaller than unity in absolute value. In linear-quadratic-Gaussian control, there arises
Matrix_difference_equation
Matrix that, squared, equals itself
matrix A {\displaystyle A} and λ {\displaystyle \lambda } its associated eigenvalue, then λ x = A x = A 2 x = A λ x = λ A x = λ 2 x , {\textstyle \lambda
Idempotent_matrix
Type of differential equation
there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. The theory of elliptic, parabolic
Partial_differential_equation
quantum algorithms for optimization problems Quantum phase estimation algorithm: estimates the phase of an eigenvalue of a unitary operator Quantum singular
List_of_algorithms
feedback controller designed to minimize a quadratic cost, is optimal for the stochastic control problem with output measurements. When process and observation
Separation_principle
Every polynomial has a real or complex root
vector space of matrices. The eigenvalues of A are precisely the poles of R(z). Since, by assumption, A has no eigenvalues, the function R(z) is an entire
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Algorithm that estimates unknowns from a series of measurements over time
statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed
Kalman_filter
{\left(\mathbf {r} \right)}=E\psi {\left(\mathbf {r} \right)},} which is an eigenvalue equation. Very often, only numerical solutions to the Schrödinger equation
List of quantum-mechanical systems with analytical solutions
List_of_quantum-mechanical_systems_with_analytical_solutions
Type of mathematical equation
+c_{n}e^{\lambda _{n}t}\mathbf {u} _{n}~,} where λ1, λ2, …, λn are the eigenvalues of A; u1, u2, …, un are the respective eigenvectors of A; and c1, c2
Matrix_differential_equation
Concept in regression analysis mathematics
denote w ^ ( λ ) {\displaystyle {\hat {w}}(\lambda )} ) we may use the eigenvalue decomposition of the covariance matrix 1 n X T X = Q diag ( α 1 , … ,
Regularized_least_squares
some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that
Periodic points of complex quadratic mappings
Periodic_points_of_complex_quadratic_mappings
Topics referred to by the same term
problem (NCP), finding a vector meeting certain conditions based on a given smooth mapping Nonlinear eigenproblem (AKA nonlinear eigenvalue problem)
Nonlinearity_(disambiguation)
Special state of wave and quantum systems in physics
symmetries of propagating modes in the continuum. Arise when the eigenvalue problem is solved by the Separation of Variables Method, and the wave function
Bound_state_in_the_continuum
Study of high-dimensional data
deviate from the unit eigenvalues of Σ {\displaystyle \Sigma } . Further information on the asymptotic behaviour of the eigenvalues of Σ ^ {\displaystyle
High-dimensional_statistics
Second-order deterministic global optimization algorithm
relaxation for nonlinear functions of general form by superposing them with a quadratic of sufficient magnitude, called α, such that the resulting superposition
ΑΒΒ
Number, approximately 3.14
of the Dirichlet eigenvalue problem in one dimension, the Poincaré inequality is the variational form of the Neumann eigenvalue problem, in any dimension
Pi
Continuous (non-quantized) quantities in quantum information science
problems for which quantum algorithms have been studied include finding matrix eigenvalues, phase estimation, the Sturm–Liouville eigenvalue problem,
Continuous-variable quantum information
Continuous-variable_quantum_information
Collection of mathematical theories
theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure
Spectral_theory
Polynomial function of degree 4
that the depressed equation is bi-quadratic, and may be solved by an easier method (see above). This was not a problem at the time of Ferrari, when one
Quartic_function
Mathematical structures that allow quantum mechanics to be explained
were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Matrix used in finite element analysis
Au = F. The stiffness matrix is symmetric, i.e. Aij = Aji, so all its eigenvalues are real. Moreover, it is a strictly positive-definite matrix, so that
Stiffness_matrix
Vera N. (1961). "On some algorithms for the solution of the complete eigenvalue problem". USSR Computational Mathematics and Mathematical Physics. 1 (3):
Timeline_of_algorithms
Representation of angular momentum tensor product states important to physics
generalizes the mere two labels for SU(2) multiplets, namely the eigenvalues of its quadratic Casimir and of I3. Since [ I ^ 3 , Y ^ ] = 0 {\displaystyle [{\hat
Clebsch–Gordan coefficients for SU(3)
Clebsch–Gordan_coefficients_for_SU(3)
Optimized math routines developed by Intel
precision data. LAPACK consists of tuned LU, Cholesky and QR factorizations, eigenvalue and least squares solvers. MKL also includes Sparse BLAS, ScaLAPACK, Sparse
Math_Kernel_Library
Special class of quantum states
{\displaystyle d_{j}\geq 0} are known as symplectic eigenvalues. Now, if we consider a purely quadratic Hamiltonian H ^ = 1 2 r ^ T H r ^ {\displaystyle
Gaussian_state
Information held in the state of a quantum system
is not an eigenstate in the other basis. According to the eigenstate–eigenvalue link, an observable is well-defined (definite) when the state of the system
Quantum_information
QUADRATIC EIGENVALUE-PROBLEM
QUADRATIC EIGENVALUE-PROBLEM
Boy/Male
Hindu, Indian
Problem
Girl/Female
Muslim/Islamic
Away from all Problems
Boy/Male
Indian, Tamil
People with this Name are Preferably Intelligent and Very Generous; Highly Knowledgeable in Problem Solving Skills
Boy/Male
Muslim
Problem solver
Girl/Female
Indian, Telugu
Destroyer of Problems
Girl/Female
Bengali, Indian
Eternity; Problem Solver
Boy/Male
Arabic, Indian, Muslim
Problem Solver
QUADRATIC EIGENVALUE-PROBLEM
QUADRATIC EIGENVALUE-PROBLEM
Boy/Male
English American
Blend of Ray and Shawn.
Female
Finnish
Short form of Finnish Marjaana, JAANA means "obstinacy, rebelliousness" or "their rebellion."Â
Girl/Female
Hindu
Liberated, Pearl
Girl/Female
Tamil
Light, Glow, Shine
Girl/Female
Tamil
Vidyasri | விதà¯à®¯à®¾à®¸à®°à¯€
Wisdom, Knowledge, Learning, Goddess Durga
Girl/Female
Australian, Indian, Punjabi, Sikh
Quite and Gentle
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lord Vishnu
Girl/Female
Hindu
Goddess Saraswati or one who knows all Vedas, Pious
Surname or Lastname
English (Devon)
English (Devon) : probably a variant of Hingston. The name in this spelling has died out in England.
Boy/Male
Indian
King of Kings
QUADRATIC EIGENVALUE-PROBLEM
QUADRATIC EIGENVALUE-PROBLEM
QUADRATIC EIGENVALUE-PROBLEM
QUADRATIC EIGENVALUE-PROBLEM
QUADRATIC EIGENVALUE-PROBLEM
a.
To square; to agree; to suit; to correspond; -- followed by with.
a.
Of or pertaining to a square, or to squares; resembling a quadrate, or square; square.
n.
A biquadratic equation.
n.
A quadrat.
p. pr. & vb. n.
of Quadrate
a.
Of or pertaining to the quadrate and jugal bones.
a.
A quadrate; a square.
a.
Of or pertaining to the biquadrate, or fourth power.
pl.
of Quadratrix
a.
The quadrate bone.
pl.
of Quadratrix
n.
A biquadrate.
n.
A curve made use of in the quadrature of other curves; as the quadratrix, of Dinostratus, or of Tschirnhausen.
a.
Quadrate; square.
v. t.
To adjust (a gun) on its carriage; also, to train (a gun) for horizontal firing.
a.
Pertaining to terms of the second degree; as, a quadratic equation, in which the highest power of the unknown quantity is a square.
a.
Tetragonal.
n.
That branch of algebra which treats of quadratic equations.
imp. & p. p.
of Quadrate
n.
Same as Quadrate.