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alternate case: system of linear equations
Basis pursuit
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is usually applied in cases where there is an underdetermined system of linear equations y = Ax that must be exactly satisfied, and the sparsest solutionLlewellyn Thomas (888 words) [view diff] exact match in snippet view article find links to article
theory), and devising an efficient method for solving tridiagonal system of linear equations (Thomas algorithm). Born in London, he studied at Cambridge UniversityMAX-3LIN-EQN (110 words) [view diff] exact match in snippet view article find links to article
problem in Computational complexity theory where the input is a system of linear equations (modulo 2). Each equation contains at most 3 variables. The problemRado's theorem (Ramsey theory) (606 words) [view diff] exact match in snippet view article
Let A x = 0 {\displaystyle A\mathbf {x} =\mathbf {0} } be a system of linear equations, where A {\displaystyle A} is a matrix with integer entries. ThisRandom walker algorithm (2,256 words) [view diff] exact match in snippet view article find links to article
These probabilities may be determined analytically by solving a system of linear equations. After computing these probabilities for each pixel, the pixelAlgebraic reconstruction technique (521 words) [view diff] exact match in snippet view article find links to article
reconstruction process. ART can be considered as an iterative solver of a system of linear equations A x = b {\displaystyle Ax=b} , where: A {\displaystyle A} is aFinite-difference frequency-domain method (938 words) [view diff] exact match in snippet view article find links to article
frequency-domain response problem leads to an A x = b {\displaystyle Ax=b} system of linear equations as described above. An early description of a frequency-domainGuyan reduction (861 words) [view diff] exact match in snippet view article find links to article
length of the displacement vector. By partitioning the above system of linear equations with regards to loaded (master) and unloaded (slave) degrees ofReduction (mathematics) (725 words) [view diff] exact match in snippet view article
is prone to some error in the solution. Consider the following system of linear equations in an FEA problem: [ K 11 K 12 K 21 K 22 ] [ x 1 x 2 ] = [ F 1Retrieval Data Structure (1,514 words) [view diff] exact match in snippet view article find links to article
{\displaystyle (h(x))_{x\in S}} are linearly independent the following system of linear equations has a solution Z ∈ { 0 , 1 } m × r {\displaystyle Z\in \{0,1\}^{m\timesGosper's algorithm (582 words) [view diff] exact match in snippet view article find links to article
finding that there is no such ƒ) is then a matter of solving a system of linear equations. Gosper's algorithm can be used to discover Wilf–Zeilberger pairsGeopositioning (1,211 words) [view diff] exact match in snippet view article find links to article
observations to compute a position fix is equivalent to solving a system of linear equations. Navigation systems use regression algorithms such as least squaresAdditive Schwarz method (1,251 words) [view diff] exact match in snippet view article find links to article
usually not possible to split our grid of 64 points (a 64×64 system of linear equations) into two grids of 32 points (two 32×32 systems of linear equations)Affine Grassmannian (manifold) (599 words) [view diff] exact match in snippet view article
n}x_{n}&=a_{n-k,n+1}.\end{aligned}}} These determine a rank n−k system of linear equations on Rn+1 a 11 x 1 + ⋯ + a 1 n x n = a 1 , n + 1 x n + 1 ⋮ a n −Validated numerics (1,341 words) [view diff] exact match in snippet view article find links to article
numerical linear algebra Validating numerical solutions of a given system of linear equations Validating numerically obtained eigenvalues Rigorously computingLinear least squares (5,375 words) [view diff] exact match in snippet view article find links to article
squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of theNewton polynomial (5,932 words) [view diff] exact match in snippet view article find links to article
leads to a problem in linear algebra where we have to solve a system of linear equations. Using a standard monomial basis for our interpolation polynomialScientific programming language (756 words) [view diff] exact match in snippet view article find links to article
mathematical constructs. For example, the following Julia code solves a system of linear equations: A = rand(20, 20) # A is a 20x20 matrix b = rand(20) # b is aProny's method (1,285 words) [view diff] exact match in snippet view article find links to article
_{m}}} values, the F n {\displaystyle F_{n}} values are part of a system of linear equations that may be used to solve for the B m {\displaystyle \mathrm {B}MAX-3SAT (1,450 words) [view diff] exact match in snippet view article find links to article
equated to "=1" as usual, one could find a proof π by solving a system of linear equations (see MAX-3LIN-EQN) implying P = NP. If z ∈ L, a fraction ≥ (1Gebhart factor (762 words) [view diff] exact match in snippet view article find links to article
_{j=1}^{N_{s}}(B_{ij})=1} Several approaches exists to describe this as a system of linear equations that can be solved by Gaussian elimination or similar methodsLeslie Fox (1,228 words) [view diff] exact match in snippet view article find links to article
by finite difference method and thus reducing the problem to a system of linear equations was the same. Careful analysis of the errors was a theme of manyOrdinary least squares (9,135 words) [view diff] exact match in snippet view article find links to article
mathematicians, OLS is an approximate solution to an overdetermined system of linear equations Xβ ≈ y, where β is the unknown. Assuming the system cannot beTriangular matrix (3,152 words) [view diff] exact match in snippet view article find links to article
inverting the matrix. The matrix equation Lx = b can be written as a system of linear equations ℓ 1 , 1 x 1 = b 1 ℓ 2 , 1 x 1 + ℓ 2 , 2 x 2 = b 2 ⋮ ⋮ ⋱ ⋮ ℓ mTaxicab geometry (2,507 words) [view diff] exact match in snippet view article find links to article
orientation in taxicab geometry. In solving an underdetermined system of linear equations, the regularization term for the parameter vector is expressedThe Nine Chapters on the Mathematical Art (2,338 words) [view diff] exact match in snippet view article find links to article
unknowns, and the most complex example analyzes the solution to a system of linear equations with up to 5 unknowns. The word jiu, or "9", means more than justBeam and Warming scheme (1,063 words) [view diff] exact match in snippet view article find links to article
n}-A_{i-1}^{n}u_{i-1}^{n}\right)\end{aligned}}} This resulted system of linear equations can be solved using the modified tridiagonal matrix algorithmA5/1 (2,725 words) [view diff] exact match in snippet view article find links to article
of 240.16 (the units are in terms of number of solutions of a system of linear equations which are required). In 2000, Alex Biryukov, Adi Shamir and DavidMöller–Trumbore intersection algorithm (1,585 words) [view diff] exact match in snippet view article find links to article
}&=-tD+u(v_{2}-v_{1})+v(v_{3}-v_{1})\end{aligned}}} This is a system of linear equations with three equations (one each for x {\displaystyle x} , y {\displaystyleOptical flow (3,109 words) [view diff] exact match in snippet view article find links to article
discrete pixels, these equations are discretised. Doing so yields a system of linear equations which can be solved for ( u , v ) {\displaystyle (u,v)} at eachIncidence (geometry) (1,733 words) [view diff] exact match in snippet view article
that is the simultaneous solution (up to a scalar factor) of the system of linear equations: a1x + b1y + c1z = 0 and a2x + b2y + c2z = 0. The solution ofCoefficients of potential (1,191 words) [view diff] exact match in snippet view article find links to article
capacitance on a two-conductor system. For a two-conductor system, the system of linear equations is ϕ 1 = p 11 Q 1 + p 12 Q 2 ϕ 2 = p 21 Q 1 + p 22 Q 2 . {\displaystyleEquivalent impedance transforms (3,538 words) [view diff] exact match in snippet view article find links to article
The entries of the matrix are chosen so that the matrix forms a system of linear equations in the mesh voltages and currents (as defined for mesh analysis):Input–output model (4,053 words) [view diff] exact match in snippet view article find links to article
want 7 units of good A and 4 units of good B. Then solving the system of linear equations derived above gives us x = ( I − A ) − 1 y = [ 19.19 12.97 ]Orthogonal matrix (4,802 words) [view diff] exact match in snippet view article find links to article
S symmetric positive-semidefinite Consider an overdetermined system of linear equations, as might occur with repeated measurements of a physical phenomenonVolterra integral equation (1,496 words) [view diff] exact match in snippet view article find links to article
i , s j ) {\displaystyle K_{ij}=K(t_{i},s_{j})} , we have the system of linear equations: x 0 = f 0 x 1 = f 1 + Δ s 2 ( K 10 x 0 + K 11 x 1 ) x 2 = f 2Ellipsoid method (3,705 words) [view diff] exact match in snippet view article find links to article
constraints, which can be solved by any method for solving a system of linear equations. Step 3: the decision problem can be reduced to a different optimizationLinear differential equation (4,754 words) [view diff] exact match in snippet view article find links to article
are partial derivatives. A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficientsFangcheng (mathematics) (1,083 words) [view diff] exact match in snippet view article
a unique solution. This is the earliest known reference to a system of linear equations in which the number of unknowns exceeds the number of equationsMaximum cut (3,125 words) [view diff] exact match in snippet view article find links to article
Thomassé, S.; Yeo, A. (2014), "Satisfying more than half of a system of linear equations over GF(2): A multivariate approach", J. Comput. Syst. Sci., 80Schur complement (2,901 words) [view diff] exact match in snippet view article find links to article
Laplacian matrix. The Schur complement arises naturally in solving a system of linear equations such as [ A B C D ] [ x y ] = [ u v ] {\displaystyleAerodynamic potential-flow code (1,541 words) [view diff] exact match in snippet view article find links to article
= {\displaystyle m_{i}=} These terms can be used to create a system of linear equations which can be solved for all the unknown values of λ {\displaystyleThermal simulations for integrated circuits (2,063 words) [view diff] exact match in snippet view article find links to article
Finally, a direct or iterative method is employed to solve the system of linear equations. For the thermal case, FEM method is more suitable due to theMediant (mathematics) (2,038 words) [view diff] exact match in snippet view article
{\displaystyle \lambda _{1},\,\lambda _{2}} must be integers, solving the system of linear equations a ′ = λ 1 a + λ 2 b {\displaystyle a'=\lambda _{1}a+\lambda _{2}b}Reed–Solomon error correction (12,395 words) [view diff] exact match in snippet view article find links to article
locations. There is a linear recurrence relation that gives rise to a system of linear equations. Solving those equations identifies those error locations Xk.Unique games conjecture (3,066 words) [view diff] exact match in snippet view article find links to article
constraints over alphabet of size k is NP-hard. Consider the following system of linear equations over the integers modulo k: a 1 x 1 ≡ b 1 ⋅ x 2 + c 1 ( mod kBoolean satisfiability problem (5,045 words) [view diff] exact match in snippet view article find links to article
This is in P, since an XOR-SAT formula can also be viewed as a system of linear equations mod 2, and can be solved in cubic time by Gaussian elimination;Mehrotra predictor–corrector method (1,729 words) [view diff] exact match in snippet view article find links to article
scaling direction. This is achieved by solving the following system of linear equations J ( x , λ , s ) [ Δ x aff Δ λ aff Δ s aff ] = − F ( x , λ , sComputational fluid dynamics (8,648 words) [view diff] exact match in snippet view article find links to article
algebraic equations. Applying a Newton or Picard iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefiniteInfo-metrics (2,101 words) [view diff] exact match in snippet view article find links to article
and James Mammen. "The simplest solution to an underdetermined system of linear equations". In Information Theory, 2006 IEEE International Symposium onMatrix differential equation (2,732 words) [view diff] exact match in snippet view article find links to article
above equations equal 1. Thus we may construct the following system of linear equations, 1 = 2 A + B {\displaystyle 1=2A+B} 1 = A + 2 B . {\displaystyleElastic collision (5,826 words) [view diff] exact match in snippet view article find links to article
by the collision. Now the above formulas follow from solving a system of linear equations for v A 2 , v B 2 , {\displaystyle v_{A2},v_{B2},} regarding mRobert J. Marks II (3,574 words) [view diff] exact match in snippet view article find links to article
of them can be obtained from the remaining ones by solving a system of linear equations". Ill-posed sampling (The Cheung-Marks Theorem). The samplingLaplace transform (9,453 words) [view diff] exact match in snippet view article find links to article
convolution equations, the Laplace transform converts it into a system of linear equations for π n ( s ) = L ( P n ) ( s ) , {\displaystyle \pi _{n}(s)={\mathcalStochastic dominance (3,659 words) [view diff] exact match in snippet view article find links to article
u ( x ) {\displaystyle u(x)} is nondecreasing and concave. A system of linear equations can test whether a given solution if efficient for any such utilityChinese mathematics (7,992 words) [view diff] exact match in snippet view article find links to article
or what is now known as linear algebra. Chapter seven solves system of linear equations with two unknowns using the false position method, similar toMathematical economics (10,736 words) [view diff] exact match in snippet view article find links to article
function with respect to quantity supplied for each firm left a system of linear equations, the simultaneous solution of which gave the equilibrium quantityGlobal Positioning System (19,124 words) [view diff] case mismatch in snippet view article find links to article
1994. Oszczak, B., "New Algorithm for GNSS Positioning Using System of Linear Equations", Proceedings of the 26th International Technical Meeting of TheSensor array (2,823 words) [view diff] exact match in snippet view article find links to article
its derivative (which is linear), let it equal zero and solve a system of linear equations. In ML beamformers the quadratic penalty function is used to theCarl Friedrich Gauss (17,929 words) [view diff] exact match in snippet view article find links to article
letter to Gerling from 1823, he described a solution of a 4x4 system of linear equations with the Gauss-Seidel method – an "indirect" iterative methodDirac equation (13,120 words) [view diff] exact match in snippet view article find links to article
{\displaystyle \gamma ^{\mu }} , solving this is a matter of solving a system of linear equations. It is a representation-free property of gamma matrices that theTrigonometry of a tetrahedron (3,048 words) [view diff] exact match in snippet view article find links to article
faces of the tetrahedron, one obtains the following homogeneous system of linear equations: { − Δ 1 + Δ 2 c 34 + Δ 3 c 24 + Δ 4 c 23 = 0 Δ 1 c 34 − Δ 2 +Gottfried Wilhelm Leibniz (19,219 words) [view diff] exact match in snippet view article find links to article
if machines were used." Leibniz arranged the coefficients of a system of linear equations into an array, now called a matrix, in order to find a solutionCausal model (6,142 words) [view diff] exact match in snippet view article find links to article
introduced principles for detecting and evaluating mediation in a system of linear equations. As of 2014 their paper was the 33rd most-cited of all time.: 324List of eponymous laws (10,525 words) [view diff] exact match in snippet view article find links to article
In linear algebra, an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the systemSteinitz's theorem (5,973 words) [view diff] exact match in snippet view article find links to article
of a drawing of a graph. The method continues by setting up a system of linear equations in the vertex coordinates, according to which each remaining vertexHistory of computed tomography (2,659 words) [view diff] exact match in snippet view article find links to article
Mathématiques. 35: 355–7. Kaczmarz S., "Approximate solution of system of linear equations. Int. J. Control. 1993; 57-9. Cormack AM (1963). "RepresentationScale-invariant feature transform (9,232 words) [view diff] exact match in snippet view article find links to article
A{\hat {\mathbf {x} }}=A^{T}\mathbf {b} .} The solution of the system of linear equations is given in terms of the matrix ( A T A ) − 1 A T {\displaystyleMedical image computing (8,475 words) [view diff] exact match in snippet view article find links to article
estimate of the diffusion tensor can be found by simply solving a system of linear equations at each location independently. However, as the volume is assumedHistory of algebra (17,047 words) [view diff] exact match in snippet view article find links to article
geometrical associations. Leibniz realized that the coefficients of a system of linear equations could be arranged into an array, now called a matrix, which canFlux balance analysis (5,950 words) [view diff] exact match in snippet view article find links to article
reactions than metabolites and this gives an under-determined system of linear equations containing more variables than equations. The standard approachList of publications in mathematics (10,413 words) [view diff] exact match in snippet view article find links to article
the earliest description of Gaussian elimination for solving system of linear equations, it also contains method for finding square root and cubic rootHückel method (7,290 words) [view diff] exact match in snippet view article find links to article
n ) {\displaystyle (c_{1},c_{2},\ldots ,c_{n})} to the above system of linear equations can only exist if the coefficient matrix [ H i j − E S i j ] {\displaystyleList of Chinese inventions (32,605 words) [view diff] exact match in snippet view article find links to article
towns and villages. Gaussian elimination: The Chinese solved a system of linear equations. The rectangle of coefficients was reduced to triangular and thenTransfer length method (5,102 words) [view diff] exact match in snippet view article find links to article
corresponding to a different gap length d i {\displaystyle d_{i}} - a system of linear equations can be obtained, which can be written in matrix-vector form. [Flow cytometry bioinformatics (8,040 words) [view diff] exact match in snippet view article find links to article
signal to remove from each channel can be computed by solving a system of linear equations based on this data to produce a spillover matrix, which when invertedGeometric rigidity (5,449 words) [view diff] exact match in snippet view article find links to article
matrix. The Jacobian, or derivative, of the rigidity map yields a system of linear equations of the form ( p ( u ) − p ( v ) ) ⋅ ( p ′ ( v ) − p ′ ( u ) )Geiringer–Laman theorem (5,841 words) [view diff] exact match in snippet view article find links to article
an easier approach, because infinitesimal rigidity involves a system of linear equations, rather than quadratic in the case of regular rigidity. In particularPseudo-range multilateration (9,850 words) [view diff] exact match in snippet view article find links to article
non-linear equations (1, 2, 3). The TDOA problem can be turned into a system of linear equations when there are three or more receivers, which can reduce the computation