World Library  
Flag as Inappropriate
Email this Article

Overdetermined system

Article Id: WHEBN0010160606
Reproduction Date:

Title: Overdetermined system  
Author: World Heritage Encyclopedia
Language: English
Subject: Linear least squares (mathematics), Spinors in three dimensions, System of polynomial equations, Locating engine, Reed–Solomon error correction
Publisher: World Heritage Encyclopedia

Overdetermined system

In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns.[1] The terminology can be described in terms of the concept of constraint counting. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom.

Therefore the critical case occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint. The overdetermined case occurs when the system has been overconstrained — that is, when the equations outnumber the unknowns. In contrast, the underdetermined case occurs when the system has been underconstrained — that is, when the number of equations is fewer than the number of unknowns.

Systems of equations

An example in two dimensions

#1 A system of three linearly independent equations, three lines, three intersections
#2 A system of three linearly independent equations, three lines (two parallel), two intersections
#3 A system of three linearly independent equations, three lines (three parallel), no intersections
#4 A system of three equations (one equation linearly dependent on the others), three lines (two collinear), one intersection
#5 A system of three equations (one equation linearly dependent on the others), three lines, one intersection
#6 A system of three equations (two equations linearly dependent on the other), three lines, an infinitude of intersections

Consider the system of 3 equations and 2 unknowns (x1 and x2), which is overdetermined because 3>2, and which corresponds to Diagram #1:




There is one solution for each pair of linear equations: for the first and second equations (0.2, −1.4), for the first and third (−2/3, 1/3), and for the second and third (1.5, 2.5). However there is no solution that satisfies all three simultaneously. Diagrams #2 and 3 show other configurations that are inconsistent because no point is on all of the lines. Systems of this variety are deemed inconsistent.

The only cases where the overdetermined system does in fact have a solution are demonstrated in Diagrams #4, 5, and 6. These exceptions can occur only when the overdetermined system contains enough linearly dependent equations that the number of independent equations does not exceed the number of unknowns. Linear dependence means that some equations can be obtained from linearly combining other equations. For example, y = x + 1 and 2y = 2x + 2 are linearly dependent equations because the second one can be obtained by taking twice the first one.

Matrix form

Any system of linear equations can be written as a matrix equation. The previous system of equations can be written as follows:

\begin{bmatrix} 2 & 1 \\ -3 & 1 \\ -1 & 1 \\ \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ \end{bmatrix} = \begin{bmatrix} -1 \\ -2 \\ 1 \\ \end{bmatrix}

Notice that the rows of the matrix (corresponding to equations) outnumber the columns (corresponding to unknowns), meaning that the system is overdetermined. In linear algebra the concepts of row space, column space and null space are important for determining the properties of matrices. The informal discussion of constraints and degrees of freedom above relates directly to these more formal concepts.

Homogeneous case

The homogeneous case (in which all constant terms are zero) is always consistent (because there is a trivial, all-zero solution). There are two cases, depending on the number of linearly dependent equations: either there is just the trivial solution, or there is the trivial solution plus an infinite set of other solutions.

Consider the system of linear equations: Li = 0 for 1 ≤ iM, and variables X1, X2, ..., XN, where each Li is a weighted sum of the Xis. Then X1 = X2 = ... = XN = 0 is always a solution. When M < N the system is underdetermined and there are always an infinitude of further solutions. In fact the dimension of the space of solutions is always at least NM.

For MN, there may be no solution other than all values being 0. There will be an infinitude of other solutions only when the system of equations has enough dependencies (linearly dependent equations) that the number of independent equations is at most N − 1. But with MN the number of independent equations could be as high as N, in which case the trivial solution is the only one.

Non-homogeneous case

In systems of linear equations, Li=ci for 1 ≤ iM, in variables X1, X2, ..., XN the equations are sometimes linearly dependent; in fact the number of linearly independent equations cannot exceed N+1. We have the following possible cases for an overdetermined system with N unknowns and M equations (M>N).

  • M = N+1 and all M equations are linearly independent. This case yields no solution. Example: x = 1, x = 2.
  • M > N but only K equations (K < M and KN+1) are linearly independent. There exist three possible sub-cases of this:
    • K = N+1. This case yields no solutions. Example: 2x = 2, x = 1, x = 2.
    • K = N. This case yields either a single solution or no solution, the latter occurring when the coefficient vector of one equation can be replicated by a weighted sum of the coefficient vectors of the other equations but that weighted sum applied to the constant terms of the other equations does not replicate the one equation's constant term. Example with one solution: 2x = 2, x = 1. Example with no solution: 2x + 2y = 2, x + y = 1, x + y = 3.
    • K < N. This case yields either infinitely many solutions or no solution, the latter occurring as in the previous sub-case. Example with infinitely many solutions: 3x + 3y = 3, 2x + 2y = 2, x + y = 1. Example with no solution: 3x + 3y + 3z = 3, 2x + 2y + 2z = 2, x + y + z = 1, x + y + z = 4.

These results may be easier to understand by putting the augmented matrix of the coefficients of the system in row echelon form by using Gaussian elimination. This row echelon form is the augmented matrix of a system of equations that is equivalent to the given system (it has exactly the same solutions). The number of independent equations in the original system is the number of non-zero rows in the echelon form. The system is inconsistent (no solution) if and only if the last non-zero row in echelon form has only one non-zero entry that is in the last column (giving an equation 0 = c where c is a non-zero constant). Otherwise, there is exactly one solution when the number of non-zero rows in echelon form is equal to the number of unknowns, and there are infinitely many solutions when the number of non-zero rows is lower than the number of variables.

Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions.

Exact solutions

All exact solutions can be obtained, or it can be shown that none exist, using matrix algebra. See System of linear equations#Matrix solution.

Approximate solutions

The method of ordinary least squares can be used to find an approximate solution to overdetermined systems. For the system Ax = b, the least squares formula is obtained from the problem

\min_{x}\Vert Ax - b \Vert,

the solution of which can be written with the normal equations,[2]

x =(A^{\mathrm{T}}A)^{-1}A^{\mathrm{T}}b,

where {\mathrm{T}} indicates a matrix transpose, provided (A^{\mathrm{T}}A)^{-1} exists (that is, provided A has full column rank). With this formula an approximate solution is found when no exact solution exists, and it gives an exact solution when one does exist. However, to achieve good numerical accuracy, using the QR factorization of A to solve the least squares problem is preferred.[3]

In general use

The concept can also be applied to more general systems of equations, such as systems of polynomial equations or partial differential equations. In the case of the systems of polynomial equations, it may happen that an overdetermined system has a solution, but that no one equation is a consequence of the others and that, when removing any equation, the new system has more solutions. For example, (x-1)(x-2)=0, (x-1)(x-3)=0 has the single solution x=1, but each equation by itself has two solutions.

See also


  1. ^ "Overdetermined equations". 
  2. ^ Anton, Howard; Rorres,, Chris (2005). Elementary Linear Algebra (9th ed.). John Wiley and Sons, Inc.  
  3. ^ Trefethen, Lloyd; Bau, III, David (1997). Numerical Linear Algebra.  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.