# Difference between revisions of "Pseudo-Inverse Algorithm"

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** a [[Courrieu's Fast Moore-Penrose Inverse Algorithm]] ([[2008_FastComputationofMoorePenroseIn|Courrieu (2008)]]), | ** a [[Courrieu's Fast Moore-Penrose Inverse Algorithm]] ([[2008_FastComputationofMoorePenroseIn|Courrieu (2008)]]), | ||

** a [[Fast B-Spline Pseudo-inversion Algorithm]] ([[Tristan & Arribas, 2007]]), | ** a [[Fast B-Spline Pseudo-inversion Algorithm]] ([[Tristan & Arribas, 2007]]), | ||

− | ** a [[Herron's Pseudo-Inverse Algorithm]] ([[Herron, 1966]]). | + | ** a [[Herron's Pseudo-Inverse Algorithm]] ([[#1966|Herron, 1966]]). |

* <B>Counter-Example(s):</B> | * <B>Counter-Example(s):</B> | ||

** a [[Matrix Multiplication Algorithm]] such as: | ** a [[Matrix Multiplication Algorithm]] such as: | ||

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* ([[2008_FastComputationofMoorePenroseIn|Courrieu, 2008]]) ⇒ [[author::Pierre Courrieu]]. ([[year::2008]]). “[https://arxiv.org/pdf/0804.4809.pdf Fast Computation of Moore-Penrose Inverse Matrices ].” In: Neural Information Processing - Letters and Reviews Journal, 8. | * ([[2008_FastComputationofMoorePenroseIn|Courrieu, 2008]]) ⇒ [[author::Pierre Courrieu]]. ([[year::2008]]). “[https://arxiv.org/pdf/0804.4809.pdf Fast Computation of Moore-Penrose Inverse Matrices ].” In: Neural Information Processing - Letters and Reviews Journal, 8. | ||

** QUOTE: The [[Moore-Penrose inverse]], also called [[Pseudoinverse]], or [[Generalized Inverse]], allows for solving [[least square system]]s, even with [[rank deficient matrice]]s, in such a way that each [[column vector]] of the solution has a [[minimum norm]], which is the desired property stated above. | ** QUOTE: The [[Moore-Penrose inverse]], also called [[Pseudoinverse]], or [[Generalized Inverse]], allows for solving [[least square system]]s, even with [[rank deficient matrice]]s, in such a way that each [[column vector]] of the solution has a [[minimum norm]], which is the desired property stated above. | ||

+ | |||

+ | === 1966 === | ||

+ | * (Herron, 1966) ⇒ [[Christopher R. Herron]] (1966). [https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19670014642.pdf "Computing the pseudo-inverse"]. | ||

+ | ** QUOTE: An [[orthogonalization algorithm]] for producing the [[pseudoinverse]] of a [[matrix]] is described, and a [[FORTRAN]] [[program]] which realizes the [[algorithm]] is given in detail. | ||

=== 1965 === | === 1965 === |

## Revision as of 13:00, 13 July 2019

A Pseudo-Inverse Algorithm is a Matrix Decomposition Algorithm that can solve a least square system such that each column vector of the solution has a minimum norm.

**AKA:**Pseudo-Inverse, Pseudoinverse Algorithm, Moore-Penrose Inverse Algorithm, Generalized Inverse Algorithm.**Example(s):**- a Bose-Nguyen Generalized Inverse Algorithm (Bose & Nguyen, 2016),
- a Weighted Pseudo-Inverse Algorithm (Shao et al., 2015),
- a Online PseudoInverse Update Method (OPIUM) Algorithm (Tapson & Schaik, 2013),
- a Courrieu's Fast Moore-Penrose Inverse Algorithm (Courrieu (2008)),
- a Fast B-Spline Pseudo-inversion Algorithm (Tristan & Arribas, 2007),
- a Herron's Pseudo-Inverse Algorithm (Herron, 1966).

**Counter-Example(s):****See:**Function Fitting Algorithm, Pseudoinverse Matrix, Matrix Decomposition, QR Factorization, Singular Value Decomposition.

## References

### 2019

- (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Moore–Penrose_inverse Retrieved:2019-7-11.
- In mathematics, and in particular linear algebra, a
**pseudoinverse***A*^{+}of a matrix*A*is a generalization of the inverse matrix.The most widely known type of matrix pseudoinverse is the**Moore–Penrose inverse**,which was independently described by E. H. Moore^{[1]}in 1920, Arne Bjerhammar^{[2]}in 1951, and Roger Penrose^{[3]}in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse.A common use of the pseudoinverse is to compute a "best fit" (least squares) solution to a system of linear equations that lacks a unique solution (see below under § Applications).

Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra.

The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition.

- In mathematics, and in particular linear algebra, a

- ↑ Moore, E. H. (1920). "On the reciprocal of the general algebraic matrix". Bulletin of the American Mathematical Society. 26 (9): 394–95. doi:10.1090/S0002-9904-1920-03322-7.
- ↑ Bjerhammar, Arne (1951). "Application of calculus of matrices to method of least squares; with special references to geodetic calculations". Trans. Roy. Inst. Tech. Stockholm. 49.
- ↑ Penrose, Roger (1955). "A generalized inverse for matrices". Proceedings of the Cambridge Philosophical Society. 51 (3): 406–13. doi:10.1017/S0305004100030401.

### 2016

- (Kadiam Bose & Nguyen, 2016) ⇒ S. Kadiam Bose, and D. T. Nguyen. (2016). “Efficient Generalized Inverse for Solving Simultaneous Linear Equations.” In: Journal of Applied Mathematics and Physics, 4. doi:10.4236/jamp.2016.41003
- QUOTE: The generalized (or pseudo) inverse of a matrix is an extension of the ordinary/regular square (non-singular) matrix inverse, which can be applied to any matrix (such as singular, rectangular etc.). The generalized inverse has numerous important engineering and sciences applications. Over the past decades, generalized inverses of matrices and its applications have been investigated by many researchers 1-6. Generalized inverse is also known as “Moore-Penrose inverse” or “g-inverse” or “pseudo-inverse” etc.
In this paper we introduce an efficient (in terms of computational time and computer memory requirement) generalized inverse formulation to solve SLE with full or deficient rank of the coefficient matrix. The coefficient matrix can be singular/non-singular, symmetric/unsymmetric, or square/rectangular. Due to popular MATLAB software, which is widely accepted by researchers and educators worldwide, the developed code from this work is written in MATLAB language.

- QUOTE: The generalized (or pseudo) inverse of a matrix is an extension of the ordinary/regular square (non-singular) matrix inverse, which can be applied to any matrix (such as singular, rectangular etc.). The generalized inverse has numerous important engineering and sciences applications. Over the past decades, generalized inverses of matrices and its applications have been investigated by many researchers 1-6. Generalized inverse is also known as “Moore-Penrose inverse” or “g-inverse” or “pseudo-inverse” etc.

### 2008

- (Courrieu, 2008) ⇒ Pierre Courrieu. (2008). “Fast Computation of Moore-Penrose Inverse Matrices .” In: Neural Information Processing - Letters and Reviews Journal, 8.
- QUOTE: The Moore-Penrose inverse, also called Pseudoinverse, or Generalized Inverse, allows for solving least square systems, even with rank deficient matrices, in such a way that each column vector of the solution has a minimum norm, which is the desired property stated above.

### 1966

- (Herron, 1966) ⇒ Christopher R. Herron (1966). "Computing the pseudo-inverse".
- QUOTE: An orthogonalization algorithm for producing the pseudoinverse of a matrix is described, and a FORTRAN program which realizes the algorithm is given in detail.

### 1965

- (Golub & Kahan, 1965) ⇒ G. Golub and W. Kahan. (1965). “Calculating the Singular Values and Pseudo-Inverse of a Matrix.” In: Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis, 2(2).

### 1955

- (Penrose, 1955) ⇒ Roger Penrose (1955, July). "A generalized inverse for matrices". In Mathematical proceedings of the Cambridge philosophical society (Vol. 51, No. 3, pp. 406-413). Cambridge University Press.

Author | S. Kadiam Bose +, D. T. Nguyen + and Pierre Courrieu + |

year | 2016 + and 2008 + |