Open In App

Moore – Penrose Pseudoinverse in R Programming

Improve
Improve
Like Article
Like
Save
Share
Report

The concept used to generalize the solution of a linear equation is known as Moore – Penrose Pseudoinverse of a matrix. Moore – Penrose inverse is the most widely known type of matrix pseudoinverse. In linear algebra pseudoinverse A^{+}    of a matrix A is a generalization of the inverse matrix. The most common use of pseudoinverse is to compute the best fit solution to a system of linear equations that lacks a unique solution. The term generalized inverse is sometimes used as a synonym of pseudoinverse. R Language provides a very simple method to calculate Moore – Penrose Pseudoinverse. The pseudoinverse is used as follows: 
 

x = A^{+} b


 

where, 
A+: Single value decomposition used to calculate the pseudoinverse or the generalized inverse of a numerical matrix 
x and b: vectors 
 


 

Note: Moore – Penrose pseudoinverse solves the problem in the least squared error sense. In general, there is no exact solution to overdetermined problems. So if you cross check the solution you will not get the exact required b but an approx value of b. 
 


 

Implementation in R


R provides two functions ginv() which is available in MASS library and pinv() which is available in pracma library to compute the Moore-Penrose generalized inverse of a matrix. These two functions return an arbitrary generalized inverse of a matrix, using gaussianElimination.
 

Syntax: 
ginv(A) 
pinv(A)
Parameter: 
A: numerical matrix 
 


Example 1: 
Consider below 3 linear equations: 
 

x_1 + 3x_2 = 17\\ 5x_1 + 7x_2 = 19\\ 11x_1 + 13x_2 = 23


Equivalently one can write above equations in matrix form as shown below: 
 

A{x} = {b}\\


 

\begin{bmatrix} 1 & 3 \\ 5 & 7 \\ 11 & 13 \\ \end{bmatrix} % \begin{bmatrix} x_1\\ x_2\\ \end{bmatrix} = \begin{bmatrix} 17\\ 19\\ 23\\ \end{bmatrix}


# Using ginv() 
 

Python3

# R program to illustrate
# solve a linear matrix
# equation of metrics using
# moore – Penrose Pseudoinverse
 
# Importing library for
# applying pseudoinverse
library(MASS)
 
# Representing A in
# matrix form in R
A = matrix(
  c(1, 5, 11, 3, 7, 13),
  nrow = 3,            
  ncol = 2,            
)
cat("A = :\n")
print(A)
 
# Representing b in
# matrices form in R
b = matrix(
  c(17, 19, 23),
  nrow = 3,            
  ncol = 1,            
)
cat("b = :\n")
print(b)
 
# Calculating x using ginv()
cat("Solution of linear equations
    using pseudoinverse:\n")
x = ginv(A) %*% b
print(x)

                    

Output: 
 

A = :
     [, 1] [, 2]
[1, ]    1    3
[2, ]    5    7
[3, ]   11   13
b = :
     [, 1]
[1, ]   17
[2, ]   19
[3, ]   23
Solution of linear equations 
    using pseudoinverse:
          [, 1]
[1, ] -7.513158
[2, ]  8.118421


# Using pinv()
 

Python3

# R program to illustrate
# solve a linear matrix
# equation of metrics using
# moore – Penrose Pseudoinverse
 
# Importing library for
# applying pseudoinverse
library(pracma)
 
# Representing A in
# matrices form in R
A = matrix(
  c(1, 5, 11, 3, 7, 13),
  nrow = 3,         
  ncol = 2,         
)
cat("A = :\n")
print(A)
 
# Representing b in
# matrices form in R
b = matrix(
  c(17, 19, 23),
  nrow = 3,         
  ncol = 1,         
)
cat("b = :\n")
print(b)
 
# Calculating x using pinv()
cat("Solution of linear equations
    using pseudoinverse:\n")
x = pinv(A) %*% b
print(x)

                    

Output: 
 

A = :
     [, 1] [, 2]
[1, ]    1    3
[2, ]    5    7
[3, ]   11   13
b = :
     [, 1]
[1, ]   17
[2, ]   19
[3, ]   23
Solution of linear equations 
    using pseudoinverse:
          [, 1]
[1, ] -7.513158
[2, ]  8.118421


Example 2: 
Similarly, let we have linear equations in matrix form as shown below: 
 

\begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 1 \\ \end{bmatrix} % \begin{bmatrix} x_1\\ x_2\\ x_3\\ \end{bmatrix} = \begin{bmatrix} 2\\ 1\\ \end{bmatrix}


# Using ginv() 
 

Python3

# R program to illustrate
# solve a linear matrix
# equation of metrics using
# moore – Penrose Pseudoinverse
 
# Importing library for
# applying pseudoinverse
library(MASS)
 
# Representing A in
# matrics form in R
A = matrix(
  c(1, 0, 2, 0, 3, 1),
  ncol = 3,
  byrow = F
)
cat("A = :\n")
print(A)
 
# Representing b in
# matrics form in R
b = matrix(
  c(2, 1),
)
cat("b = :\n")
print(b)
 
# Calculating x using ginv()
cat("Solution of linear equations
    using pseudoinverse:\n")
x = ginv(A) %*% b
print(x)

                    

Output: 
 

A = :
     [, 1] [, 2] [, 3]
[1, ]    1    2    3
[2, ]    0    0    1
b = :
     [, 1]
[1, ]    2
[2, ]    1
Solution of linear equations 
    using pseudoinverse:
     [, 1]
[1, ] -0.2
[2, ] -0.4
[3, ]  1.0


# Using pinv()
 

Python3

# R program to illustrate
# solve a linear matrix
# equation of metrics using
# moore – Penrose Pseudoinverse
 
# Importing library for
# applying pseudoinverse
library(pracma)
 
# Representing A in
# matrics form in R
A = matrix(
  c(1, 0, 2, 0, 3, 1),
  ncol = 3,
  byrow = F
)
cat("A = :\n")
print(A)
 
# Representing b in
# matrics form in R
b = matrix(
  c(2, 1),
)
cat("b = :\n")
print(b)
 
# Calculating x using pinv()
cat("Solution of linear equations
    using pseudoinverse:\n")
x = pinv(A) %*% b
print(x)

                    

Output: 
 

A = :
     [, 1] [, 2] [, 3]
[1, ]    1    2    3
[2, ]    0    0    1
b = :
     [, 1]
[1, ]    2
[2, ]    1
Solution of linear equations 
    using pseudoinverse:
     [, 1]
[1, ] -0.2
[2, ] -0.4
[3, ]  1.0


 



Last Updated : 26 Dec, 2022
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads