Generalized Inverse

Description

Japanese help file: Ginv_JPH

Usage

Ginv(
  X,
  MP = 0,
  method = 1,
  orth = 0,
  MP1 = 0,
  method1 = 1,
  MP2 = 0,
  method2 = 1,
  nopermute = 0,
  eps = 1e-08,
  MASS = 1,
  tol = .Machine$double.eps,
  chkginv = 0,
  epsc = 1e-08
)

Arguments

Details

Calculates Generalized Inverse of the input matrix X.

The input matrix X must be a real matrix.

The combination of MP= and method= options defines the method to calculate the Moor-Penrose and a generalized invers as follows.
Denoting a generalized inverse of X as ginv(X),

MP=1
 1 svd: If X=U %*% D %*% t(V), MP(X)= V %*% ginv(D) %*% t(U)
 2 QR:  If X=Q %*% R,
            MP(X)=t(R)%*%inv(R%*%t(R))%*%inv(t(Q)%*%Q)%*%t(Q)
 3 ginv1-1: MP(X)=t(X)%*%X%*%ginv(t(X)%*%X%*%t(X)%*%X)%*%t(X)
 3 ginv1-2: MP(X)=t(X)%*%ginv(X%*%t(X)%*%X%*%t(X))%*%X%*%t(X)
 4 ginv2: MP(X)=t(X)%*%ginv(X%*%t(X))%*%X%*%ginv(t(X)%*%X)%*%t(X)

MP=0
 1 Sweep: Make X square by augmenting 0's and use matSwp
 2 QR: If X=Q %*% R, ginv(X)=ginv(R)%*%t(Q)
 3 rank factorization: If X=B %*% C, ginv(X)=ginv(C)%*%ginv(B)
 4 ginv1-1: ginv(X)=ginv(t(X)%*%X) %*% t(X)
 4 ginv1-2: ginv(X)=X %*% ginv(X %*% t(X) )

Since Moor-Penrose inverse is unique, if MP=1 the above four methods should result in the same matrix.
If MP=0 the result may vary.

When MP=1, method=1, MASS=1, the algorithm should be equivalent to ginv of MASS package.

orth is available in the following combinations of (MP,method).
 MP=1, method=2
 MP=2, method=3

MP1 is available in the following combinations of (MP,method).
 MP=0, method=2
 MP=0, method=2, 3, 4

method1 is available in the following combinations of (MP,method).
 MP=1, method=3, 4
 MP=0, method=2, 3, 4

MP2 is available in the following combinations of (MP,method).
 MP=0, method=3

method2 is available in the following combinations of (MP,method).
 MP=1, method=4
 MP=0, method=3

Note that, in the older version prior to 20200703, MP=0 meant always use matSwp to calculate g-inverse. Since it is possible to calculate Moor-Penrose inverse using matSwp, this option is misleading!!

Also, in the old version, MASS=0 was the default setting and if MP=0, a generalized inverse of a rectangular matrix cannot be calculated using matSwp. (Caused error.) In this version, MASS=1 is the default setting and if MP=0, the Moor-Penrose inverse of a rectangular matrix will be calculated using matSwp.

Note that the following matrices generated from ginvX are also a generalized inverses of X satisfying condition g1 because

 (I-X%*%ginvX)) %*% X = 0,  X %*% (I-ginvX%*%X))  = 0,
 Q(X) %*% X = 0

. See the examples below.

 ginvX  = Ginv(X, ....)
 ginvX1 = ginvX + Y %*% (I-X%*%ginvX)
 ginvX2 = ginvX + (I-ginvX%*%X) %*% Y
 ginvX3 = ginvX + Y %*% Q( X, Ginv=1 )
 

where Q(X, Ginv=1) is the projection operator to the orthogonal complement of the column space of X calculated with Ginv=1 option and Y is any ncol(X) x nrow(X) matrix.

Examples

set.seed(1701)

nr=9
nc=6
num=0:9
A=matrix(sample(num,nr*nc,replace=1),nr,nc)

A=cbind(0,A[,1],0,A[,2],0,A[,1]+A[,2],0)
A=rbind(0,A[1,],0,A[2,],0,A[3:nrow(A),],0)
rownames(A)=paste("r",1:nrow(A),sep="")
colnames(A)=paste("c",1:ncol(A),sep="")

### A=1:3
### A=t(A)

MP=1
for( method in 1:4 ){
 Print(MP,method)
 ginvA=Ginv( A, MP=MP, method=method, chkginv=2 )
 Print(A,round(ginvA, 5))
}
MP=0
for( method in 1:4 ){
  Print(MP,method)
  ginvA=Ginv( A, MP=MP, method=method, chkginv=2 )
  Print(A,round(ginvA, 5))
}


# generating generalized inverses from a generalized inverse
set.seed(1701)
Y=matrix( sample( 0:9, ncol(A)*nrow(A), replace=1), ncol(A), nrow(A) )
ginvA=Ginv( A, MP=0, method=1 )
Print(ginvA)

ginvA2=ginvA + Y%*%(diag(nrow(A))-A%*%ginvA)
temp=chkginv( A, ginvA2, print=2 )
Print(ginvA2)

ginvA3=ginvA + (diag(ncol(A))-ginvA%*%A)%*%Y
temp=chkginv( A, ginvA3, print=2 )
Print(ginvA3)

ginvA4 =ginvA  +  Y%*%Q( A, Ginv=1 )
temp=chkginv( A, ginvA4, print=1)
Print(ginvA4)