Ginv {lazy.mat}

R Documentation

Generalized Inverse

Description

Calculates Generalized Inverse of the input matrix X.

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

X	An input matrix
MP	= 1 to calculate Moor-Penrose inverse.
method	See the descriptions below.
orth	Option for QRGS function to calculate QR or rank factorization.
MP1	The value of MP option when this function is used recursively.
method1	The value of method option when this function is used recursively.
MP2	The value of MP option when this function is used recursively.
method2	The value of method option when this function is used recursively.
nopermute	= 1 not allow the permutation of rows/cols by matSwp
eps	Tolerance for pivot in matSwp
MASS	= 0 not to use MASS::ginv equivalent settings of tol
tol	A relative tolerance to detect zero singular values.
chkginv	= 1 to check if the result is valid.
epsc	tolerance for chkinv.

Details

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)

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