1. Introduction
Throughout this paper,
denotes the set of real
matrices. For a matrix
,
is the transpose of
A,
denotes the rank of
A,
is the spectral norm of
A, and
is the Frobenius norm of
A. For a vector
a,
is its
∞-norm, and
the 2-norm. The notation
is a matrix whose components are the absolute values of the corresponding components of
A. For any matrix
A, the following four equations uniquely define the Moore–Penrose inverse
of
A [
1]:
The generalized inverse
is defined by
where
,
denotes weight matrix and
is the orthogonal projection onto the null space of
C and
may not have full rank and
J is a signature matrix defined by
The generalized inverse
originated from the equality constrained indefinite least squares problem (EILS), which is stated as follows [
2,
3,
4,
5]:
where
and
. The EILS problem has a unique solution:
under the following condition:
The above condition implies
then (
5) ensures the existence and uniqueness of generalized inverse
(see [
2,
6]). The generalized inverse
has significant applications in the study of EILS algorithms, the analysis of large-scale structure, error analysis, perturbation theory, and the solution of the EILS problem [
2,
3,
4,
5,
7,
8,
9,
10]. The EILS problem was first demonstrated by Bojanczyk et al. [
5]. Additionally, we reveal some detailed work on the perturbation analysis of this problem. The perturbation theory of the EILS problem was discussed by Wang [
11] and extended by Shi and Liu [
8] based on the hyperbolic MGS elimination method. Diao and Zhou [
12] recovered the linearized estimate of the backward error of this problem. Later, Li et al. [
13] investigated the componentwise condition numbers for the EILS problem. Recently, Wang and Meng [
14] studied the condition numbers and normwise perturbation analysis of the EILS problem.
Componentwise perturbation analysis has received significant attention in recent years; for references, see [
15,
16,
17,
18,
19]. The motivation for studying componentwise perturbation analysis is reasonable for research because, if the perturbation in the input data is measured componentwise rather than by norm, it may help us to measure the sensitivity of a function more accurately [
15], and improve the exactness and effectiveness of the EILS solution computation. It has attracted many authors’ attention to consider the componentwise perturbation analysis in which the least squares problem [
16] and the weighted least squares problem [
17] are included. In this article, we continue the research on componentwise perturbation analysis of the EILS problem. We can recover the componentwise perturbation bounds of the indefinite least squares problem with the intermediate result.
The generalized inverse
reduce to
K-weighted pseudoinverse
when
and
K has a full row rank. This pseudoinverse was expanded to the MK-weighted pseudoinverse
by Wei and Zhang [
6], which describes its structure and uniqueness. Its algorithm was developed by Elden [
20]. According to Wei [
21], the expression of
based on GSVD was investigated. A perturbation equation for
was given by Gulliksson et al. [
22]. The condition numbers for the
K-weighted pseudoinverse
and their statistical estimate were recently provided by Mahvish et al. [
23].
The condition number is a well-known research topic in numerical analysis that estimates the worst-case sensitivity of input data to small perturbations on it (see [
24,
25,
26] and references therein). The normwise condition number [
25] has the disadvantage of disregarding the scaling structure of both input and output data. To address this issue, the terms mixed and componentwise condition numbers are introduced [
26]. Mixed condition numbers employ componentwise error analysis for input data and normwise error analysis for output data. On the other hand, the componentwise condition numbers employ componentwise error analysis for input and output data. In fact, due to rounding errors and data storage difficulties, it is more practical to estimate input errors componentwise rather than normwise. However, the condition numbers of the generalized inverse
have not been discussed until now. Inspired by this, we attempt to present the explicit expressions of normwise, mixed and componentwise condition numbers for the generalized inverse
, as well as their statistical estimation due to their importance in EILS research.
The rest of this manuscript is organized as follows:
Section 2 provides some preliminaries that will be helpful for the upcoming discussions. With the intermediate result, i.e., the derivative of
, we can recover the explicit expression of condition numbers for the solution of the EILS problem in
Section 3.
Section 4 will present the componentwise perturbation analysis for the EILS problem. In
Section 5, we propose the first two algorithms for the normwise condition number by using the probabilistic spectral norm estimator [
27] and the small-sample statistical condition estimation [
28] method. Additionally, we construct the third algorithm for the mixed and componentwise condition numbers by using the small-sample statistical condition estimation [
28] method. To check the efficiency of these algorithms, we demonstrate them through numerical experiments in
Section 6.
2. Preliminaries
In this part, we introduce some definitions and important results, which will be used in the upcoming sections.
Firstly, we define the entrywise division between two vectors
and
by
with
Following [
1,
26,
29], the componentwise distance between
v and
w is defined by
Note that when
,
gives the relative distance from
v to
w with respect to
w, while the absolute distance for
. We describe the distance between the matrices
as follows:
In order to define the mixed and componentwise condition numbers, we also need to define the set
and
for given
Definition 1 ([
29])
. Let be a continuous mapping defined on an open set , and such that .- (i)
The normwise condition number of ℵ at v is given by - (ii)
The mixed condition number of ℵ at v is given by - (iii)
The componentwise condition number of ℵ at v is given by
When the map
ℵ in Definition 1 is Fréchet differentiable, the following lemma given in [
29] makes the computation of condition numbers easier.
Lemma 1 ([
29])
. Under the assumptions of Definition 1, and supposing ℵ is differentiable at v, we havewhere stands for the Fréchet derivative of ℵ at v. To obtain the explicit expressions of the above condition numbers, we need some properties of the Kronecker product [
30] between
X and
Y:
where the matrix
Z has a suitable dimension, and
is the vec-permutation matrix, which depends only on the dimensions
m and
n.
Now, we present the following two lemmas, which will be helpful for obtaining condition numbers and their upper bounds.
Lemma 2 ([
31], p. 174, Theorem 5)
. Let S be an open subset of , and let be a matrix function defined and times (continuously) differentiable on S. If is constant on then is k times (continuously) differentiable on S, and Lemma 3 ([
1])
. For any matrices and with dimensions making the following well definedwe haveand 3. Condition Numbers
First, we define a mapping
by
Here, , , and for matrix , and
Then, using Definition 1, we present the definitions of the normwise, mixed, and componentwise condition numbers for generalized inverse
as given in [
32]:
With the help of the
operator, Frobenius, spectral, and Max norms, we can rewrite the definitions of normwise, mixed and componentwise condition numbers as follows:
In the following, we find the expression of the Fréchet derivative of at u.
Lemma 4. Let the mapping ϕ be continuous. Then, the Fréchet differential at u is:where Proof. Differentiating both sides of (
2), we obtain
From ([
3], Theorem 2.2), we obtain
Thus, substituting (
20) into (
19) and differentiating both sides of the equation, we can deduce
Further, using (
9), we have
Noting (
20), (
2), and
, the previous equation may be expressed as
Further, by the fact
, (
21), and
the above equation may be simplified as follows:
Considering
, we obtain
Substituting this fact into (
23) implies
We can rewrite the above equation by using (
2) and (
20) as
By applying “vec” operator on (
25), and using (
6) and (
7), we obtain
That is,
Thus, we have obtained the required result by using the definition of Fréchet derivative. □
Remark 1. Setting , , and C as full row rank, we have andwhere the latter is just the result of ([23], Lemma 3.1), with which we can recover the condition numbers for K-weighted pseudoinverse [23]. Using the straightforward results of Lemmas 1 and 4, we derive the following condition numbers for
Theorem 1. The normwise, mixed and componentwise condition numbers for defined in (11)–(13) are Next, we provide easier computable upper bounds by minimizing the cost of computing the above condition numbers. The estimation of the upper bounds will be demonstrated by numerical experiments in
Section 6.
Corollary 1. The upper bounds of normwise, mixed and componentwise condition numbers for are Proof. For any two matrices
X and
Y, it is well-known that
. With the help of Theorem 1, and (
8), we obtain
Secondly, by using Lemma 3 and Theorem 1, we obtain
and finally, we have
□
Remark 2. Using the GHQR factorization [3] on A and C in (2) and (5):where and and a -orthogonal matrix, (i.e., ), and are lower triangular and non-singular. We havewhere , ; and are, respectively, the submatrices of U and H obtained by taking the first r columns. Putting all the above terms into (18) leads to Remark 3. We can obtain
using the
expression, where (
4) is the solution of EILS problem (
3). By differentiating (
4), we obtain
Thus, using (
20), we obtain
Substituting (
25) into above equation and using (
9), we have
which together with (
20)–(
22) give
Noting (
24), the above equation can be rewritten as
Further, by (
20) and (
4), we have
where
,
. By utilizing operator “vec” on (
30), and using (
6) and (
7), we obtain
From the above result, we can recover the condition numbers of the EILS problem provided in [
3,
13,
14]. Further, we observe that
. Applying the same procedure, we can determine
and condition numbers for residuals of EILS.
4. Componentwise Perturbation Analysis
In the following section, we derive a componentwise perturbation analysis of the augmented system for the EILS problem.
Let the perturbations
,
,
and
satisfy
,
and
for a small
and
. Suppose that the perturbed augmented system is
Denoting
and the perturbations
When
A is of full column rank and
C has full row rank,
S is invertible. It can be verified that
If the spectral radius
then
is invertible. Clearly, the condition
implies (
31). The following results [
24] are important for Theorem 2.
Lemma 5. The perturbed system of a linear system is defined as follows:where is the solution to the perturbed system, when the perturbations and are sufficiently small such that is invertible, the perturbation in the solution v satisfieswhich impliesFurthermore, when the spectral radius , we have Now, we have the following bounds for the perturbations in the equality constrained indefinite least squares solution and residual.
Theorem 2. Under the above assumption, for any satisfying the condition (32), when the componentwise perturbations , and , the error in the solution is bounded byand error in the residual is bounded by Proof. Since the condition (
32) implies (
31), applying (
33) in Lemma 5, we obtain
Finally, using the conditions
,
and
, and the explicit form of
, the upper bounds (
34) and (
35) can be obtained. □
Furthermore, we can obtain the componentwise perturbation bounds of the indefinite least squares solution and its residual.
Remark 4. Assume that C is a zero matrix, , and . Using the above notations, for any , if the componentwise perturbations satisfy and , then the error in the solution is bounded byand the error in the residual is bounded by 5. Statistical Condition Estimates
This section proposes three algorithms for estimating the normwise, mixed and componentwise condition numbers for the generalized inverse
Algorithm 1 is based on a probabilistic condition estimator method [
27] and utilized to examine the normwise condition number for K-weighted pseudoinverse
[
23], ILS problem [
33], constrained and weighted least squares problem [
34] and Tikhonov regularization of total least squares problem [
35]. Based on the SSCE method [
28], we develop Algorithm 2 to estimate the normwise condition number; for details, see [
23,
33,
36,
37,
38].
Algorithm 1: Probabilistic condition estimator for the normwise condition number |
|
Algorithm 2: Small-sample statistical condition estimation method for the normwise condition number |
Generate matrices with each entry in and Orthonormalize the following matrix
to obtain by modified Gram-Schmidt orthogonalization process. Each can be converted into the corresponding matrices by applying the unvec operation. Let . Approximate and by
For compute
Compute the absolute condition vector by
where the square operation is applied to each entry of and the square root is also applied componentwise. Estimate the normwise condition number ( 26) by
where .
|
To estimate the mixed and componentwise condition numbers, we need the following SSCE method, which is from [
28] and has been applied to many problems (see, e.g., [
23,
32,
33,
34,
35]).