Features Of Inverse to Block - Tridiagonal Matrices with Zero
Leading Block Angular Minors
Turdimukhammad Rakhmonov
Tashkent University of Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan
https://doi.org/10.5281/zenodo.10471069
Keywords:
Matrix, minor, representations, system solutions, block-tridiagonal matrix, linear algebra, computational
mathematics, ill-posed system.
Abstract:
The article reports about construction of a previously unknown structure and shows the results of the study of
direct representations of inverse to block-tridiagonal matrices with zero (or close to zero) leading block-
angular minors.The study of the structure of the inverse matrix are related, firstly, to the solution of a system
of linear algebraic equations, and secondly, to the inversion of a matrix of the indicated type. The task of
inversing a block-tridiagonal type matrix arises while solving the problem of processing and analyzing
experimental data of high-energy physics. At the same time, the special-type likelihood function is minimized,
the parameters of which are the kinematic parameters of the secondary particles. The experimental data
obtained by measurements contain errors. Therefore, stringent requirements are set to numerical methods for
the stability of intermediate computer calculations.
INTRODUCTION
This article is devoted to the construction and
analysis of the structure of direct representations of
inverse to block-tridiagonal matrices with zero (or
close to zero) leading block-angular minors. The
study of the properties of the explicit type of inverse
matrices and knowledge of their matrix structure is
necessary when constructing direct (non-iteration)
methods for solving poorly conditioned systems of
linear algebraic equations. It is known that solutions
of poorly conditioned systems of linear algebraic
equations with block - tridiagonal matrices arise when
the following problems are formulated: in the process
of numerical solution of boundary value problems of
mathematical physics; processing and analysis of
experimental data in high-energy physics; modeling
and numerical solution of geophysics problems,etc.
(see, for example, [1 - 6]).
So, let
𝐶
be a block-three-diagonal matrix of the
general type
𝐶 =
[
𝑞
1
𝑟
2
𝑝
2
𝑞
2
𝑟
3
𝑝
3
𝑞
3
𝑟
4
−
−
−
𝑝
𝑚−1
𝑞
𝑚−1
𝑟
𝑚
𝑝
𝑚
𝑞
𝑚
]
,
(1)
where
{𝑞
𝜉
}
𝜉=1
𝑚
- diagonal elements - blocks of the
matrix
𝐶
, which are generally square matrices of
various dimensions, at
{𝑝
𝜉
}
𝜉=2
𝑚
and
{𝑟
𝜉
}
𝜉= 2
𝑚
- under
(above) diagonal elements - blocks of the matrix
𝐶
,
which are generally rectangular matrices, the
dimensions of which are determined by the
dimensions of the corresponding square matrices
{𝑞
𝜉−1
𝑎𝑛𝑑 𝑞
𝜉
}
𝜉=2
𝑚
.
Tridiagonal and block - tridiagonal
matrices play an independent role in solving many
problems of computational mathematics and
mathematical physics. Indeed, the numerical solution
of boundary value problems of mathematical physics
leads, in the general case, to the solution of algebraic
problems with tridiagonal tape and block tridiagonal
matrices. It is known (see, for example, [1]) that the
factorized representation of the error matrix taking
into account multiple Coulomb scattering, energy
losses, as well as hardware errors and arbitrary track
splitting when processing information in high-energy
physics experiments contains matrices of the form
𝐶(1)
. Moreover, the methods of constructing and the
efficiency of computational algorithms for solving
the above basic problems of linear algebra (with such
matrices)
are
largely
determined
by
the
representation’s characteristic of them and their
inverse matrices, as well as representations of their
leading angular minors and determinants. Extensive
literature is devoted to the method of solving systems
of linear algebraic equations with tridiagonal
matrices. These methods are largely associated with
the construction of economical algorithms for the
characteristic types of systems that arise when solving
problems of mathematical physics. The main task in
their development is to minimize the number of
arithmetic operations taking into account the specifics
of the equations or to ensure reliability, i.e. stability
to the accumulation of rounding errors when
implemented on a computer.
MATERIALS AND METHODS
Direct representations of inverse to block-
tridiagonal matrices with zero leading block-angular
minors.In the cases [1, 8 - 9], the results have been
obtained on various direct factorized representations
of
𝐵
𝑖𝑗
- elements - blocks of the inverse matrix to
𝐶(1)
if it has only one upper or lower or at the same time
one upper and one lower zero (or close to zero)
leading block - corner minors (Under the leading
{𝛥
1
𝜉
}
𝜉=1
𝑚
- upper and
{𝛥
𝜉
𝑚
}
𝜉=1
𝑚
- lower block - corner
minors
𝐶(1)
, where the determinants of the leading
truncated submatrices
𝐶
1
𝑘−1
and
𝐶
𝑘+2
𝑚
starting with
𝑞
1
and
𝑞
𝑚
respectively) are understood. In this case, we
have used, introduced by us [9] in the case of the
vanishing of some of their leading upper (lower)
block - angular minors of nondegenerate matrices
𝐶(1)
, generalized matrix (processes) - sequences of
the type:
{
𝐼𝑓
𝑑𝑒𝑡( 𝛬
𝜉
) ≠ 0, 𝑡ℎ𝑒𝑛 𝛬
𝜉+1
= 𝑞
𝜉
− 𝑝
𝜉
𝛬
𝜉
−1
𝑟
𝜉
, 𝛬
2
= 𝑞
1
,
(𝑑𝑒𝑡( 𝑞
1
) ≠ 0), 𝜉 = 2, . . . , 𝑚.
𝐼𝑓
𝑑𝑒𝑡( 𝛬
𝜉
) = 0 𝑓𝑜𝑟 𝑎𝑛𝑦𝜉 𝑓𝑟𝑜𝑚 (3 ≤ 𝜉 ≤ 𝑚 − 1),
𝑡ℎ𝑒𝑛 𝛬
𝜉+1
=? , 𝑏𝑢𝑡 𝛬
𝜉+2
= 𝑞
𝜉+1
, 𝑤ℎ𝑒𝑟𝑒 𝑑𝑒𝑡( 𝑞
𝜉+1
) ≠ 0
(2)
{
𝐼𝑓
𝑑𝑒𝑡( 𝐺
𝜉
) ≠ 0, 𝑡ℎ𝑒𝑛 𝐺
𝜉−1
= 𝑞
𝜉
− 𝑟
𝜉+1
𝐺
𝜉
−1
𝑝
𝜉+1
,
𝐺
𝑚−1
= 𝑞
𝑚
, (𝑑𝑒𝑡( 𝑞
𝑚
) ≠ 0), 𝜉 = 𝑚 − 1, . . . ,1.
𝐼𝑓
𝑑𝑒𝑡( 𝐺
𝜉
) = 0 𝑓𝑜𝑟 𝑎𝑛𝑦 𝜉 𝑓𝑟𝑜𝑚 (2 ≤ 𝜉 ≤ 𝑚 − 2),
𝑡ℎ𝑒𝑛 𝐺
𝜉−1
=? , 𝑏𝑢𝑡 𝐺
𝜉−2
= 𝑞
𝜉−1
, 𝑤ℎ𝑒𝑟𝑒 𝑑𝑒𝑡( 𝑞
𝜉−1
) ≠ 0
(3)
It was established in [9] that if one of the
following conditions is satisfied for sequences of
matrices
{𝛬}(2)
and
{𝐺}(3)
:
𝐼. {
[𝑑𝑒𝑡( 𝛬
𝑘+1
) = 0 𝑓𝑜𝑟𝑎𝑛𝑦 𝑘 𝑓𝑟𝑜𝑚 (2 ≤ 𝑘 ≤ 𝑚 − 2),
𝑏𝑢𝑡 {𝑑𝑒𝑡( 𝛬
𝜉
) ≠ 0}
𝜉=2
𝑘
, {𝑑𝑒𝑡( 𝛬
𝜉
) ≠ 0}
𝜉=𝑘+3
𝑚+1
] →
[𝛥
1
𝑘
= 0 𝑓𝑜𝑟𝑎𝑛𝑦 𝑘 𝑓𝑟𝑜𝑚 (2 ≤ 𝑘 ≤ 𝑚 − 2)];
𝐼𝐼. {
[𝑑𝑒𝑡( 𝐺
𝑘
) = 0 𝑓𝑜𝑟𝑎𝑛𝑦 𝑘 𝑓𝑟𝑜𝑚 (2 ≤ 𝑘 ≤ 𝑚 − 2),
𝑏𝑢𝑡 {𝑑𝑒𝑡( 𝐺
𝜇
) ≠ 0}
𝜇=𝑘+1
𝑚−1
, {𝑑𝑒𝑡( 𝐺
𝜇
) ≠ 0}
𝜇=0
𝑘−2
] →
[𝛥
𝑘+1
𝑚
= 0 𝑓𝑜𝑟𝑎𝑛𝑦 𝑘𝑓𝑟𝑜𝑚 (2 ≤ 𝑘 ≤ 𝑚 − 2)];
𝐼𝐼𝐼.
{
[𝑑𝑒𝑡( 𝐺
𝑘
) = 0 𝑓𝑜𝑟𝑎𝑛𝑦 𝑘 𝑓𝑟𝑜𝑚 (2 ≤ 𝑘 ≤ 𝑚 − 2),
𝑏𝑢𝑡 {𝑑𝑒𝑡( 𝐺
𝜉
) ≠ 0}
𝜉= 𝑘+1
𝑚−1
, {𝑑𝑒𝑡( 𝐺
𝜉
) ≠ 0}
𝜉= 0
𝑘−2
]𝑎𝑛𝑑
[𝑑𝑒𝑡( 𝛬
𝑘+1
) = 0𝑓𝑜𝑟𝑎𝑛𝑦 𝑘 𝑓𝑟𝑜𝑚 (2 ≤ 𝑘 ≤ 𝑚 − 2),
𝑏𝑢𝑡 {𝑑𝑒𝑡( 𝛬
𝜉
) ≠ 0}
𝜉=2
𝑘
, {𝑑𝑒𝑡( 𝛬
𝜉
) ≠ 0}
𝜉=𝑘+3
𝑚+1
] →
[𝛥
1
𝑘
= 0𝑎𝑛𝑑 𝛥
𝑘+1
𝑚
= 0 𝑎𝑡𝑡ℎ𝑒𝑠𝑎𝑚𝑒 𝑡𝑖𝑚𝑒𝑓𝑜𝑟 𝑎𝑛𝑦 𝑘 𝑓𝑟𝑜𝑚 (2 ≤ 𝑘 ≤ 𝑚 − 2)],
then the inverse matrix
𝐵 = 𝐶
−1
has the following structure
𝐵 =
[
[
[𝐵
𝑖𝑗
]
11
]
𝐵
1 𝑘
𝐵
2 𝑘
⋯
𝐵
𝑘−1𝑘
𝐵
1 𝑘+1
𝐵
2 𝑘+1
⋯
𝐵
𝑘−1𝑘+1
[
[𝐵
𝑖𝑗
]
12
]
𝐵
𝑘
1
⋯ 𝐵
𝑘
𝑘−1
(𝐵
𝑘𝑘
= 𝑋
𝑘𝑘
) (𝐵
𝑘𝑘+1
= 𝑋
𝑘𝑘+1
)
𝐵
𝑘 𝑘+2
⋯ 𝐵
𝑘𝑚
𝐵
𝑘+1 1
⋯ 𝐵
𝑘+1 𝑘−1
(𝐵
𝑘+1𝑘
= 𝑋
𝑘+1𝑘
) (𝐵
𝑘+1𝑘+1
= 𝑋
𝑘+1𝑘+1
)
𝐵
𝑘+1 𝑘+2
⋯ 𝐵
𝑘+1 𝑚
[
[𝐵
𝑖𝑗
]
21
]
𝐵
𝑘+2 𝑘
𝐵
𝑘+3 𝑘
⋯
𝐵
𝑚
𝑘
𝐵
𝑘+2 𝑘+1
𝐵
𝑘+3 𝑘+1
⋯
𝐵
𝑚
𝑘+1
[
[𝐵
𝑖𝑗
]
22
]
]
(4)
While for
𝐵
𝑖𝑗
- elements of blocks of
𝐵(4)
, the
following direct factorized representations take place:
Representation 1.1 (on condition I)
[𝐵
𝑖𝑗
(𝛬)]
11
=
{
𝐵̃
𝑖𝑗
(𝛬) + 𝜏
𝑖
𝑋
𝑘𝑘
𝑧
𝑗
,
1 ≤ 𝑖 < 𝑗 ≤ 𝑘 − 1,
𝐵̃
𝑖𝑖
(𝛬) + 𝜏
𝑖
𝑋
𝑘𝑘
𝑧
𝑖
,
1 ≤ (𝑖 = 𝑗) ≤ 𝑘 − 1,
𝐵̃
𝑖𝑗
(𝛬) + 𝜏
𝑖
𝑋
𝑘𝑘
𝑧
𝑗
,
1 ≤ 𝑗 < 𝑖 ≤ 𝑘 + 1;
[𝐵
𝑖𝑗
(𝛬)]
22
=
{
𝐵̃
𝑖𝑗
(𝛬) + 𝜉
𝑖
𝑋
𝑘+1 𝑘+1
𝑓
𝑗
,
𝑘 + 2 ≤ 𝑖 < 𝑗 ≤ 𝑚;
𝐵̃
𝑖𝑖
(𝛬) + 𝜉
𝑖
𝑋
𝑘+1 𝑘+1
𝑓
𝑖
,
𝑘 + 2 ≤ (𝑖 = 𝑗) ≤ 𝑚;
𝐵̃
𝑖𝑗
(𝛬) + 𝜉
𝑖
𝑋
𝑘+1 𝑘+1
𝑓
𝑗
,
𝑘 + 2 ≤ 𝑗 < 𝑖 ≤ 𝑚.
(5)
{
{
𝐵
𝑘𝑗
= 𝑋
𝑘𝑘
𝑧
𝑗
;
𝐵
𝑘+1 𝑗
= 𝑋
𝑘+1 𝑘
𝑧
𝑗
,
1 ≤ 𝑗 ≤ 𝑘 − 1.
{
𝐵
𝑖 𝑘
= 𝜏
𝑖
𝑋
𝑘𝑘
;
𝐵
𝑖 𝑘+1
= 𝜏
𝑖
𝑋
𝑘 𝑘+1
,
1 ≤ 𝑖 ≤ 𝑘 − 1.
{
{
𝐵
𝑘 𝑗
= 𝑋
𝑘 𝑘+1
𝑓
𝑗
;
𝐵
𝑘+1 𝑗
= 𝑋
𝑘+1 𝑘
𝑓
𝑗
,
𝑘 + 2 ≤ 𝑗 ≤ 𝑚.
{
𝐵
𝑖 𝑘
= 𝜉
𝑖
𝑋
𝑘+1𝑘
;
𝐵
𝑘+1 𝑗
= 𝜉
𝑖
𝑋
𝑘+1 𝑘+1
,
𝑘 + 2 ≤ 𝑖 ≤ 𝑚.
(6)
{
[𝐵
𝑖 𝑗
(𝛬)]
21
= 𝜉
𝑖
𝑋
𝑘+1 𝑘
𝑧
𝑗
,
𝑘 + 2 ≤ 𝑖 ≤ 𝑚,
1 ≤ 𝑗 ≤ 𝑘 − 1;
{
[𝐵
𝑖 𝑗
(𝛬)]
12
= 𝜏
𝑖
𝑋
𝑘+1 𝑘
𝑓
𝑗
,
1 ≤ 𝑖 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑗 ≤ 𝑚;
(7)
Here in (4) - (7) aredefined:
{
{𝜏
𝑖
= 𝛱
𝜉=𝑖+1
𝑘
𝑐
𝜉
}
𝑖=1
𝑘−1
, {𝜉
𝑖
= −𝐵̃
𝑖𝑖
(𝛬)𝛱
𝜉=𝑘+3
𝑖
𝛽
𝜉
𝑝
𝑘+2
}
𝑖=𝑘+2
𝑚
,
{𝑧
𝑗
= 𝛱
𝜉=𝑗+1
𝑘
𝛽
𝜉
}
𝑗=1
𝑘−1
, {𝑓
𝑗
= −𝑟
𝑘+2
𝛱
𝜉=𝑘+3
𝑗
𝑐
𝜉
𝐵̃
𝑗𝑗
(𝛬)}
𝑗=𝑘+2
𝑚
.
( 8)
Here
𝐵̃
𝑖𝑗
(𝛬)
- elements - blocks of truncated
inverse
submatrices
𝐵̃
1
𝑘−1
= [𝐶
1
𝑘−1
(𝛬)]
−1
and
𝐵̃
𝑘+2
𝑚
= [𝐶
𝑘+2
𝑚
(𝛬)]
−1
, as functions of a sequence of
{𝛬}
matrices, respectively, can be represented as
𝐵̃
𝑖𝑗
(𝛬) =
{
𝐵̃
𝑖𝑗
(𝛬) = 𝑐
𝑖+1
𝐵̃
𝑖+1𝑗
(𝛬),
1 ≤ 𝑖 < 𝑗 ≤ 𝑘 − 1; 𝑘 + 2 ≤ 𝑖 < 𝑗 ≤ 𝑚,
𝐵̃
𝑖𝑖
(𝛬) = 𝛬
𝑖+1
−1
+ 𝑐
𝑖+1
𝐵̃
𝑖+1 𝑖+1
(𝛬)𝛽
𝑖+1
,
1 ≤ (𝑖 = 𝑗) ≤ 𝑘 − 1; 𝑘 + 2 ≤ (𝑖 = 𝑗) ≤ 𝑚,
𝐵̃
𝑖𝑗
(𝛬) = 𝐵̃
𝑖 𝑖+1
(𝛬)𝛽
𝑖+1
,
1 ≤ 𝑗 < 𝑖 ≤ 𝑘 − 1; 𝑘 + 2 ≤ 𝑗 < 𝑖 ≤ 𝑚.
(9)
Representation 1.2 (oncondition II)
[𝐵
𝑖𝑗
(𝐺)]
11
=
{
𝐵̃
𝑖𝑗
(𝐺) + 𝑓̂
𝑖
𝑋
𝑘𝑘
𝜉̂
𝑗
,
1 ≤ 𝑖 < 𝑗 ≤ 𝑘 − 1,
𝐵̃
𝑖𝑗
(𝐺) + 𝑓̂
𝑖
𝑋
𝑘𝑘
𝜉̂
𝑖
,
1 ≤ (𝑖 = 𝑗) ≤ 𝑘 − 1,
𝐵̃
𝑖𝑗
(𝐺) + 𝑓̂
𝑖
𝑋
𝑘𝑘
𝜉̂
𝑗
,
1 ≤ 𝑗 < 𝑖 ≤ 𝑘 − 1;
[𝐵
𝑖𝑗
(𝐺)]
22
=
{
𝐵̃
𝑖𝑗
(𝐺) + 𝑧̂
𝑖
𝑋
𝑘+1 𝑘+1
𝜏̂
𝑗
,
𝑘 + 2 ≤ 𝑖 < 𝑗 ≤ 𝑚;
𝐵̃
𝑖𝑗
(𝐺) + 𝑧̂
𝑖
𝑋
𝑘+1 𝑘+1
𝜏̂
𝑖
,
𝑘 + 2 ≤ (𝑖 = 𝑗) ≤ 𝑚;
𝐵̃
𝑖𝑗
(𝐺) + 𝑧̂
𝑖
𝑋
𝑘+1 𝑘+1
𝜏̂
𝑗
,
𝑘 + 2 ≤ 𝑗 < 𝑖 ≤ 𝑚.
(10)
{
{
𝐵
𝑘𝑗
= 𝑋
𝑘𝑘
𝜉̂
𝑗
,
𝐵
𝑘+1 𝑗
= 𝑋
𝑘+1 𝑘
𝜉̂
𝑗
,
1 ≤ 𝑗 ≤ 𝑘 − 1.
{
𝐵
𝑖 𝑘
= 𝑓̂
𝑖
𝑋
𝑘𝑘
,
𝐵
𝑖 𝑘+1
= 𝑓̂
𝑖
𝑋
𝑘 𝑘+1
,
1 ≤ 𝑖 ≤ 𝑘 − 1.
,
{
{
𝐵
𝑘𝑗
= 𝑋
𝑘𝑘+1
𝜏̂
𝑗
,
𝐵
𝑘+1 𝑗
= 𝑋
𝑘+1 𝑘+1
𝜏̂
𝑗
,
𝑘 + 2 ≤ 𝑗 ≤ 𝑚.
{
𝐵
𝑖 𝑘
= 𝑧̂
𝑖
𝑋
𝑘+1𝑘
,
𝐵
𝑘+1 𝑗
= 𝑧̂
𝑖
𝑋
𝑘+1 𝑘+1
,
𝑘 + 2 ≤ 𝑖 ≤ 𝑚.
(11)
{
[𝐵
𝑖 𝑗
(𝐺)]
12
= 𝑓̂
𝑖
𝑋
𝑘𝑘+1
𝜏̂
𝑗
,
1 ≤ 𝑖 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑗 ≤ 𝑚.
{
[𝐵
𝑖 𝑗
(𝐺)]
21
= 𝑧̂
𝑖
𝑋
𝑘+1𝑘
𝜉̂
𝑗
,
𝑘 + 2 ≤ 𝑖 ≤ 𝑚,
1 ≤ 𝑗 ≤ 𝑘 − 1.
(12)
Where
{
{𝜉̂
𝑗
= −𝑝
𝑘
𝛱
𝜉=𝑗+1
𝑘−2
𝑐
𝜉
𝐵̃
𝑗𝑗
(𝐺)}
𝑗=1
𝑘−1
,
{𝜏̂
𝑗
= 𝛱
𝜉=𝑘+2
𝑗
𝛽̂
𝜉
}
𝑗=𝑘+2
𝑚
,
{𝑓̂
𝑖
= −𝐵̃
𝑗𝑗
(𝐺)𝛱
𝜉=𝑖+1
𝑘−1
𝛽
𝜉
𝑟
𝑘
}
𝑖=1
𝑘−1
,
{𝑧̂
𝑖
= 𝛱
𝜉=𝑘+2
𝑖
𝑐̂
𝜉
}
𝑖=𝑘+1
𝑚
.
(13)
Here
𝐵̃
𝑖𝑗
(𝐺)
- elements - blocks of truncated
submatrices
𝐵̃
1
𝑘−1
= [𝐶
1
𝑘−1
(𝐺)]
−1
and
𝐵̃
𝑘+2
𝑚
=
[𝐶
𝑘+2
𝑚
(𝐺)]
−1
, as functions of a sequence of
{𝐺}
matrices, respectively, can be represented as
𝐵̃
𝑖𝑗
(𝐺) =
{
𝐵̃
𝑖𝑗
(𝐺) = 𝐵̃
𝑖𝑗−1
(𝐺)𝛽̂
𝑗
,
𝑖𝑓 1 ≤ 𝑖 < 𝑗 ≤ 𝑘 − 1; 𝑘 + 2 ≤ 𝑖 < 𝑗 ≤ 𝑚,
𝐵̃
𝑖𝑖
(𝐺) = 𝐺
𝑖−1
−1
+ 𝑐̂
𝑖
𝐵̃
𝑖−1 𝑖−1
(𝐺)𝛽̂
𝑖
,
𝑖𝑓 1 ≤ (𝑖 = 𝑗) ≤ 𝑘 − 1; 𝑘 + 2 ≤ (𝑖 = 𝑗) ≤ 𝑚,
𝐵̃
𝑖𝑗
(𝐺) = 𝑐̂
𝑖
𝐵̃
𝑖−1 𝑗
(𝐺),
𝑖𝑓 1 ≤ 𝑗 < 𝑖 ≤ 𝑘 − 1; 𝑘 + 2 ≤ 𝑗 < 𝑖 ≤ 𝑚.
(14)
Representation 1.3 (on condition III)
[𝐵
𝑖𝑗
(𝛬, 𝐺)]
11
=
{
𝐵̃
𝑖𝑗
(𝛬, 𝐺) + 𝑓̂
𝑖
𝑋
𝑘𝑘
𝑧
𝑗
,
1 ≤ 𝑖 < 𝑗 ≤ 𝑘 − 1,
𝐵̃
𝑖𝑖
(𝛬, 𝐺) + 𝑓̂
𝑖
𝑋
𝑘𝑘
𝑧
𝑖
,
1 ≤ (𝑖 = 𝑗) ≤ 𝑘 − 1,
𝐵̃
𝑖𝑗
(𝛬, 𝐺) + 𝑓̂
𝑖
𝑋
𝑘𝑘
𝑧
𝑗
,
1 ≤ 𝑗 < 𝑖 ≤ 𝑘 − 1.
[𝐵
𝑖𝑗
(𝛬, 𝐺)]
22
=
{
𝐵̃
𝑖𝑗
(𝛬, 𝐺) + 𝑧̂
𝑖
𝑋
𝑘+1 𝑘+1
𝑓
𝑗
,
𝑘 + 2 ≤ 𝑖 < 𝑗 ≤ 𝑚,
𝐵̃
𝑖𝑖
(𝛬, 𝐺) + 𝑧̂
𝑖
𝑋
𝑘+1 𝑘+1
𝑓
𝑖
,
𝑘 + 2 ≤ (𝑖 = 𝑗) ≤ 𝑚,
𝐵̃
𝑖𝑗
(𝛬, 𝐺) + 𝑧̂
𝑖
𝑋
𝑘+1 𝑘+1
𝑓
𝑗
,
𝑘 + 2 ≤ 𝑗 < 𝑖 ≤ 𝑚.
(15)
{
{
𝐵
𝑘𝑗
= 𝑋
𝑘𝑘
𝑧
𝑗
,
𝐵
𝑘+1 𝑗
= 𝑋
𝑘+1 𝑘
𝑧
𝑗
,
𝑖𝑓 1 ≤ 𝑗 ≤ 𝑘 − 1.
{
𝐵
𝑖 𝑘
= 𝑓̂
𝑖
𝑋
𝑘𝑘
,
𝐵
𝑖 𝑘+1
= 𝑓̂
𝑖
𝑋
𝑘 𝑘+1
,
𝑖𝑓 1 ≤ 𝑖 ≤ 𝑘 − 1.
,
{
{
𝐵
𝑘𝑗
= 𝑋
𝑘𝑘+1
𝑓
𝑗
,
𝐵
𝑘+1 𝑗
= 𝑋
𝑘+1 𝑘+1
𝑓
𝑗
,
𝑖𝑓 𝑘 + 2 ≤ 𝑗 ≤ 𝑚.
{
𝐵
𝑖 𝑘
= 𝑧̂
𝑖
𝑋
𝑘+1𝑘
,
𝐵
𝑖 𝑘+1
= 𝑧̂
𝑖
𝑋
𝑘+1 𝑘+1
,
𝑖𝑓 𝑘 + 2 ≤ 𝑖 ≤ 𝑚.
(
16)
{
[𝐵
𝑖 𝑗
(𝛬, 𝐺)]
12
= 𝑓̂
𝑖
𝑋
𝑘 𝑘+1
𝑓
𝑗
,
1 ≤ 𝑖 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑗 ≤ 𝑚.
{
[𝐵
𝑖 𝑗
(𝛬, 𝐺)]
21
= 𝑧̂
𝑖
𝑋
𝑘+1 𝑘
𝑧
𝑗
,
𝑘 + 2 ≤ 𝑖 ≤ 𝑚,
1 ≤ 𝑗 ≤ 𝑘 − 1.
(17)
where
{
{𝑓̂
𝑖
= −𝐵̃
𝑖𝑖
(𝛬, 𝐺)𝛱
𝜉=𝑖+1
𝑘−1
𝛽̂
𝜉
𝑟
𝑘
}
𝑖=1
𝑘−1
,
{𝑧̂
𝑖
= 𝛱
𝜉=𝑘+2
𝑖
𝑐̂
𝜉
}
𝑖=𝑘+2
𝑚
,
{𝑧
𝑗
= 𝛱
𝜉=𝑗+1
𝑘
𝛽
𝜉
}
𝑗=1
𝑘−1
,
{𝑓
𝑗
= −𝑟
𝑘+2
𝛱
𝜉=𝑘+3
𝑐
𝜉
𝐵̃
𝑗𝑗
(𝛬, 𝐺)}
𝑗=𝑘+2
𝑚
.
(18)
Here
𝐵̃
𝑖𝑗
(𝛬, 𝐺)
are block elements of truncated submatrices
𝐵̃
1
𝑘−1
= [𝐶
1
𝑘−1
(𝛬, 𝐺)]
−1
and
𝐵̃
𝑘+2
𝑚
=
[𝐶
𝑘+2
𝑚
(𝛬, 𝐺)]
−1
, as functions of a sequence of
{𝛬}
and
{𝐺}
matrices, respectively, can be represented as
𝐵̃
𝑖𝑗
(𝛬, 𝐺) =
{
𝛱
𝜉=𝑖+1
𝑗
𝑐
𝜉
𝐵̃
𝑗𝑗
(𝛬, 𝐺),
1 ≤ 𝑖 < 𝑗 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑖 < 𝑗 ≤ 𝑚.
𝐵̃
𝑖𝑖
(𝛬, 𝐺)𝛱
𝜉=𝑗+1
𝑖
𝛽
𝜉
,
1 ≤ 𝑗 < 𝑖 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑗 < 𝑖 ≤ 𝑚.
𝐵̃
𝑖𝑗
(𝛬, 𝐺) =
{
𝐵
̃
𝑖𝑖
(𝛬, 𝐺)𝛱
𝜉=𝑖+1
𝑗
𝛽̂
𝜉
,
1 ≤ 𝑖 < 𝑗 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑖 < 𝑗 ≤ 𝑚.
𝛱
𝜉=𝑗+1
𝑖
𝑐̂
𝜉
𝐵̃
𝑗𝑗
(𝛬, 𝐺),
1 ≤ 𝑗 < 𝑖 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑗 < 𝑖 ≤ 𝑚.
(19)
𝐵̃
𝑖𝑗
(𝛬, 𝐺) =
{
𝛱
𝜉=𝑖+1
𝑗
𝑐
𝜉
𝐵̃
𝑗𝑗
(𝛬, 𝐺),
1 ≤ 𝑖 < 𝑗 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑖 < 𝑗 ≤ 𝑚.
𝛱
𝜉=𝑗+1
𝑖
𝑐̂
𝜉
𝐵̃
𝑗𝑗
(𝛬, 𝐺),
1 ≤ 𝑗 < 𝑖 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑗 < 𝑖 ≤ 𝑚.
𝐵̃
𝑖𝑗
(𝛬, 𝐺) =
{
𝐵
̃
𝑖𝑖
(𝛬, 𝐺)𝛱
𝜉=𝑖+1
𝑗
𝛽̂
𝜉
,
1 ≤ 𝑖 < 𝑗 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑖 < 𝑗 ≤ 𝑚.
𝐵̃
𝑖𝑖
(𝛬, 𝐺)𝛱
𝜉=𝑗+1
𝑖
𝛽
𝜉
,
1 ≤ 𝑗 < 𝑖 ≤ 𝑘 − 1,
𝑘 + 2 ≤ 𝑗 < 𝑖 ≤ 𝑚.
(20)
Here in (8) - (20)
{
𝛽
𝜉+1
= −(𝑝
𝜉+1
𝛬
𝜉+1
−1
), 𝑐
𝜉+1
= − (𝛬
𝜉+1
−1
𝑟
𝜉+1
),
1 ≤ 𝜉 ≤ 𝑘 − 1; 𝑘 + 2 ≤ 𝜉 ≤ 𝑚 − 1;
𝛽̂
𝜉+1
= − (𝑝
𝜉+1
𝐺
𝜉+1
−1
), 𝑐̂
𝜉+1
= − (𝐺
𝜉+1
−1
𝑟
𝜉+1
),
1 ≤ 𝜉 ≤ 𝑘 − 2; 𝑘 + 1 ≤ 𝜉 ≤ 𝑚 − 1;
𝐵̃
𝜉𝜉
= (𝛬
𝜉+1
+ 𝐺
𝜉−1
− 𝑞
𝜉
)
−1
, 1 ≤ 𝜉 ≤ 𝑘 − 1; 𝑘 + 2 ≤ 𝜉 ≤ 𝑚,
𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 𝑜𝑓 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠 {𝛬} 𝑎𝑛𝑑 {𝐺}
𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑖𝑛 𝑎𝑐𝑐𝑜𝑑𝑎𝑛𝑐𝑒 𝑤𝑖𝑡ℎ (2) ÷ (3).
(21)
At the same time,
unknown
{[𝑋
𝑘𝑘
(𝛬), 𝑋
𝑘𝑘+1
(𝛬), 𝑋
𝑘+1𝑘
(𝛬), 𝑋
𝑘+1𝑘+1
(𝛬)],
1
1
1
1
( ),
( ),
( ),
( ) ,
kk
kk
k
k
k
k
X
G X
G X
G X
G
+
+
+ +
[𝑋
𝑘𝑘
(𝛬, 𝐺), 𝑋
𝑘𝑘+1
(𝛬, 𝐺), 𝑋
𝑘+1𝑘
(𝛬, 𝐺), 𝑋
𝑘+1𝑘+1
(𝛬, 𝐺)]}
- elements - blocks in
𝐵(4)
, as functions of the
sequence
{𝛬}
or
{𝐺}
, or simultaneously as functions of
the
sequences
{𝛬}
and
{𝐺}
under
appropriate
conditions, can be found from the following matrix
equations:
On condition I
[
𝛬
𝑘+1
𝑟
𝑘+1
𝑝
𝑘+1
(𝑞
𝑘+1
− 𝑄
𝑘+1
)
] [
𝑋
𝑘𝑘
(𝛬)
𝑋
𝑘𝑘+1
(𝛬)
𝑋
𝑘+1𝑘
(𝛬)
𝑋
𝑘+1𝑘+1
(𝛬)
]
=
= [
𝐸
𝑘𝑘
𝐸
𝑘+1𝑘+1
]
,
Where
𝑸
𝒌+𝟏
(𝜦) = 𝒓
𝒌+𝟐
𝑩
̃
𝒌+𝟐𝒌+𝟐
(𝑮)𝒑
𝒌+𝟐
. Where
in
(𝑞
𝑘+1
− 𝑄
𝑘+1
) = 𝐺
𝑘
,
if
{𝑑𝑒𝑡( 𝐺
𝜉
) ≠ 0}
𝜉=𝑘+1
𝑚−1
.
On condition II
[
(𝑞
𝑘
− 𝑄
𝑘
)
𝑟
𝑘+1
𝑝
𝑘+1
𝐺
𝑘
] [
𝑋
𝑘𝑘
(𝐺)
𝑋
𝑘𝑘+1
(𝐺)
𝑋
𝑘+1𝑘
(𝐺)
𝑋
𝑘+1𝑘+1
(𝐺)
] =
= [
𝐸
𝑘𝑘
𝐸
𝑘+1𝑘+1
]
,
Where
𝑸
𝒌
(𝑮) = 𝒑
𝒌
𝑩
̃
𝒌+𝟏𝒌+𝟏
(𝜦)𝒓
𝒌
. Where
in
(𝑞
𝑘
− 𝑄
𝑘
) = 𝛬
𝑘+1
,
if
{𝑑𝑒𝑡( 𝛬
𝜉
) ≠ 0}
𝜉=2
𝑘
.
On condition III
[
𝛬
𝑘+1
𝑟
𝑘+1
𝑝
𝑘+1
𝐺
𝑘
] [
𝑋
𝑘𝑘
(𝐺, 𝛬)
𝑋
𝑘𝑘+1
(𝐺, 𝛬)
𝑋
𝑘+1𝑘
(𝐺, 𝛬)
𝑋
𝑘+1𝑘+1
(𝐺, 𝛬)
] =
= [
𝐸
𝑘𝑘
𝐸
𝑘+1𝑘+1
]
.
Remark.
The
validity
of
the
above
representations for
𝐵̃
𝑖𝑗
elements - blocks of the inverse
matrix
𝐵 = 𝐶
−1
was established [9] basing on
verification of the basic equalities of
𝐵С =
Е
= СВ
. In this case, one should use the
following matrix equalities [9]:
{
𝑐̂
𝜉+1
𝐵̃
𝜉 𝜉
= 𝐵̃
𝜉+1𝜉+1
𝛽
𝜉+1
,
𝐵̃
𝜉 𝜉
𝛽̂
𝜉+1
= 𝑐
𝜉+1
𝐵̃
𝜉+1𝜉+1
, 1 ≤ 𝜉 ≤ 𝑘 − 2;
𝑘 + 1 ≤ 𝜉 ≤ 𝑚 − 1,
where are the matrices
{𝑐̂
𝜉+1
, 𝛽
𝜉+1
, 𝑐
𝜉+1
, 𝛽̂
𝜉+1
, 𝐵̃
𝜉
}
- defined in (21).
RESULTS AND DISCUSSIONS
The practical use of existing software packages
and a comparative analysis of the calculation results
based on them show [7] that in the case of poor
conditioning
(especially
pathologically
poor
conditioning , there is matrix condition number -
relative error of calculations with real numbers of a
given computer) the task of developing an effective
software package has not yet been completely solved.
The algorithms of machine-independent software
packages JINRLINPACK were created based on the
new abovementioned representations (methods) [8]..
CONCLUSIONS
Thus, the previously unknown structure of
matrices which are inverse to
𝐵 = 𝐶
−1
case has been
studied for one upper or lower or simultaneously
upper and lower zero (or close to zero) leading block
- angular minors of the matrix
𝐶(1)
. Moreover, for
the elements
𝐵̃
𝑖𝑗
of the truncated submatrices
𝐵̃
1
𝑘−1
=
[𝐶
1
𝑘−1
]
−1
and
𝐵̃
𝑘+2
𝑚
= [𝐶
𝑘+2
𝑚
]
−1
,
various
representations are obtained. It should be noted, that
these results we have been used while constructing a
solution to ill-conditioned systems of linear algebraic
equations of a general form [7-9].
𝐴𝑍 = 𝐹,
(22)
where
𝐴 = (𝑎
𝑖𝑗
)
is a square of order m or a
rectangular dimension of
𝑚 × 𝑛
is a real matrix of
general
form,
𝑖 = 1, … , 𝑚, 𝑗 = 1, … , 𝑛,
𝑍 =
(𝑧
1
, . . . , 𝑧
𝑛
)
𝑇
is the desired
𝐹 = (𝑓
1
, . . . , 𝑓
𝑚
)
𝑇
is the
given n and m - dimensional vectors, respectively.
Numerical experiments have shown that new
programs have on average better basic indicators than
similar programs from the most famous packages
CERNLIB, NAG, LIBJINR, LINA.
REFERENCES
[1]
T.T.
Rakhmonov,
M.I.
Fazylov,
G.A.
Kulabdullaev. On role of matrix algegra in software
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