solving systems of linear equations using matrix inverse methods
Tài liệu Đề tài " Isomonodromy transformations of linear systems of difference equations" pptx
Ngày tải lên :
14/02/2014, 17:20
... there
are matrix elements in A
1
, ,A
n
. Let us call P (z) the monodromy matrix of
(1).
Other results of Birkhoff show that for any periodic matrix P of a specific
form, there exists an equation of the ... Sp(S)=Sp(Y ), Sp(T ) = Sp(X).
The matrix elements of S and T are rational functions of the matrix elements
of X and Y which do not depend on the spectra of X and Y .
Proof 1. Lemma 3.3 proves the uniqueness ... Y
σ(1)
) ···(z −Y
σ(N)
).
The matrix elements of {Y
i
} are rational functions of the matrix elements of
{X
i
} which do not depend on the spectra of {X
i
}.
Proof. The existence and rationality...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
Ngày tải lên :
15/12/2013, 04:15
... columns
of the matrix inverse of A (Đ2.1 and Đ2.3).
ã Calculation of the determinant of a square matrix A (Đ2.3).
If M<N,orifM=Nbut the equations are degenerate, then there are
effectively fewer equations ... simplification is that the matrix A is held
constant, while the right-hand sides, the bs, are changed (Đ2.1Đ2.10).
ã Calculation of the matrix A
−1
which is the matrix inverse of a square
matrix A, i.e., ... equations as unknowns, and there is a good
chance of solving for a unique solution set of x
j
’s. Analytically, there can fail to
be a unique solution if one or more of the M equations is a linear...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
Ngày tải lên :
15/12/2013, 04:15
... C. 1971,
Linear Algebra
,vol.IIof
Handbook for Automatic Com-
putation
(New York: Springer-Verlag).
Westlake, J.R. 1968,
A Handbook of Numerical Matrix Inversion and Solution of Linear Equations
(New ... inverting a matrix, Gauss-Jordan elimination is about as efficient as any
other method. For solving sets of linear equations, Gauss-Jordan elimination
produces both the solution of the equations ... =1,2,3,4) of the vector solution of the jth
right-hand side (j =1,2,3), the one whose coefficients are b
ij
,i =1,2,3,4;and
that the matrix of unknown coefficients y
ij
is the inverse matrix of a
ij
.Inother
words,...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
Ngày tải lên :
15/12/2013, 04:15
... is not used for typical systems of linear equations. However, we will
meet special cases where QR is the method of choice.
100
Chapter 2. Solution of Linear Algebraic Equations
Sample page from ... involve solvinga successionof linear systems each of which
differs only slightly from its predecessor. Instead of doing O(N
3
) operations each time
to solve the equations from scratch, one can often ... solve linear systems. In many applications only the
part (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv.
98
Chapter 2. Solution of Linear Algebraic Equations
Sample...
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
Ngày tải lên :
24/12/2013, 12:16
... to a factor 1.5 by not computing the inverse matrix
as part of the Gauss-Jordan scheme.)
For computing the inverse matrix (which we can view as the case of M = N
right-hand sides, namely the ... York:
McGraw-Hill), Program B-2, p. 298.
Westlake, J.R. 1968,
A Handbook of Numerical Matrix Inversion and Solution of Linear Equations
(New York: Wiley).
Ralston, A., and Rabinowitz, P. 1978,
A ... 42
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C)...
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Tài liệu Solution of Linear Algebraic Equations part 4 docx
Ngày tải lên :
24/12/2013, 12:16
... Johnson, C.R. 1985,
Matrix Analysis
(Cambridge: Cambridge University Press).
2.4 Tridiagonal and Band Diagonal Systems
of Equations
The special case of a system of linear equations that is tridiagonal, ... columns of
B instead of with the unit vectors that would give A’s inverse. This saves a whole
matrix multiplication, and is also more accurate.
Determinant of a Matrix
The determinant of an LU ... modify the loop of the above fragment and (e.g.) divide by powers of ten,
to keep track of the scale separately, or (e.g.) accumulate the sum of logarithms of
the absolute values of the factors...
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Tài liệu Solution of Linear Algebraic Equations part 5 docx
Ngày tải lên :
24/12/2013, 12:16
... Johnson, C.R. 1985,
Matrix Analysis
(Cambridge: Cambridge University Press).
2.4 Tridiagonal and Band Diagonal Systems
of Equations
The special case of a system of linear equations that is tridiagonal, ... Prentice-Hall), Chapters 9, 16, and 18.
Westlake, J.R. 1968,
A Handbook of Numerical Matrix Inversion and Solution of Linear Equations
(New York: Wiley).
Stoer, J., and Bulirsch, R. 1980,
Introduction ... limitations of bandec, and the above
routine does take advantage of the opportunity. In general, when TINY is returned as a
diagonal element of U, then the original matrix (perhaps as modified by roundoff...
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Tài liệu Solution of Linear Algebraic Equations part 12 pdf
Ngày tải lên :
24/12/2013, 12:16
... submatrices. Imagine doing the inversionof a very large matrix, of order
N =2
m
, recursively by partitions in half. At each step, halving the order doubles
the number of inverse operations. But this means ... 104
Chapter 2. Solution of Linear Algebraic Equations
Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
Copyright (C) ... it is that factor at each hierarchical level of the recursion. In total it reduces
the process of matrix multiplication to order N
log
2
7
instead of N
3
.
What about all the extra additions in...
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Tài liệu Solution of Linear Algebraic Equations part 6 pptx
Ngày tải lên :
24/12/2013, 12:16
... n]
of the linear set of equations A · X = B.Thematrix
a[1 n][1 n]
, and the vectors
b[1 n]
and
x[1 n]
are input, as is the dimension
n
.
Also input is
alud[1 n][1 n]
,theLU decomposition of
a
as ... decomposition of A is itself not exact.
A different analytical approach starts with some matrix B
0
that is assumed to be an
approximate inverse of the matrix A,sothatB
0
·Ais approximately the identity matrix ... x.
2.5 Iterative Improvement of a Solution to
Linear Equations
Obviously it is not easy to obtain greater precision for the solution of a linear
set than the precision of your computer’s floating-point...
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Tài liệu Solution of Linear Algebraic Equations part 7 docx
Ngày tải lên :
24/12/2013, 12:16
... Sparse Linear Systems
A system of linear equations is called sparse if only a relatively small number
of its matrix elements a
ij
are nonzero. It is wasteful to use general methods of
linear ... making
the same permutation of the columns of U,elementsofW,andcolumnsofV(or
rows of V
T
), or (ii) forming linear combinations of any columns of U and V whose
corresponding elements of W happen to be ... throwing away one linear combination of the set of equations that
we are trying to solve. The resolution of the paradox is that we are throwing away
precisely a combination of equations that is...
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Tài liệu Solution of Linear Algebraic Equations part 8 docx
Ngày tải lên :
24/12/2013, 12:16
... Sparse Linear Systems
A system of linear equations is called sparse if only a relatively small number
of its matrix elements a
ij
are nonzero. It is wasteful to use general methods of
linear ... applications.)
ã Each of the rst N locations of ija stores the index of the array sa that contains
the first off-diagonal element of the corresponding row of the matrix. (If there are
no off-diagonal elements ... case of a tridiagonal matrix was treated specially, because that
particular type of linear system admits a solution in only of order N operations,
rather than of order N
3
for the general linear...
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Enclosing solutions of systems of equations
Ngày tải lên :
12/01/2014, 22:20
... Interval-based Solving of Hybrid
Constraint Systems. 3rd IFAC Conference on Analysis and Design of Hybrid Systems
(ADHS09) (to appear)
10. Kăuhn, W.: Rigorously Computed Orbits of Dynamical Systems ... syst`emes hybrides, equations differentielles ordinaires, anal-
yse par intervalles.
References
1. Hansen, E. and Sengupta, S.: Bounding solutions of systems of equations using
interval analysis. ... BIT 21, 203–211 (1981)
2. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Univ. Press,
(1990)
3. Goldsztejn, A.: A Comparison of the Hansen-Sengupta and Frommer-Lang-Schnurr
Existence...
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