powerseries solution of odes
Consolidation Chareteristics based on a direct analytical solution of the Terzaghi Theory
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21/03/2013, 14:09
... coefficient of consolidation and end of
primary settlement based on a direct solution of the
Terzaghi theory. This new method determines the
coefficient of consolidation utilizing the entire range of ... Fulfillment of the
Requirements for the Degree of Master of Science in
Civil Engineering at Jordan University of Science and
Technology, 167.
Al-Zoubi, M.S. 2004a. Coefficient of Consolidation ... mm
Figure (1): Graphical solution of Eq. 8 using two sets of selected data points.
δ
ti
, mm
0.0 0.5 1.0 1.5 2.0 2.5
δ
pi
, mm
0.0
0.5
1.0
1.5
2.0
2.5
Solution of Eq. 8 where the
third point...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
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... can be either no
solution, or else more than one solution vector x. In the latter event, the solution
space consists of a particular solution x
p
added to any linear combination of
(typically) ... that direct solution of the
normal equations (2.0.4) is not generally the best way to find least-squares solutions.
Some other topics in this chapter include
• Iterative improvement of a solution ... Sets of Equations
If N = M then there are as many 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...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
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... any two rows of A and the corresponding rows of the b’s
and of 1, does not change (or scramble in any way) the solution x’s and
Y. Rather, it just corresponds to writing the same set of linear equations
in ... the identity matrix, of course).
• Interchanging any two columns of A gives the same solution set only
if we simultaneously interchange corresponding rows of the x’s and of
Y. In other words, ... out of the operands of the operator.
It should not take you long to write out equation (2.1.1) and to see that it simply
states that x
ij
is the ith component (i =1,2,3,4) of the vector solution...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
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... (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 page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... America).
x[i]=sum/p[i];
}
}
A typicaluseof choldcand cholslis in theinversionof covariancematrices describing
the fit of data to a model; see, e.g., §15.6. In this, and many other applications,one often needs
L
−1
. ... decomposition, it 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...
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
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... is called backsubstitution.Thecom-
bination of Gaussian elimination and backsubstitution yields a solution to the set
of equations.
The advantage of Gaussian elimination and backsubstitutionover ... 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 ... increasing numbers of
predictable zeros reduce the count to one-third), and
1
2
N
2
M times, respectively.
Each backsubstitution of a right-hand side is
1
2
N
2
executions of a similar loop (one
multiplication...
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Tài liệu Solution of Linear Algebraic Equations part 4 docx
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24/12/2013, 12:16
... 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 ... 1967,
Computer Solution of Linear Algebraic Systems
(Engle-
wood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18.
Westlake, J.R. 1968,
A Handbook of Numerical Matrix Inversion and Solution of Linear ... 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...
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Tài liệu Solution of Linear Algebraic Equations part 5 docx
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24/12/2013, 12:16
... 1967,
Computer Solution of Linear Algebraic Systems
(Engle-
wood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18.
Westlake, J.R. 1968,
A Handbook of Numerical Matrix Inversion and Solution of Linear ... would each use
4 real multiplies, while the solution of a 2N × 2N problem involves 8 times the work of
an N × N one. If you can tolerate these factor -of- two inefficiencies, then equation (2.3.18)
is ... 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
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... 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 ... 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 ... complicated nature
of the recursive Strassen algorithm, you will find that LU decomposition is in no
immediate danger of becoming obsolete.
If, on the other hand, you like this kind of fun, then try...
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Tài liệu Solution of Linear Algebraic Equations part 6 pptx
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... improved solution 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 ... J. 1985, in Proceedings of the Seventeenth Annual ACM Symposium on
Theory of Computing (New York: Association for Computing Machinery). [1]
2.5 Iterative Improvement of a Solution to Linear Equations
57
Sample ... than the square root of your computer’s roundoff error, then after one
application of equation (2.5.10) (that is, going from x
0
≡ B
0
·b to x
1
) the first neglected term,
of order R
2
, will...
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Tài liệu Solution of Linear Algebraic Equations part 7 docx
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... North America).
A
⋅
x = b
SVD solution
of A
⋅
x = c
solutions of
A
⋅
x = c′
solutions of
A
⋅
x = d
null
space
of A
SVD solution of
A
⋅
x = d
range of A
d
c
(b)
(a)
A
x
b
c′
Figure ... 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 ... particular solution closest to zero, as shown. The point c lies outside of the range
of A,soA·x=chas no solution. SVD finds the least-squares best compromise solution, namely a
solution of A · x...
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Tài liệu Solution of Linear Algebraic Equations part 8 docx
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... applications.)
• Each of the first 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 ... condition number of the matrix A
T
· A is the square of the condition number of
A (see §2.6 for definition of condition number). A large condition number both increases the
number of iterations required, ... Toeplitz
Matrices
In §2.4 the 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...
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06 ON TAYLOR MODEL BASED INTEGRATION OF ODES
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12/01/2014, 21:46
... uniqueness of the solution of an IVP for an ODE, Reliable Computing, 7 (2001), pp. 449–465.
[40] M. Neher, Geometric series bounds for the local errors of Taylor methods for linear n-th order ODEs, ... series (with respect to t, a, and b) of the solution of (4.1) is employed. The third-order Taylor
polynomial serves as an approximate solution. The truncation error of the series is enclosed by a
suitable ... with
success to a variety of problems, including global optimization [34], verified multidimensional integration
[7], and the verified solution of ODEs and DAEs [6, 13].
2.4. Representation of Intervals by...
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10 on taylor model based integration of ODEs neuer 2006
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... Taylor Models
Verified Integration of ODEs
Taylor Model Methods for ODEs
Verified Integration of Linear ODEs
Introduction
Interval Methods for ODEs
Verified Integration of ODEs
Interval IVP:
u
= f (t, ... Integration of ODEs
Interval Arithmetic and Taylor Models
Verified Integration of ODEs
Taylor Model Methods for ODEs
Verified Integration of Linear ODEs
Introduction
Interval Methods for ODEs
Verified ... Integration of ODEs
Interval Arithmetic and Taylor Models
Verified Integration of ODEs
Taylor Model Methods for ODEs
Verified Integration of Linear ODEs
Introduction
Interval Methods for ODEs
Verified...
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Tài liệu Solution of Linear Algebraic Equations part 9 docx
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... Toeplitz
Matrices
In §2.4 the 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 ... actually two distinct sets of solutions to the
original linear problem for a nonsymmetric matrix, namely right-hand solutions (which we
have been discussing) and left-hand solutions z
i
. The formalism ... saysthat A
jk
is exactly the inverse of the matrix of componentsx
k−1
i
,which
appears in (2.8.2), with the subscript as the column index. Therefore the solution of (2.8.2)
is just that matrix inverse...
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- sceptical solution of these doubts
- fast solution methods of chemical
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