powerseries solution of odes
Consolidation Chareteristics based on a direct analytical solution of the Terzaghi Theory
... 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. δti , mm0.0 0.5 1.0 1.5 2.0 2.5δpi , mm0.00.51.01.52.02.5 Solution of Eq. 8 where thethird point...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
... 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 xpadded to any linear combination of (typically) ... that direct solution of thenormal 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 EquationsIf N = M then there are as many equations as unknowns, and there is a goodchance of solving for a unique solution set of xj’s. Analytically, there can fail tobe a unique solution...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
... any two rows of A and the corresponding rows of the b’sand of 1, does not change (or scramble in any way) the solution x’s andY. Rather, it just corresponds to writing the same set of linear equationsin ... the identity matrix, of course).• Interchanging any two columns of A gives the same solution set onlyif 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 simplystates that xijis 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
... (2.10.4) of the algorithm is needed, so we separate it off into its own routine rsolv.98Chapter 2. Solution of Linear Algebraic EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... America).x[i]=sum/p[i];}}A typicaluseof choldcand cholslis in theinversionof covariancematrices describingthe fit of data to a model; see, e.g., §15.6. In this, and many other applications,one often needsL−1. ... decomposition, it is not used for typical systems of linear equations. However, we willmeet special cases where QR is the method of choice.100Chapter 2. Solution of Linear Algebraic EquationsSample page...
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
... 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 ... 42Chapter 2. Solution of Linear Algebraic EquationsSample 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), and12N2M times, respectively.Each backsubstitution of a right-hand side is12N2executions of a similar loop (onemultiplication...
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Tài liệu Solution of Linear Algebraic Equations part 4 docx
... 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 wholematrix multiplication, and is also more accurate.Determinant of a MatrixThe determinant of an LU...
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Tài liệu Solution of Linear Algebraic Equations part 5 docx
... 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 use4 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 aboveroutine does take advantage of the opportunity. In general, when TINY is returned as adiagonal 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
... 104Chapter 2. Solution of Linear Algebraic EquationsSample 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 orderN =2m, recursively by partitions in half. At each step, halving the order doublesthe number of inverse operations. But this ... complicated nature of the recursive Strassen algorithm, you will find that LU decomposition is in noimmediate 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
... improved solution x.2.5 Iterative Improvement of a Solution toLinear EquationsObviously it is not easy to obtain greater precision for the solution of a linearset than the precision of your ... J. 1985, in Proceedings of the Seventeenth Annual ACM Symposium onTheory of Computing (New York: Association for Computing Machinery). [1]2.5 Iterative Improvement of a Solution to Linear Equations57Sample ... than the square root of your computer’s roundoff error, then after oneapplication of equation (2.5.10) (that is, going from x0≡ B0·b to x1) the first neglected term, of order R2, will...
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Tài liệu Solution of Linear Algebraic Equations part 7 docx
... North America).A ⋅ x = bSVD solution of A ⋅ x = csolutions of A ⋅ x = c′solutions of A ⋅ x = dnullspace of ASVD solution of A ⋅ x = drange of Adc(b)(a)Axbc′Figure ... makingthe same permutation of the columns of U,elementsofW,andcolumnsofV(orrows of VT), or (ii) forming linear combinations of any columns of U and V whosecorresponding 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
... applications.)• Each of the first N locations of ija stores the index of the array sa that containsthe first off-diagonal element of the corresponding row of the matrix. (If there areno off-diagonal ... condition number of the matrix AT· A is the square of the condition number of A (see §2.6 for definition of condition number). A large condition number both increases thenumber of iterations required, ... ToeplitzMatricesIn §2.4 the case of a tridiagonal matrix was treated specially, because thatparticular type of linear system admits a solution in only of order N operations,rather than of order N3for the...
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06 ON TAYLOR MODEL BASED INTEGRATION OF ODES
... 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 Taylorpolynomial serves as an approximate solution. The truncation error of the series is enclosed by asuitable ... withsuccess 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
... Taylor ModelsVerified Integration of ODEs Taylor Model Methods for ODEs Verified Integration of Linear ODEs IntroductionInterval Methods for ODEs Verified Integration of ODEs Interval IVP:u= f (t, ... Integration of ODEs Interval Arithmetic and Taylor ModelsVerified Integration of ODEs Taylor Model Methods for ODEs Verified Integration of Linear ODEs IntroductionInterval Methods for ODEs Verified ... Integration of ODEs Interval Arithmetic and Taylor ModelsVerified Integration of ODEs Taylor Model Methods for ODEs Verified Integration of Linear ODEs IntroductionInterval Methods for ODEs Verified...
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Tài liệu Solution of Linear Algebraic Equations part 9 docx
... ToeplitzMatricesIn §2.4 the case of a tridiagonal matrix was treated specially, because thatparticular type of linear system admits a solution in only of order N operations,rather than of order N3for the ... actually two distinct sets of solutions to theoriginal linear problem for a nonsymmetric matrix, namely right-hand solutions (which wehave been discussing) and left-hand solutions zi. The formalism ... saysthat Ajkis exactly the inverse of the matrix of componentsxk−1i,whichappears 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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