solution of system of linear algebraic equations

... trade@cup.cam.ac.uk (outside North America). Chapter 2. Solution of Linear Algebraic Equations 2.0 Introduction A set of linear algebraic equations looks like this: a 11 x 1 + a 12 x 2 + a 13 x 3 + ... 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 ... linear combinations of each other, some of the equations may be so close to linearly dependent that roundoff errors in the machine render them linearly dependent at some stage in the solution process....

Ngày tải lên: 15/12/2013, 04:15

... of N ì N matrices, with M sets of right-hand side vectors, in completely analogous fashion. The routine implemented below is, of course, general. 38 Chapter 2. Solution of Linear Algebraic Equations Sample ... Moler, C.B. 1967, Computer Solution of Linear Algebraic Systems (Engle- wood Cliffs, NJ: Prentice-Hall). Wilkinson, J.H., and Reinsch, C. 1971, Linear Algebra ,vol.IIof Handbook for Automatic ... about as efficient as any other method. For solving sets of linear equations, Gauss-Jordan elimination produces both the solution of the equations for one or more right-hand side vectors b, and...

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... 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 ... 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 ... purposes, because of its greater diagnostic capability in pathological cases.) Updating a QR decomposition Some numericalalgorithms involve solvinga successionof linear systems each of which differs...

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... 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 ... 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 ... 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...

Ngày tải lên: 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 Equations (New ... 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, that is, has nonzero elements only ... (i=j;i<=n;i++) { This is i = j of equation (2.3.12) and i = j +1 N of equation (2.3.13).sum=a[i][j]; for (k=1;k<j;k++) 44 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL...

Ngày tải lên: 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 Equations (New ... 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, that is, has nonzero elements only ... Johns Hopkins University Press), Đ 4.3. 50 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...

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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...

Ngày tải lên: 24/12/2013, 12:16

... 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 ... verify convergence can be reassuring. 58 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 ... 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...

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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 ... 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...

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 ... 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 ... 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...

Ngày tải lên: 24/12/2013, 12:16

... 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 ... Vandermonde problems in double precision. 94 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 ... are too complicated to include here. 90 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...

Ngày tải lên: 21/01/2014, 18:20

... 1967, Computer Solution of Linear Algebraic Systems (Engle- wood Cliffs, NJ: Prentice-Hall), Đ19. [1] Westlake, J.R. 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New ... for fast solution of the symmetric Toeplitz problem, by a bordering method,thatis, 98 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC ... typical systems of linear equations. However, we will meet special cases where QR is the method of choice. 2.7 Sparse Linear Systems 89 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...

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... through the evaporation of lighter oils and the emulsification, as well as the dissolution of the heavier oils: ( ) ( ) QhDvh t h =∇∇−∇+ ∂ ∂  , (1) Development of system of hydrodynamic-environmental ... layer. So the system of models should be based on the 3D thermo- hydrodynamical models for each particular environment that contains interested matter. Development of system of hydrodynamic-environmental ... concentration can be taken as the average value of the thickness of the surface sediment. So we can build the model of oil in the sediment with the laws of dynamical transformation mainly between...

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... performance of the visual system. It is assumed to be a subjectively and objectively measurable criterion that is of particular value of ascertaining long- term adaptive processes of the visual system. ... discussion of how our visual system processes the binocular disparity to generate the 3D sensation. First, we introduce several basic concepts and definitions of functions of the human visual system ... horizontal and 1/3 vertical resolution with each view of 640x360 pixels, (d) Full resolution for each view with each view of 1920x1080 pixels 137 Figure 8-3 : Equipment setup of Experiment 1 139 Figure...

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... a part of speech on the basis of the part -of- speech implications of the affixes and the length of the remaining kernel. An accuracy of 95 per cent is achieved from the point of view of inclusive ... 90-95 per cent of the cases from the point of view of in- clusive part of speech, with something less than 5 per cent variation, depending on whether part of speech implications of affixes are ... loss from the point of view of inclusive part of speech. Rule A, it will be remembered, requires the removal of affixes from the kernel of the word. If this kernelizing of the word is omitted,...

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