solution 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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... 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 NUMERICAL ... 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 ... solve the next set of linear equations. The LU decomposition is complicated to update because of pivoting. However, QR turns out to be quite simple for a very common kind of update, A → A + s...

Ngày tải lên: 15/12/2013, 04:15

... 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 ... (i.e., selection of a salubrious pivot element for the division in equation 2.3.13) is absolutely essential for the stability of Crout’s 50 Chapter 2. Solution of Linear Algebraic Equations Sample ... (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 ... (j=LMAX(1,1-k);j<=tmploop;j++) b[i] += a[i][j]*x[j+k]; } } 54 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 ... 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...

Ngày tải lên: 24/12/2013, 12:16

... 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 ... 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...

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 ... form the sequence of improved estimates x k+1 = x k + α k p k (2.7.37) 76 Chapter 2. Solution of Linear Algebraic Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... 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 ... 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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... collection of events, sigma-algebra A filtration E(X) expectation of X E(X |A) conditional expectation of X under A P (A) probability of A P (A |B) probability of A conditioned on B ∈ element of ∈ ... approxi- mation of continuous solutions of SDEs. The discrete time approximation of SDEs with jumps represents the focus of the monograph. The reader learns about powerful numerical methods for the solution ... context of derivative pricing. The book does not claim to be a complete account of the state of the art of the subject. Rather it attempts to provide a systematic framework for an understanding of...

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