least squares solution of linear systems
Tài liệu Solution of Linear Algebraic Equations part 1 docx
Ngày tải lên :
15/12/2013, 04:15
... 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 xp added to any linear combination of (typically) N ... defined in the least- squares sense, i.e., that the sum of the squares of the differences between the left- and right-hand sides of equation (2.0.1) be minimized, then the overdetermined linear problem ... Introduction Much of the sophistication of complicated linear equation-solving packages” is devoted to the detection and/or correction of these two pathologies As you work with large linear sets of equations,...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
Ngày tải lên :
15/12/2013, 04:15
... 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 ... the solution set is unchanged and in no way scrambled if we replace any row in A by a linear combination of itself and any other row, as long as we the same linear combination of the rows of the ... 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, this...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
Ngày tải lên :
15/12/2013, 04:15
... a succession of linear systems each of which differs only slightly from its predecessor Instead of doing O(N ) operations each time to solve the equations from scratch, one can often update a ... 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 Sample page from NUMERICAL RECIPES IN C: THE ART OF ... to 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⊗t...
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
Ngày tải lên :
24/12/2013, 12:16
... (2.2.4) is called backsubstitution The combination of Gaussian elimination and backsubstitution yields a solution to the set of equations The advantage of Gaussian elimination and backsubstitution ... A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) Suppose we are able to write the matrix A as a product of two matrices, L·U=A (2.3.1) where L is lower ... and the increasing numbers of predictable zeros reduce the count to one-third), and N M times, respectively Each backsubstitution of a right-hand side is N executions of a similar loop (one multiplication...
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Tài liệu Solution of Linear Algebraic Equations part 4 docx
Ngày tải lên :
24/12/2013, 12:16
... Computer Solution of Linear Algebraic Systems (Englewood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations ... America) Determinant of a Matrix 50 Chapter Solution of Linear Algebraic Equations A quick-and-dirty way to solve complex systems is to take the real and imaginary parts of (2.3.16), giving A·x−C·y=b ... 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
... easy to obtain greater precision for the solution of a linear set than the precision of your computer’s floating-point word Unfortunately, for large sets of linear equations, it is not always easy ... giving δb The linear set with this right-hand side is inverted, giving δx This is subtracted from the first guess giving an improved solution x 2.5 Iterative Improvement of a Solution to Linear Equations ... in §2.3 is lubksb 55 2.5 Iterative Improvement of a Solution to Linear Equations A δx x+ b δx δb A−1 Figure 2.5.1 Iterative improvement of the solution to A · x = b The first guess x + δx is multiplied...
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Tài liệu Solution of Linear Algebraic Equations part 12 pdf
Ngày tải lên :
24/12/2013, 12:16
... “7/8”; it is that factor at each hierarchical level of the recursion In total it reduces the process of matrix multiplication to order N log2 instead of N What about all the extra additions in (2.11.3)–(2.11.4)? ... email to trade@cup.cam.ac.uk (outside North America) c22 = Q1 + Q3 − Q2 + Q6 104 Chapter Solution of Linear Algebraic Equations CITED REFERENCES AND FURTHER READING: Strassen, V 1969, Numerische ... submatrices Imagine doing the inversion of a very large matrix, of order N = 2m , recursively by partitions in half At each step, halving the order doubles the number of inverse operations But this means...
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Tài liệu Solution of Linear Algebraic Equations part 6 pptx
Ngày tải lên :
24/12/2013, 12:16
... inverse of the matrix A, so that B0 · A is approximately the identity matrix Define the residual matrix R of B0 as 58 Chapter Solution of Linear Algebraic Equations We can define the norm of a matrix ... modeling of data SVD methods are based on the following theorem of linear algebra, whose proof is beyond our scope: Any M × N matrix A whose number of rows M is greater than or equal to its number of ... than the square root of your computer’s roundoff error, then after one application of equation (2.5.10) (that is, going from x0 ≡ B0 · b to x1 ) the first neglected term, of order R2 , will be...
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Tài liệu Solution of Linear Algebraic Equations part 7 docx
Ngày tải lên :
24/12/2013, 12:16
... Value Decomposition A A⋅x = b (a) null space of A solutions of A⋅x = d solutions of A ⋅ x = c′ SVD solution of A ⋅ x = c range of A d c′ c SVD solution of A⋅x = d (b) Figure 2.6.1 (a) A nonsingular ... same permutation of the columns of U, elements of W, and columns of V (or rows of VT ), or (ii) forming linear combinations of any columns of U and V whose corresponding elements of W happen to ... particular solution closest to zero, as shown The point c lies outside of the range of A, so A · x = c has no solution SVD finds the least- squares best compromise solution, namely a solution of A ·...
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Tài liệu Solution of Linear Algebraic Equations part 8 docx
Ngày tải lên :
24/12/2013, 12:16
... 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 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 for the general linear ... In the unlikely event that this causes loss of precision in the second of these equations, you can make a different choice 76 Chapter Solution of Linear Algebraic Equations Here A is, as usual,...
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Tài liệu Solution of Linear Algebraic Equations part 9 docx
Ngày tải lên :
21/01/2014, 18:20
... 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 zi The formalism ... forms] Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) [2] von Mises, R 1964, Mathematical Theory of Probability and Statistics (New ... square root” of the matrix A The components of LT are of course related to those of L by LT = Lji ij (2.9.3) Writing out equation (2.9.2) in components, one readily obtains the analogs of equations...
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Tài liệu Solution of Linear Algebraic Equations part 10 docx
Ngày tải lên :
21/01/2014, 18:20
... 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 for the general linear ... 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 zi The formalism ... forms] Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) [2] von Mises, R 1964, Mathematical Theory of Probability and Statistics (New...
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Tài liệu Đề tài " Isomonodromy transformations of linear systems of difference equations" pptx
Ngày tải lên :
14/02/2014, 17:20
... Annals of Mathematics, 160 (2004), 1141–1182 Isomonodromy transformations of linear systems of difference equations By Alexei Borodin Abstract We introduce and study “isomonodromy” transformations of ... expressed through solution of an associated isomonodromy problem for a linear system of differential equations with rational coefficients The goal of this paper is to develop a general theory of “isomonodromy” ... solutions of isomonodromy problems for such systems of difference equations In the case of one-interval gap probability this has been done (in a different language) in [Bor], [BB] One example of...
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Báo cáo hóa học: " Research Article Nonexpansive Matrices with Applications to Solutions of Linear Systems by Fixed Point Iterations Teck-Cheong Lim" docx
Ngày tải lên :
21/06/2014, 20:20
... which approaches to ∞ as k → ∞ This completes the proof We now consider applications of preceding theorems to approximation of solution of a linear system Ax b, where A ∈ Mn and b a given vector ... eigenvalue of B Proof As in the proof of Theorem 2, a and b are equivalent For < μ < 1, denote μI − μ B by B μ Suppose now that a holds Let λ be an eigenvalue of B Then μ 1− μ λ is an eigenvalue of ... The index of the eigenvalue of Q−1 A is the R Q−1 A −1 index of eigenvalue of B I − Q A Thus by Lemma 5, Cn Q−1 R A ⊕ N A For every vector v ∈ Cn , let v r and v n denote the component of v in...
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Báo cáo hóa học: " Research Article NLOS Identification and Weighted Least-Squares Localization for UWB Systems Using Multipath Channel Statistics" potx
Ngày tải lên :
22/06/2014, 19:20
... local minima of the loss function [37] Alternatively, it is possible to use the techniques as proposed in [27, 38] in order to obtain a linear set of equations from the nonlinear least- squares model ... respectively The kurtosis of a certain data is defined as the ratio of the fourth-order moment of the data to the square of the second-order moment (i.e., the variance) of the data As stated in ... matrix of size (N − 1), and r is the FT chosen in order to obtain the linear model.3 Therefore, the WLS solution is given by x = AT W A −1 AT W2 p (19) The mean square error (MSE) of the WLS solution...
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Analysis and Control of Linear Systems - Chapter 0 pptx
Ngày tải lên :
09/08/2014, 06:23
... variability of the behavior of the processes, more than from their possible non-linearity It is necessary to know the basics of linear automated systems before learning about the theory of non -linear systems ... Analysis and control of linear systems analysis and control of linear systems/ edited by Philippe de Larminat p cm ISBN-13: 978-1-905209-35-4 ISBN-10: 1-905209-35-5 Linear control systems Automatic ... 2.1.3 Case of linear systems 2.1.4 Case of continuous and invariant linear systems 2.2 Resolving the equation of state ...
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Analysis and Control of Linear Systems - Chapter 1 pptx
Ngày tải lên :
09/08/2014, 06:23
... and Control of Linear Systems In general, the Dirac impulse is a very simplified model of any impulse phenomenon centered in t = t o , with a shorter period than the time range of the systems in ... Characteristics of continuous systems The input-output behavior of a system may be characterized by different relations with various degrees of complexity In this work, we will deal only with linear systems ... Analysis and Control of Linear Systems Table 1.1 sums up the features of a system’s transfer function, the existence conditions of its frequency response and the possibility of performing a harmonic...
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Analysis and Control of Linear Systems - Chapter 3 pps
Ngày tải lên :
09/08/2014, 06:23
... 94 Analysis and Control of Linear Systems NOTE 3.3.– the class of rational systems that can be described by [3.16] or [3.18] is a sub-class of DLTI systems To be certain of this, let us consider ... development of X (z) by polynomial division according to 86 Analysis and Control of Linear Systems the decreasing powers of z −1 or apply the method of deviations, starting from the definition of the ... and Control of Linear Systems 3.2.2 Delay and lead operators The concept of an operator is interesting because it enables a compact formulation of the description of signals and systems The manipulation...
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Analysis and Control of Linear Systems - Chapter 4 ppt
Ngày tải lên :
09/08/2014, 06:23
... 110 Analysis and Control of Linear Systems The object of this chapter is to describe certain structural properties of linear systems that condition the resolution of numerous control problems ... illustrated with respect to the existence of solutions, the existence of stabilizing solutions and flexibilities offered in terms of poles positions (concept of fixed poles) This is illustrated in ... a basis of the canceller at the left of W (i.e a maximal solution of equation WtW = {0}), a basis of LV is obtained by directly preserving only the independent columns of LV A basis of L–1W is...
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Analysis and Control of Linear Systems - Chapter 5 pptx
Ngày tải lên :
09/08/2014, 06:23
... proximity of ν0 152 Analysis and Control of Linear Systems Figure 5.1 Typical power spectrum of an MA (left) or AR (right) model In the case of a single denominator (nc = 0), we talk of an AR ... ∀κ ∈ Z [5.18] 146 Analysis and Control of Linear Systems are independent of index k, i.e independent of the time origin σy = ryy [0] is the r [κ] variance of the signal considered yy2 is the correlation ... Analysis and Control of Linear Systems A discrete-time deterministic signal y[k], k ∈ Z is, by definition, a sequence of complex numbers: y = y[k] k∈Z In short, we often speak of a discrete signal...
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