8 solution of linear differential equations and the matrix exponential

ONE PARAMETER FAMILY OF LINEAR DIFFERENCE EQUATIONS AND THE STABILITY PROBLEM FOR THE NUMERICAL pdf

Ngày tải lên : 23/06/2014, 00:20

... to the number of roots of the characteristic polynomial inside the unit circle and the number of conditions at the end of the interval of integration is equal to the number of roots outside the ... becomes of ﬁrst order, it follows that detC(0) = This implies the thesis Theorem 3.6 Any solution of the diﬀerence equation (3.4) can be expressed as linear combination of ρk (z) and σk (z) Proof The ... In the discrete case it essentially states that the number of initial conditions should be equal to the number of roots of the characteristic polynomial inside the unit circle and, of course, the...

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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 ... of the last off-diagonal element of the last row (It can be read to determine the number of nonzero elements in the matrix, or the number of elements in the arrays sa and ija.) Location N + of ... adds the components of v to the ith row (Recall that u ⊗ v is a matrix whose i, jth element is the product of the ith component of u and the jth component of v.) If v is a unit vector ej , then...

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Báo cáo " On the set of periods for periodic solusions of some linear differential equations on the multidimensional sphere $S^n$ " docx

Ngày tải lên : 14/03/2014, 13:20

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Báo cáo "On the asymptotic behavior of delay differential equations and its relationship with C0 - semigoup " potx

Ngày tải lên : 22/03/2014, 11:20

... satisfies the hypothesis of theorem (2.3) then the solution of Eq.(12) in the case µ = is asymptotic equivalence to the solution of Eq.(12) in the case µ = Ackowledgments This paper is based on the ... such that the solution of Eq.(4) in the case µ = will be asymptotic equivalence to the solution of Eq.(4) in the case µ = 0(in the following we will give µ = 1) .The obtained result of this part ... an extention of Levinson’s theorem to the case of linear delay differential equations under nonlinear perturbation (see [1, 13, 14]) Let’s consider the two following differential equations: dx(t)...

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The numerical solution of delay-differential equations P1

Ngày tải lên : 11/10/2014, 04:02

... university of Manchester In 1 986 he obtained the degree of MSc in Numerical Analysis and Computation and held the position of temporary lecturer for the academic year 1 988 -89 within the above ... In this thesis we consider the development of numerical software for the solution of such problems Our discussion opens with a brief introduction to the theory of delay -differential equations ... delay -differential equations 18 1.1.1 Introduction p Delay -differential equations In this thesis we are concerned with the numerical solution of delay -differential equations (DDE's) Delay-differential...

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The numerical solution of delay-differential equations P2

Ngày tải lên : 11/10/2014, 04:03

... listing page number 1 58 140 95 100 67 127 73 52 75 165 58 106 131 103 177 149 152 172 179 45 123 29 180 1 18 1 38 170 116 175 154 1 78 17 61 68 1 68 56 163 49 50 387 3 78 356 3 58 342 372 345 334 346 ... correspond to the page numbers used in the thesis The 'numbers of statements' (1) include non-executable statements but not comment lines The total number of lines including comment lines is 84 98 DO2QFQ ... will be the basis of our result 297 PROPOSITION PROOF: This result follows from the properties of Ei As we have already observed El is a tree and so can contain no loops Furthermore since there...

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Tài liệu Solution of Linear Algebraic Equations part 1 docx

Ngày tải lên : 15/12/2013, 04:15

... be in the nullspace of the matrix A) The task of ﬁnding the solution space of A involves • Singular value decomposition of a matrix A This subject is treated in §2.6 In the opposite case there ... 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 also the matrix inverse ... the matrix inverse of A (§2.1 and §2.3) • Calculation of the determinant of a square matrix A (§2.3) If M < N , or if M = N but the equations are degenerate, then there are effectively fewer equations...

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Tài liệu Solution of Linear Algebraic Equations part 2 ppt

Ngày tải lên : 15/12/2013, 04:15

... right-hand side (j = 1, 2, 3), the one whose coefﬁcients are bij , i = 1, 2, 3, 4; and that the matrix of unknown coefﬁcients yij is the inverse matrix of aij In other words, the matrix solution of ... as long as we the same linear combination of the rows of the b’s and (which then is no longer the identity matrix, of course) • Interchanging any two columns of A gives the same solution set only ... corresponding rows of the x’s and of Y In other words, this interchange scrambles the order of the rows in the solution If we this, we will need to unscramble the solution by restoring the rows to their original...

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Tài liệu Solution of Linear Algebraic Equations part 11 ppt

Ngày tải lên : 15/12/2013, 04:15

... b[]) Solves the set of n linear equations R · x = b, where R is an upper triangular matrix stored in a and d a[1 n][1 n] and d[1 n] are input as the output of the routine qrdcmp and are not modiﬁed ... Chapter Solution of Linear Algebraic Equations We will make use of QR decomposition, and its updating, in §9.7 CITED REFERENCES AND FURTHER READING: Golub, G.H., and Van Loan, C.F 1 989 , Matrix ... a[1 n][1 n], c[1 n], and d[1 n] are input as the output of the routine qrdcmp and are not modiﬁed b[1 n] is input as the right-hand side vector, and is overwritten with the solution vector on output...

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Tài liệu Solution of Linear Algebraic Equations part 3 pdf

Ngày tải lên : 24/12/2013, 12:16

... §2.2.1 Westlake, J.R 19 68, 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 ... computing the inverse matrix as part of the Gauss-Jordan scheme.) For computing the inverse matrix (which we can view as the case of M = N right-hand sides, namely the N unit vectors which are the ... backsubstitution The combination of Gaussian elimination and backsubstitution yields a solution to the set of equations The advantage of Gaussian elimination and backsubstitution over Gauss-Jordan...

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Tài liệu Solution of Linear Algebraic Equations part 4 docx

Ngày tải lên : 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 and ... Computer Solution of Linear Algebraic Systems (Englewood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18 Westlake, J.R 19 68, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations ... b to get the real part of the solution vector, and (iii) backsubstitute d to get the imaginary part of the solution vector If the matrix itself is complex, so that you want to solve the system...

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Tài liệu Solution of Linear Algebraic Equations part 5 docx

Ngày tải lên : 24/12/2013, 12:16

... gives the determinant) are returned in the ﬁrst column of A’s storage space The following routine, bandec, is the band-diagonal analog of ludcmp in §2.3: 54 Chapter Solution of Linear Algebraic Equations ... Given the arrays a, al, and indx as returned from bandec, and given a right-hand side vector b[1 n], solves the band diagonal linear equations A · x = b The solution vector x overwrites b[1 n] The ... b[i]=dum/a[i][1]; if (l < mm) l++; } } The routines bandec and banbks are based on the Handbook routines bandet1 and bansol1 in [1] CITED REFERENCES AND FURTHER READING: Keller, H.B 19 68, Numerical Methods for...

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Tài liệu Solution of Linear Algebraic Equations part 12 pdf

Ngày tải lên : 24/12/2013, 12:16

... as the reciprocal if the a’s and c’s are scalars, but as matrix inversion if the a’s and c’s are themselves submatrices Imagine doing the inversion of a very large matrix, of order N = 2m , recursively ... matrices a11 a21 a12 a22 and c11 c21 c12 c22 (2.11.5) are inverses of each other Then the c’s can be obtained from the a’s by the following operations (compare equations 2.7.22 and 2.7.25): R1 = Inverse(a11 ... sixteenths, etc And note the key point: The savings is not just a factor “7 /8 ; it is that factor at each hierarchical level of the recursion In total it reduces the process of matrix multiplication...

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Tài liệu Solution of Linear Algebraic Equations part 6 pptx

Ngày tải lên : 24/12/2013, 12:16

... 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 as the largest amplification of length that it ... Improves a solution vector x[1 n] of the linear set of equations A · X = B The matrix a[1 n][1 n], and the vectors b[1 n] and x[1 n] are input, as is the dimension n Also input is alud[1 n][1 n], the ... calculate the right-hand side in double precision, since there will be a lot of cancellation in the subtraction of b Then, we need only solve (2.5.4) for the error δx, then subtract this from the wrong...

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Tài liệu Solution of Linear Algebraic Equations part 7 docx

Ngày tải lên : 24/12/2013, 12:16

... Formally, the condition number of a matrix is deﬁned as the ratio of the largest (in magnitude) of the wj ’s to the smallest of the wj ’s A matrix is singular if its condition number is inﬁnite, and ... a solution When the vector b on the right-hand side is not zero, the important question is whether it lies in the range of A or not If it does, then the singular set of equations does have a solution ... 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...

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Tài liệu Solution of Linear Algebraic Equations part 9 docx

Ngày tải lên : 21/01/2014, 18:20

... other special forms] Westlake, J.R 19 68, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) [2] von Mises, R 1964, Mathematical Theory of Probability and ... 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 for the left-hand ... appears in (2 .8. 2), with the subscript as the column index Therefore the solution of (2 .8. 2) is just that matrix inverse times the right-hand side, N wj = Ajk qk (2 .8. 5) k=1 As for the transpose...

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Tài liệu Solution of Linear Algebraic Equations part 10 docx

Ngày tải lên : 21/01/2014, 18:20

... other special forms] Westlake, J.R 19 68, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) [2] von Mises, R 1964, Mathematical Theory of Probability and ... 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 for the left-hand ... appears in (2 .8. 2), with the subscript as the column index Therefore the solution of (2 .8. 2) is just that matrix inverse times the right-hand side, N wj = Ajk qk (2 .8. 5) k=1 As for the transpose...

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Tài liệu Numerical Solution of Stochastic Differential Equations with Jumps in Finance pdf

Ngày tải lên : 19/02/2014, 22:20

... Feymano Kac formula and the existence and uniqueness of solutions of SDEs These tools and results provide the basis for the application and numerical solution of stochastic differential equations with ... definition of the strong error on the left hand side of (0.0.1) one notices that strong schemes provide pathwise approximations of the original solution X of the given SDE These methods are therefore ... strong mathematical background and mathematicians may omit the introductory Chap The following flowchart focuses on the theoretical aspects of the numerical approximation of solutions of stochastic...

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galois theory of linear differential equations - m. van der put, m. singer

Ngày tải lên : 31/03/2014, 15:56

... basis of V over C and a basis of M over k and let F denote the matrix which expresses the ﬁrst basis into the second basis Then R is generated over k by the entries of F and the inverse of the ... translates into ZL = (ra,b )GL A proof of Lemma 1.29 ﬁnishes the proof of the theorem Proof of lemma 1.29 The proof is rather similar to the one of lemma 1.23 The only thing that we have to verify ... of (1) or (2), R is generated by the entries of a matrix F and the inverse of its determinant Further σ(F ) = F C(σ) for some constant matrix C(σ) Therefore σ(R) = R Hence σ is the image of the...

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partial differential equations and the finite element method - pave1 solin

Ngày tải lên : 31/03/2014, 15:57

... B.15 Approximate solution p)t,p the cone-sphere problem of 452 B.16 Details of the singularity of 452 B.17 The hp-mesh, global view 453 B. 18 The hp-mesh, details of the tip of the cone 453 B.19 ... (1.26), (1. 28) In the language of linear forms Let V = HA(R) We define a bilinear form a(., ) VxV+Iw and a linear form E V ' , : 16 PARTIAL DIFFERENTIAL EQUATIONS Then the weak formulation of the problem ... representation theorem (Theorem A 15) Proof: The uniqueness of solution follows immediately from the V-ellipticity of the bilinear form a We will use Theorem 1.2 to verify the existence Let A : V + V’ be the...

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