- 5.5.1: Use the Runge-Kutta-Fehlberg method with tolerance TOL = I0~4 , hma...
- 5.5.2: Use the Runge-Kutta Fehlberg Algorithm with tolerance TOL = 10~4 to...
- 5.5.3: Use the Runge-Kutta-Fehlberg method with tolerance TOL l()~6 , hmax...
- 5.5.4: Use the Runge-Kutta-Fehlberg method with tolerance TOL ItT6 , hmax ...
- 5.5.5: In the theory of the spread of contagious disease (see [Bal] or [Ba...
- 5.5.6: In the previous exercise, all infected individuals remained in the ...
- 5.5.7: The Runge-Kutta-Verner method (see [VeJ) is based on the formulas 1...
Solutions for Chapter 5.5: Error Control and the Runge-Kutta-Fehlberg Method
Full solutions for Numerical Analysis | 10th Edition
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column space C (A) =
space of all combinations of the columns of A.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Invert A by row operations on [A I] to reach [I A-I].
A symmetric matrix with eigenvalues of both signs (+ and - ).
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.