- 15.1: Express each of the rational numbers below as finite simple continu...
- 15.2: Determine the rational numbers represented by the following simple ...
- 15.3: If r = [ao; ai, az, ... , an], where r > 1, show that1- = [O;ao, a1...
- 15.4: Represent the following simple continued fractions in an equivalent...
- 15.5: Compute the convergents of the following simple continued fractions...
- 15.6: (a) If Ck = Pk/qk denotes the kth convergent of the finite simple c...
- 15.7: Evaluate Pb qko and Ck(k = 0, 1, ... , 8) for the simple continued ...
- 15.8: If Ck = Pk/qk is the kth convergent of the simple continued fractio...
- 15.9: Find the simple continued fraction representation of 3.1416, and th...
- 15.10: If Ck = Pk/qk is the kth convergent of the simple continued fractio...
- 15.11: By means of continued fractions determine the general solutions of ...
- 15.12: Assume that the continued fraction representation for the irrationa...
- 15.13: Let x be an irrational number with convergents Pnf qn. For every n ...
- 15.14: Apply the theory of this section to confirm that there exist infini...
- 15.15: The Pell numbers Pn and qn are defined byPo= 0qo = 1P1 = 1qi= 1Pn =...
- 15.16: For the Pell numbers, derive the relations below, where n :'.'.: 1:...
Solutions for Chapter 15: FINITE CONTINUED FRACTIONS
Full solutions for Elementary Number Theory | 7th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
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
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.