- 8.16.1: In 16, find the least squares line for the given data.
- 8.16.2: In 16, find the least squares line for the given data.
- 8.16.3: In 16, find the least squares line for the given data.
- 8.16.4: In 16, find the least squares line for the given data.
- 8.16.5: In 16, find the least squares line for the given data.
- 8.16.6: In 16, find the least squares line for the given data.
- 8.16.7: In an experiment, the following correspondence was found between te...
- 8.16.8: In an experiment the following correspondence was found between tem...
- 8.16.9: In 9 and 10, proceed as in Example 3 and find the least squares par...
- 8.16.10: In 9 and 10, proceed as in Example 3 and find the least squares par...
Solutions for Chapter 8.16: Method of Least Squares
Full solutions for Advanced Engineering Mathematics | 6th Edition
Remove row i and column j; multiply the determinant by (-I)i + j •
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Outer product uv T
= column times row = rank one matrix.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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