 20.5.20.1.82: Fit a straight line to the given points (x, y) by least squares. Sh...
 20.5.20.1.83: Fit a straight line to the given points (x, y) by least squares. Sh...
 20.5.20.1.84: Fit a straight line to the given points (x, y) by least squares. Sh...
 20.5.20.1.85: Fit a straight line to the given points (x, y) by least squares. Sh...
 20.5.20.1.86: Fit a straight line to the given points (x, y) by least squares. Sh...
 20.5.20.1.87: Fit a straight line to the given points (x, y) by least squares. Sh...
 20.5.20.1.88: Derive the normal equations (8).
 20.5.20.1.89: Fit a parabola (7) to the given points Cx, y) by least squares. Che...
 20.5.20.1.90: Fit a parabola (7) to the given points Cx, y) by least squares. Che...
 20.5.20.1.91: Fit a parabola (7) to the given points Cx, y) by least squares. Che...
 20.5.20.1.92: Fit (2) and (7) by lea<;t squares to (1.0,5.4), (0.5,4.1), (0,3.9...
 20.5.20.1.93: (Cubic parabola) Derive the formula for the normal equations of a c...
 20.5.20.1.94: Fit curves (2) and (7) and a cubic parabola by least squares to (2...
 20.5.20.1.95: CAS PROJECT. Least Squares. Write programs for calculating and solv...
 20.5.20.1.96: CAS EXPERIMENT. Least Squares versus Interpolation. For the given d...
 20.5.20.1.97: TEAM PROJECT. The least squares approximation of a function f(x) on...
Solutions for Chapter 20.5: Least Squares Method
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 20.5: Least Squares Method
Get Full SolutionsThis textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Chapter 20.5: Least Squares Method includes 16 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 16 problems in chapter 20.5: Least Squares Method have been answered, more than 46190 students have viewed full stepbystep solutions from this chapter.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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

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!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).