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Consider some (physical) quantity \( u \) depending on time \( t \) and a parameter vector \( a = {\left( {a}_{1},\ldots ,{a}_{n}\right) }^{T} \in {\mathbb{R}}^{n} \) in terms of a known function\n\n\[ u\left( t\right) = f\left( {t;a}\right) \]\n\nIn order to determine the values of the parameter \( a \) (representing ... | The necessary conditions for a minimum,\n\n\[ \frac{\partial g}{\partial {a}_{j}} = 0,\;j = 1,\ldots, n \]\n\nlead to the normal equations\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{m}\left\lbrack {u\left( {t}_{k}\right) - f\left( {{t}_{k};a}\right) }\right\rbrack \frac{\partial f\left( {{t}_{k};a}\right) }{\partial {a}_{j... | Yes |
We consider the system\n\n\[ \n{x}_{1} + {200}{x}_{2} = {100} \]\n\n\[ \n{x}_{1} + \;{x}_{2} = 1 \]\n\nwith the exact solution \( {x}_{1} = {100}/{199} = {0.502}\ldots ,{x}_{2} = {99}/{199} = {0.497}\ldots \) . | For the following computations we use two-decimal-digit floating-point arithmetic. Column pivoting leads to \( {a}_{11} \) as pivot element, and the elimination yields\n\n\[ \n{x}_{1} + {200}{x}_{2} = {100} \]\n\n\[ \n- {200}{x}_{2} = - {99} \]\n\nsince \( {199} = {200} \) in two-digit floating-point representation. Fr... | Yes |
Theorem 2.9 For a nonsingular matrix \( A \), Gaussian elimination (without reordering rows and columns) yields an LR decomposition. | Proof. In the first elimination step we multiply the first equation by \( {a}_{j1}/{a}_{11} \) and subtract the result from the \( j \) th equation; i.e., the matrix \( {A}_{1} = A \) is multiplied from the left by the lower triangular matrix\n\n\[ \n{L}_{1} = \left( \begin{array}{rrrr} 1 & & & \\ - \frac{{a}_{21}}{{a}... | Yes |
Theorem 3.5 The limit of a convergent sequence is uniquely determined. | Proof. Assume that \( {x}_{n} \rightarrow x \) and \( {x}_{n} \rightarrow y \) for \( n \rightarrow \infty \) . Then from the triangle inequality we obtain that\n\n\[ \parallel x - y\parallel = \begin{Vmatrix}{x - {x}_{n} + {x}_{n} - y}\end{Vmatrix} \leq \begin{Vmatrix}{x - {x}_{n}}\end{Vmatrix} + \begin{Vmatrix}{{x}_{... | Yes |
Theorem 3.7 Two norms \( \parallel \cdot {\parallel }_{a} \) and \( \parallel \cdot {\parallel }_{b} \) on a linear space \( X \) are equivalent if and only if there exist positive numbers \( c \) and \( C \) such that\n\n\[ c\parallel x{\parallel }_{a} \leq \parallel x{\parallel }_{b} \leq C\parallel x{\parallel }_{a}... | Proof. Provided that the conditions are satisfied, from \( {\begin{Vmatrix}{x}_{n} - x\end{Vmatrix}}_{a} \rightarrow 0 \) , \( n \rightarrow \infty \), it follows that \( {\begin{Vmatrix}{x}_{n} - x\end{Vmatrix}}_{b} \rightarrow 0, n \rightarrow \infty \), and vice versa.\n\nConversely, let the two norms be equivalent ... | Yes |
Theorem 3.8 On a finite-dimensional linear space all norms are equivalent. | Proof. In a linear space \( X \) with finite dimension \( n \) and basis \( {u}_{1},\ldots ,{u}_{n} \) every element can be expressed in the form\n\n\[ x = \mathop{\sum }\limits_{{j = 1}}^{n}{\alpha }_{j}{u}_{j} \]\n\nAs in Example 3.2,\n\n\[ \parallel x{\parallel }_{\infty } \mathrel{\text{:=}} \mathop{\max }\limits_{... | Yes |
Theorem 3.11 Any bounded sequence in a finite-dimensional normed space \( X \) contains a convergent subsequence. | Proof. Let \( {u}_{1},\ldots ,{u}_{n} \) be a basis of \( X \) and let \( \left( {x}_{\nu }\right) \) be a bounded sequence. Then writing\n\n\[ \n{x}_{\nu } = \mathop{\sum }\limits_{{j = 1}}^{n}{\alpha }_{j\nu }{u}_{j} \n\] \n\nand using the norm (3.2), as in the proof of Theorem 3.8 we deduce that each of the sequence... | Yes |
Theorem 3.14 For a scalar product we have the Cauchy-Schwarz inequality\n\n\[ \n{\left| \left( x, y\right) \right| }^{2} \leq \left( {x, x}\right) \left( {y, y}\right) \n\]\n\nfor all \( x, y \in X \), with equality if and only if \( x \) and \( y \) are linearly dependent. | Proof. The inequality is trivial for \( x = 0 \) . For \( x \neq 0 \) it follows from\n\n\[ \n\left( {{\alpha x} + {\beta y},{\alpha x} + {\beta y}}\right) = {\left| \alpha \right| }^{2}\left( {x, x}\right) + 2\operatorname{Re}\{ \alpha \bar{\beta }\left( {x, y}\right) \} + {\left| \beta \right| }^{2}\left( {y, y}\righ... | Yes |
Theorem 3.14 For a scalar product we have the Cauchy-Schwarz inequality\n\n\\[ \n{\\left| \\left( x, y\\right) \\right| }^{2} \\leq \\left( {x, x}\\right) \\left( {y, y}\\right) \n\\]\n\nfor all \\( x, y \\in X \\), with equality if and only if \\( x \\) and \\( y \\) are linearly dependent. | Proof. The inequality is trivial for \\( x = 0 \\) . For \\( x \\neq 0 \\) it follows from\n\n\\[ \n\\left( {{\\alpha x} + {\\beta y},{\\alpha x} + {\\beta y}}\\right) = {\\left| \\alpha \\right| }^{2}\\left( {x, x}\\right) + 2\\operatorname{Re}\\{ \\alpha \\bar{\\beta }\\left( {x, y}\\right) \\} + {\\left| \\beta \\ri... | Yes |
Theorem 3.15 A scalar product \( \left( {\cdot , \cdot }\right) \) on a linear space \( X \) defines a norm by\n\n\[ \parallel x\parallel \mathrel{\text{:=}} {\left( x, x\right) }^{1/2} \]\n\nfor all \( x \in X \) ; i.e., a pre-Hilbert space is always a normed space. | Proof. We leave it as an exercise for the reader to verify the norm axioms. The triangle inequality follows by\n\n\[ \parallel x + y{\parallel }^{2} = \left( {x + y, x + y}\right) \leq \parallel x{\parallel }^{2} + 2\parallel x\parallel \parallel y\parallel + \parallel y{\parallel }^{2} = {\left( \parallel x\parallel +... | No |
Theorem 3.17 The elements of an orthonormal system are linearly independent. | Proof. From\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{n}{\alpha }_{k}{q}_{k} = 0 \]\n\nfor the orthonormal system \( \left\{ {{q}_{1},\ldots ,{q}_{n}}\right\} \), by taking the scalar product with \( {q}_{j} \), we immediately have that \( {\alpha }_{j} = 0 \) for \( j = 1,\ldots, n \) . | Yes |
Theorem 3.18 Let \( \left\{ {{u}_{0},{u}_{1},\ldots }\right\} \) be a finite or countable number of linearly independent elements of a pre-Hilbert space. Then there exists a uniquely determined orthogonal system \( \left\{ {{q}_{0},{q}_{1},\ldots }\right\} \) of the form\n\n\[ \n{q}_{n} = {u}_{n} + {r}_{n},\;n = 0,1,\l... | Proof. Assume that we have constructed orthogonal elements of the form (3.3) with the property (3.4) up to \( {q}_{n - 1} \) . By (3.4), the \( \left\{ {{q}_{0},\ldots ,{q}_{n - 1}}\right\} \) are linearly independent, and therefore \( \begin{Vmatrix}{q}_{k}\end{Vmatrix} \neq 0 \) for \( k = 0,1,\ldots, n - 1 \) . Henc... | Yes |
Theorem 3.21 A linear operator is continuous if it is continuous at one element. | Proof. Let \( A : X \rightarrow Y \) be continuous at \( {x}_{0} \in X \) . Then for every \( x \in X \) and every sequence \( \left( {x}_{n}\right) \) with \( {x}_{n} \rightarrow x, n \rightarrow ∞ \), we have\n\n\[ A{x}_{n} = A\left( {{x}_{n} - x + {x}_{0}}\right) + A\left( {x - {x}_{0}}\right) \rightarrow A\left( {x... | Yes |
Theorem 3.23 A linear operator \( A : X \rightarrow Y \) is bounded if and only if\n\n\[ \parallel A\parallel \mathrel{\text{:=}} \mathop{\sup }\limits_{{\parallel x\parallel = 1}}\parallel {Ax}\parallel < \infty . \]\n\nThe number \( \parallel A\parallel \) is the smallest bound for \( A \) and is called the norm of \... | Proof. Assume that \( A \) is bounded with the bound \( C \) . Then\n\n\[ \mathop{\sup }\limits_{{\parallel x\parallel = 1}}\parallel {Ax}\parallel \leq C \]\n\nand, in particular, \( \parallel A\parallel \) is less than or equal to any bound for \( A \) . Conversely, if \( \parallel A\parallel < \infty \), then using ... | Yes |
Theorem 3.24 A linear operator is continuous if and only if it is bounded. | Proof. Let \( A : X \rightarrow Y \) be bounded and let \( \left( {x}_{n}\right) \) be a sequence in \( X \) with \( {x}_{n} \rightarrow 0, n \rightarrow \infty \) . Then from \( \begin{Vmatrix}{A{x}_{n}}\end{Vmatrix} \leq C\begin{Vmatrix}{x}_{n}\end{Vmatrix} \) it follows that \( A{x}_{n} \rightarrow 0 \) , \( n \righ... | Yes |
Theorem 3.27 To each matrix \( A \) there exists a unitary matrix \( Q \) such that \( {Q}^{ * }{AQ} \) is an upper triangular matrix. | Proof. Assume that it has been shown that for each \( \left( {n - 1}\right) \times \left( {n - 1}\right) \) matrix \( {A}_{n - 1} \) there exists a unitary \( \left( {n - 1}\right) \times \left( {n - 1}\right) \) matrix \( {Q}_{n - 1} \) such that \( {Q}_{n - 1}^{ * }{A}_{n - 1}{Q}_{n - 1} \) is an upper triangular mat... | Yes |
Lemma 3.28 For an \( n \times n \) matrix \( A \) and its adjoint \( {A}^{ * } \) we have that\n\n\[ \left( {{Ax}, y}\right) = \left( {x,{A}^{ * }y}\right) \]\n\nfor all \( x, y \in {\mathbb{C}}^{n} \), where \( \left( {\cdot , \cdot }\right) \) denotes the Euclidean scalar product. | Proof. Simple calculations yield\n\n\[ \left( {{Ax}, y}\right) = \mathop{\sum }\limits_{{j = 1}}^{n}{\left( Ax\right) }_{j}{\bar{y}}_{j} = \mathop{\sum }\limits_{{j = 1}}^{n}\mathop{\sum }\limits_{{k = 1}}^{n}{a}_{jk}{x}_{k}{\bar{y}}_{j} \]\n\n\[ = \mathop{\sum }\limits_{{k = 1}}^{n}\mathop{\sum }\limits_{{j = 1}}^{n}{... | Yes |
The eigenvalues of a Hermitian \( n \times n \) matrix are real, and the eigenvectors form an orthogonal basis in \( {\mathbb{C}}^{n} \) . | Proof. If \( A \) is Hermitian, i.e., if \( A = {A}^{ * } \), then the matrix \( \widetilde{A} \mathrel{\text{:=}} {Q}^{ * }{AQ} \) from Theorem 3.27 is also Hermitian, since\n\n\[ \n{\widetilde{A}}^{ * } = {\left( {Q}^{ * }AQ\right) }^{ * } = {Q}^{ * }{A}^{ * }{Q}^{* * } = {Q}^{ * }{AQ} = \widetilde{A}.\n\]\n\nTherefo... | Yes |
For an \( n \times n \) matrix \( A \) we have\n\n\[ \parallel A{\parallel }_{2} = \sqrt{\rho \left( {{A}^{ * }A}\right) } \]\n\nIf \( A \) is Hermitian, then\n\n\[ \parallel A{\parallel }_{2} = \rho \left( A\right) \] | Proof. From Lemma 3.28 we have that\n\n\[ \parallel {Ax}{\parallel }_{2}^{2} = \left( {{Ax},{Ax}}\right) = \left( {x,{A}^{ * }{Ax}}\right) \]\n\nfor all \( x \in {\mathbb{C}}^{n} \) . Hence the Hermitian matrix \( {A}^{ * }A \) is positive semidefinite and therefore has \( n \) orthonormal eigenvectors\n\n\[ {A}^{ * }A... | Yes |
Theorem 3.32 For each norm on \( {\mathbb{C}}^{n} \) and each \( n \times n \) matrix \( A \) we have that\n\n\[ \rho \left( A\right) \leq \parallel A\parallel \]\n\nConversely, to each matrix \( A \) and each \( \varepsilon > 0 \) there exists a norm on \( {\mathbb{C}}^{n} \) such that\n\n\[ \parallel A\parallel \leq ... | Proof. Let \( \lambda \) be an eigenvalue of \( A \) with eigenvector \( u \) . We may assume that \( \parallel u\parallel = 1 \) . Then the first part of the theorem follows from\n\n\[ \parallel A\parallel = \mathop{\sup }\limits_{{\parallel x\parallel = 1}}\parallel {Ax}\parallel \geq \parallel {Au}\parallel = \paral... | Yes |
Theorem 3.34 Every convergent sequence is a Cauchy sequence. | Proof. Let \( {x}_{n} \rightarrow x, n \rightarrow \infty \) . Then, for \( \varepsilon > 0 \) there exists \( N\left( \varepsilon \right) \in \mathbb{N} \) such that \( \begin{Vmatrix}{{x}_{n} - x}\end{Vmatrix} < \varepsilon /2 \) for all \( n \geq N\left( \varepsilon \right) \) . Now the triangle inequality yields\n\... | Yes |
Example 3.36 The linear space \( C\left\lbrack {a, b}\right\rbrack \) furnished with the maximum norm\n\n\[ \parallel f{\parallel }_{\infty } \mathrel{\text{:=}} \mathop{\max }\limits_{{x \in \left\lbrack {a, b}\right\rbrack }}\left| {f\left( x\right) }\right| \]\n\nis a Banach space. | Proof. The norm axioms (N1)-(N3) are trivially satisfied. The triangle inequality follows from\n\n\[ \parallel f + g{\parallel }_{\infty } = \mathop{\max }\limits_{{x \in \left\lbrack {a, b}\right\rbrack }}\left| {\left( {f + g}\right) \left( x\right) }\right| = \left| {\left( {f + g}\right) \left( {x}_{0}\right) }\rig... | Yes |
The linear space \( C\left\lbrack {a, b}\right\rbrack \) equipped with the \( {L}_{1} \) norm\n\n\[ \parallel f{\parallel }_{1} \mathrel{\text{:=}} {\int }_{a}^{b}\left| {f\left( x\right) }\right| {dx} \]\n\nis not complete. | The norm axioms are trivially satisfied. Without loss of generality we take \( \left\lbrack {a, b}\right\rbrack = \left\lbrack {0,2}\right\rbrack \) and choose\n\n\[ {f}_{n}\left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} {x}^{n}, & 0 \leq x \leq 1 \\ 1, & 1 \leq x \leq 2 \end{array}\right. \]\n\nThen for ... | Yes |
Example 3.38 The linear space \( C\left\lbrack {a, b}\right\rbrack \) equipped with the \( {L}_{2} \) norm\n\n\[ \n\parallel f{\parallel }_{2} \mathrel{\text{:=}} {\left( {\int }_{a}^{b}{\left| f\left( x\right) \right| }^{2}dx\right) }^{1/2} \n\]\n\nis not complete. | Proof. The norm is generated by the scalar product\n\n\[ \n\left( {f, g}\right) \mathrel{\text{:=}} {\int }_{a}^{b}f\left( x\right) g\left( x\right) {dx}. \n\]\n\nConsidering the same sequence as in Example 3.37, it can be seen that \( C\left\lbrack {a, b}\right\rbrack \) also is not complete with respect to the \( {L}... | Yes |
Theorem 3.39 Each finite-dimensional normed space is a Banach space. | Proof. Let \( X \) be finite-dimensional with basis \( {u}_{1},\ldots ,{u}_{n} \) and assume that \( \left( {x}_{\nu }\right) \) is a Cauchy sequence in \( X \) . We represent\n\n\[ \n{x}_{\nu } = \mathop{\sum }\limits_{{j = 1}}^{n}{\alpha }_{j\nu }{u}_{j} \n\] \n\nand recall from Theorem 3.8 that there exists \( C > 0... | Yes |
Theorem 3.44 Each contraction operator has at most one fixed point. | Proof. Assume that \( x \) and \( y \) are two different fixed points of the contraction operator \( A \) . Then\n\n\[ 0 \neq \parallel x - y\parallel = \parallel {Ax} - {Ay}\parallel \leq q\parallel x - y\parallel \]\n\nwhence \( 1 \leq q \) follows. This is a contradiction to the fact that \( A \) is a contraction op... | Yes |
Theorem 3.45 (Banach) Let \( U \) be a complete subset of a normed space \( X \) and let \( A : U \rightarrow U \) be a contraction operator. Then \( A \) has a unique fixed point. | Proof. Starting from an arbitrary element \( {x}_{0} \in U \) we define a sequence \( \left( {x}_{n}\right) \) in \( U \) by the recursion\n\n\[ \n{x}_{n + 1} \mathrel{\text{:=}} A{x}_{n},\;n = 0,1,2,\ldots \n\]\n\nThen we have\n\n\[ \n\begin{Vmatrix}{{x}_{n + 1} - {x}_{n}}\end{Vmatrix} = \begin{Vmatrix}{A{x}_{n} - A{x... | Yes |
Theorem 3.46 Let \( A \) be a contraction operator with contraction constant \( q \) mapping a complete subset \( U \) of a normed space \( X \) into itself. Then the successive approximations\n\n\[ \n{x}_{n + 1} \mathrel{\text{:=}} A{x}_{n},\;n = 0,1,2,\ldots ,\n\]\n\nwith arbitrary \( {x}_{0} \in U \) converge to the... | Proof. The a priori error estimate follows from (3.12) by passing to the limit \( m \rightarrow \infty \) . The a posteriori estimate follows from the a priori estimate applied with starting element \( {x}_{0} = {x}_{n - 1} \) . | Yes |
Theorem 3.48 Let \( B : X \rightarrow X \) be a bounded linear operator on a Banach space \( X \) with \( \parallel B\parallel < 1 \), and let \( I : X \rightarrow X \) denote the identity operator. Then \( I - B \) is bijective; i.e., for each \( z \in X \) the equation\n\n\[ x - {Bx} = z \]\n\nhas a unique solution \... | Proof. For fixed, but arbitrary, \( z \in X \) we define the operator \( A : X \rightarrow X \) by\n\n\[ {Ax} \mathrel{\text{:=}} {Bx} + z,\;x \in X. \]\n\nThen we have\n\n\[ \parallel {Ax} - {Ay}\parallel = \parallel B\left( {x - y}\right) \parallel \leq \parallel B\parallel \parallel x - y\parallel \]\n\nfor all \( x... | Yes |
Theorem 3.50 Let \( U \) be a finite-dimensional subspace of a normed space \( X \) . Then for every element in \( X \) there exists a best approximation with respect to \( U \) . | Proof. Let \( w \in X \) and choose a minimizing sequence \( \left( {u}_{n}\right) \) for \( w \) ; i.e., \( {u}_{n} \in U \) satisfies\n\n\[\n\begin{Vmatrix}{w - {u}_{n}}\end{Vmatrix} \rightarrow d \mathrel{\text{:=}} \mathop{\inf }\limits_{{u \in U}}\parallel w - u\parallel ,\;n \rightarrow \infty .\n\]\n\nBecause of... | Yes |
Theorem 3.51 Let \( U \) be a linear subspace of a pre-Hilbert space \( X \) . An element \( v \) is a best approximation to \( w \in X \) with respect to \( U \) if and only if\n\n\[ \left( {w - v, u}\right) = 0 \]\n\n(3.13)\n\nfor all \( u \in U \), i.e., if and only if \( w - v \bot U \) . To each \( w \in X \) ther... | Proof. We begin by noting the equality\n\n\[ \parallel w - u{\parallel }^{2} = \parallel w - v{\parallel }^{2} + 2\operatorname{Re}\left( {w - v, v - u}\right) + \parallel v - u{\parallel }^{2}, \]\n\n(3.14)\n\nwhich is valid for all \( u, v \in U \) . From this, sufficiency of the condition (3.13) is obvious, since \(... | Yes |
Theorem 3.52 Let \( U \) be a complete linear subspace of a pre-Hilbert space \( X \) . Then to each element \( w \in X \) there exists a unique best approximation with respect to \( U \) . The operator \( P : X \rightarrow U \) mapping \( w \in X \) onto its best approximation is a bounded linear operator with the pro... | Proof. Choose a sequence \( \left( {u}_{n}\right) \) with\n\n\[ \n{\begin{Vmatrix}w - {u}_{n}\end{Vmatrix}}^{2} \leq {d}^{2} + \frac{1}{n},\;n \in \mathbb{N},\n\]\n\n(3.15)\n\nwhere \( d \mathrel{\text{:=}} \mathop{\inf }\limits_{{u \in U}}\parallel w - u\parallel \) . Then\n\n\[ \n\parallel \left( {w - {u}_{n}}\right)... | Yes |
Corollary 3.53 Let \( U \) be a finite-dimensional linear subspace of a pre-Hilbert space \( X \) with basis \( {u}_{1},\ldots ,{u}_{n} \) . The linear combination\n\n\[ v = \mathop{\sum }\limits_{{k = 1}}^{n}{\alpha }_{k}{u}_{k} \]\n\nis the best approximation to \( w \in X \) with respect to \( U \) if and only if th... | Proof. The normal equations (3.16) obviously are equivalent to (3.13). | No |
Corollary 3.54 Let \( U \) be a finite-dimensional linear subspace of a pre-Hilbert space \( X \) with orthonormal basis \( {u}_{1},\ldots ,{u}_{n} \) . Then the orthogonal projection operator is given by\n\n\[ \n{Pw} = \mathop{\sum }\limits_{{k = 1}}^{n}\left( {w,{u}_{k}}\right) {u}_{k},\;w \in X.\n\] | Proof. This is trivial from either the orthogonality condition of Theorem 3.51 or the normal equations of Corollary 3.53. | No |
Theorem 4.1 Let \( B \) be an \( n \times n \) matrix. Then the successive approximations\n\n\[ \n{x}_{\nu + 1} \mathrel{\text{:=}} B{x}_{\nu } + z,\;\nu = 0,1,2,\ldots ,\n\]\n\nconverge for each \( z \in {\mathbb{C}}^{n} \) and each \( {x}_{0} \in {\mathbb{C}}^{n} \) if and only if\n\n\[ \n\rho \left( B\right) < 1\n\]... | Proof. If \( \rho \left( B\right) < 1 \), then by Theorem 3.32 there exists a norm \( \parallel \cdot \parallel \) on \( {\mathbb{C}}^{n} \) such that \( \parallel B\parallel < 1 \) . Now convergence follows from Theorem 3.48 together with the equivalence of all norms on \( {\mathbb{C}}^{n} \) according to Theorem 3.8.... | Yes |
Theorem 4.2 Assume that the matrix \( A = \left( {a}_{jk}\right) \) satisfies\n\n\[ \n{q}_{\infty } \mathrel{\text{:=}} \mathop{\max }\limits_{{j = 1,\ldots, n}}\mathop{\sum }\limits_{\substack{{k = 1} \\ {k \neq j} }}^{n}\left| \frac{{a}_{jk}}{{a}_{jj}}\right| < 1 \n\]\n\n(4.1)\n\nor\n\n\[ \n{q}_{1} \mathrel{\text{:=}... | Proof. The Jacobi matrix \( - {D}^{-1}\left( {{A}_{L} + {A}_{R}}\right) \) has diagonal entries zero and off-diagonal entries \( - {a}_{jk}/{a}_{jj} \) . Hence by Theorem 3.26 we have\n\n\[ \n{\begin{Vmatrix}-{D}^{-1}\left( {A}_{L} + {A}_{R}\right) \end{Vmatrix}}_{\infty } = {q}_{\infty } \n\]\n\n\[ \n{\begin{Vmatrix}-... | Yes |
Theorem 4.3 Assume that the matrix \( A = \left( {a}_{jk}\right) \) fulfills the Sassenfeld criterion\n\n\[ p \mathrel{\text{:=}} \mathop{\max }\limits_{{j = 1,\ldots, n}}{p}_{j} < 1 \]\n\nwhere the numbers \( {p}_{j} \) are recursively defined by\n\n\[ {p}_{1} \mathrel{\text{:=}} \mathop{\sum }\limits_{{k = 2}}^{n}\le... | Proof. Consider the equation\n\n\[ \left( {D + {A}_{L}}\right) x = - {A}_{R}z \]\n\nfor \( z \in {\mathbb{C}}^{n} \) with \( \parallel z{\parallel }_{\infty } = 1 \), that is,\n\n\[ {x}_{j} = - \mathop{\sum }\limits_{{k = 1}}^{{j - 1}}\frac{{a}_{jk}}{{a}_{jj}}{x}_{k} - \mathop{\sum }\limits_{{k = j + 1}}^{n}\frac{{a}_{... | Yes |
Example 4.5 The tridiagonal matrix\n\n\[ A = \left( \begin{array}{rrrrrr} 2 & - 1 & & & & \\ - 1 & 2 & - 1 & & & \\ & - 1 & 2 & - 1 & & \\ & \cdot & \cdot & \cdot & \cdot & \cdot \\ & & & - 1 & 2 & - 1 \\ & & & & - 1 & 2 \end{array}\right) \] from Example 2.1 is not strictly row-diagonally dominant, but it satisfies th... | Proof. Obviously, \( {q}_{\infty } = 1 \) ; i.e.,(4.1) is not fulfilled. We have the recursion\n\n\[ {p}_{1} = \frac{1}{2},\;{p}_{j} = \frac{1}{2}{p}_{j - 1} + \frac{1}{2},\;j = 2,\ldots, n - 1,\;{p}_{n} = \frac{1}{2}{p}_{n - 1}. \]\n\nFrom this, by induction, it follows that\n\n\[ {p}_{j} = 1 - \frac{1}{{2}^{j}},\;j =... | Yes |
Assume that the Jacobi matrix \( B \mathrel{\text{:=}} - {D}^{-1}\left( {{A}_{L} + {A}_{R}}\right) \) has real eigenvalues and spectral radius less than one. Then the spectral radius of the iteration matrix\n\n\[ I - \omega {D}^{-1}A = \left( {1 - \omega }\right) I - \omega {D}^{-1}\left( {{A}_{L} + {A}_{R}}\right) \]\... | Proof. For \( \omega > 0 \) the equation \( {Bu} = {\lambda u} \) is equivalent to\n\n\[ \left\lbrack {\left( {1 - \omega }\right) I + {\omega B}}\right\rbrack u = \left\lbrack {1 - \omega + {\omega \lambda }}\right\rbrack u. \]\n\nHence the eigenvalues \( \lambda \) of \( B \) correspond to the eigenvalues \( 1 - \ome... | Yes |
Theorem 4.11 (Kahan) A necessary condition for the SOR method to be convergent is that \( 0 < \omega < 2 \) . | Proof. Since the eigenvalues \( {\mu }_{1},\ldots ,{\mu }_{n} \) of \( B\left( \omega \right) \) are the zeros of the characteristic polynomial, they satisfy\n\n\[ \mathop{\prod }\limits_{{j = 1}}^{n}{\mu }_{j} = \det B\left( \omega \right) \]\n\n(where multiple eigenvalues are repeated according to their algebraic mul... | Yes |
Theorem 4.12 (Ostrowski) If \( A \) is Hermitian and positive definite, then the SOR method converges for all \( {x}_{0} \in {\mathbb{C}}^{n} \), all \( y \in {\mathbb{C}}^{n} \), and all \( 0 < \omega < 2 \) to the unique solution of \( {Ax} = y \) . | Proof. Let \( \mu \) be an eigenvalue of \( B\left( \omega \right) \) with eigenvector \( x \) ; i.e., \[ \left\lbrack {\left( {1 - \omega }\right) D - \omega {A}_{R}}\right\rbrack x = \mu \left( {D + \omega {A}_{L}}\right) x. \] With the aid of \[ \left( {2 - \omega }\right) D - {\omega A} - \omega \left( {{A}_{R} - {... | Yes |
Corollary 4.16 Under the assumptions of Theorem 4.15 the Gauss-Seidel method converges twice as fast as the Jacobi method. | Proof. From (4.8) we observe that \( \mu = {\lambda }^{2} \) for \( \omega = 1 \) ; i.e., we have\n\n\[ \rho \left\lbrack {B\left( 1\right) }\right\rbrack = {\left\{ \rho \left\lbrack -{D}^{-1}\left( {A}_{L} + {A}_{R}\right) \right\rbrack \right\} }^{2} \]\n\nfor the spectral radii of the Gauss-Seidel matrix \( B\left(... | Yes |
For the tridiagonal matrix \( A \) from Example 4.5 we have\n\n\[ \frac{N\left( \mathrm{{SOR}}\right) }{N\left( \mathrm{{Jacobi}}\right) } \approx \frac{\pi }{4\left( {n + 1}\right) } \] \n\nfor the optimal relaxation parameter. | Proof. Using the trigonometric addition theorem\n\n\[ \frac{1}{2}\sin \frac{{\pi j}\left( {k - 1}\right) }{n + 1} + \frac{1}{2}\sin \frac{{\pi j}\left( {k + 1}\right) }{n + 1} = \cos \frac{\pi j}{n + 1}\sin \frac{\pi jk}{n + 1}, \] \n\nit can be seen that the Jacobi matrix\n\n\[ - {D}^{-1}\left( {{A}_{L} + {A}_{R}}\rig... | Yes |
Theorem 4.18 For the spectral radius of \( T \) we have that \( \rho \left( T\right) = {0.5} \) ; i.e., the two-grid iterations converge. | Proof. We note that from (4.18) and (4.19), with \( h \) replaced by \( {2h} \), we have\n\nthat\n\[ \n{A}^{\left( 2h\right) }{v}_{j}^{\left( 2h\right) } = \frac{1}{{h}^{2}}{\sin }^{2}\left( {\pi jh}\right) {v}_{j}^{\left( 2h\right) } = \frac{4}{{h}^{2}}{c}_{j}^{2}{s}_{j}^{2}{v}_{j}^{\left( 2h\right) }, \n\]\n\nwhence\... | Yes |
We consider the best approximation of a given continuous function \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \) by a polynomial \[ p\left( x\right) = \mathop{\sum }\limits_{{k = 0}}^{n}{\alpha }_{k}{x}^{k} \] of degree \( n \) in the least squares sense, i.e., with respect to the \( {L}_{2} \) norm. | Using the monomials \( x \mapsto {x}^{k}, k = 0,1,\ldots, n \), as a basis of the subspace \( {P}_{n} \subset C\left\lbrack {0,1}\right\rbrack \) of polynomials of degree less than or equal to \( n \) (see Theorem 8.2), from Corollary 3.53 and the integrals \[ {\int }_{0}^{1}{x}^{j}{x}^{k}{dx} = \frac{1}{j + k + 1} \] ... | Yes |
Theorem 5.3 Let \( X \) and \( Y \) be Banach spaces, let \( A : X \rightarrow Y \) be a bounded linear operator with a bounded inverse \( {A}^{-1} : Y \rightarrow X \) and let \( {A}^{\delta } : X \rightarrow Y \) be a bounded linear operator such that \( \begin{Vmatrix}{A}^{-1}\end{Vmatrix}\begin{Vmatrix}{{A}^{\delta... | Proof. Writing \( {A}^{\delta } = A\left\lbrack {I + {A}^{-1}\left( {{A}^{\delta } - A}\right) }\right\rbrack \), by Theorem 3.48 we observe that the inverse operator \( {\left\lbrack {A}^{\delta }\right\rbrack }^{-1} = {\left\lbrack I + {A}^{-1}\left( {A}^{\delta } - A\right) \right\rbrack }^{-1}{A}^{-1} \) exists and... | Yes |
Theorem 5.4 Let \( A \) be an \( m \times n \) matrix of rank \( r \) . Then there exist nonnegative numbers\n\n\[ \n{\mu }_{1} \geq {\mu }_{2} \geq \cdots \geq {\mu }_{r} > {\mu }_{r + 1} = \cdots = {\mu }_{n} = 0 \n\]\n\nand orthonormal vectors \( {u}_{1},\ldots ,{u}_{n} \in {\mathbb{C}}^{n} \) and \( {v}_{1},\ldots ... | Proof. The Hermitian and semipositive definite matrix \( {A}^{ * }A \) of rank \( r \) has \( n \) orthonormal eigenvectors \( {u}_{1},\ldots ,{u}_{n} \) with nonnegative eigenvalues\n\n\[ \n{A}^{ * }A{u}_{j} = {\mu }_{j}^{2}{u}_{j},\;j = 1,\ldots, n \n\]\n\n(5.7)\n\nwhich we may assume to be ordered according to \( {\... | Yes |
Theorem 5.5 Let \( A \) be an \( m \times n \) matrix of rank \( r \) with singular system \( \left( {{\mu }_{j},{u}_{j},{v}_{j}}\right) \). The linear system\n\n\[ \n{Ax} = y \n\]\n\n(5.9)\n\nis solvable if and only if\n\n\[ \n\left( {y, z}\right) = 0 \n\]\n\n\( \left( {5.10}\right) \)\n\nfor all \( z \in {\mathbb{C}}... | Proof. Let \( x \) be a solution of (5.9) and let \( {A}^{ * }z = 0 \). Then\n\n\[ \n\left( {y, z}\right) = \left( {{Ax}, z}\right) = \left( {x,{A}^{ * }z}\right) = 0. \n\]\n\nThis implies the necessity of condition (5.10) for the solvability of (5.9).\n\nConversely, assume that (5.10) is satisfied. In terms of the ort... | Yes |
Theorem 5.7 Let \( A \) be an \( m \times n \) matrix of rank \( r \) with singular system \( \left( {{\mu }_{j},{u}_{j},{v}_{j}}\right) \) and let \( \alpha > 0 \) . Then for each \( y \in {\mathbb{C}}^{m} \) the linear system\n\n\[ \n\alpha {x}_{\alpha } + {A}^{ * }A{x}_{\alpha } = {A}^{ * }y \n\]\n\n(5.18)\n\nis uni... | Proof. For \( \alpha > 0 \) the matrix \( {\alpha I} + {A}^{ * }A \) is positive definite and therefore nonsingular. Since\n\n\[ \n\alpha {u}_{j} + {A}^{ * }A{u}_{j} = \left( {\alpha + {\mu }_{j}^{2}}\right) {u}_{j} \n\]\n\na singular system for the matrix \( {\alpha I} + {A}^{ * }A \) is given by \( \left( {\alpha + {... | Yes |
Corollary 5.8 Under the assumptions of Theorem 5.7 we have convergence:\n\n\[ \mathop{\lim }\limits_{{\alpha \rightarrow 0}}{\left( \alpha I + {A}^{ * }A\right) }^{-1}{A}^{ * }y = {A}^{ \dagger }y. \] | Proof. This is obvious from (5.13) and (5.19). | No |
Theorem 5.9 Let \( A \) be an \( m \times n \) matrix and let \( \alpha > 0 \) . Then for each \( y \in {\mathbb{C}}^{m} \) there exists a unique \( {x}_{\alpha } \in {\mathbb{C}}^{n} \) such that\n\n\[ \n{\begin{Vmatrix}A{x}_{\alpha } - y\end{Vmatrix}}_{2}^{2} + \alpha {\begin{Vmatrix}{x}_{\alpha }\end{Vmatrix}}_{2}^{... | Proof. (Compare to the proof of Theorem 3.51.) We first note the relation\n\n\[ \n\parallel {Ax} - y{\parallel }_{2}^{2} + \alpha \parallel x{\parallel }_{2}^{2} = \parallel A{x}_{\alpha } - y{\parallel }_{2}^{2} + \alpha \parallel {x}_{\alpha }{\parallel }_{2}^{2}\n\]\n\n\[ \n+ 2\operatorname{Re}\left( {x - {x}_{\alph... | Yes |
Theorem 5.10 Let \( A \) be an \( m \times n \) matrix and let \( y \in A\left( {\mathbb{C}}^{n}\right) ,{y}^{\delta } \in {\mathbb{C}}^{m} \) satisfy\n\n\[ \n{\begin{Vmatrix}{y}^{\delta } - y\end{Vmatrix}}_{2} \leq \delta < {\begin{Vmatrix}{y}^{\delta }\end{Vmatrix}}_{2}\n\]\n\nfor \( \delta > 0 \) . Then there exists... | Proof. We have to show that the function \( F : \left( {0,\infty }\right) \rightarrow \mathbb{R} \) defined by\n\n\[ \nF\left( \alpha \right) \mathrel{\text{:=}} {\begin{Vmatrix}A{x}_{\alpha } - {y}^{\delta }\end{Vmatrix}}_{2}^{2} - {\delta }^{2}\n\]\n\nhas a unique zero. In terms of a singular system, from the represe... | Yes |
Theorem 6.1 Let \( D \subset \mathbb{R} \) be a closed interval and let \( f : D \rightarrow D \) be a continuously differentiable function with the property\n\n\[ q \mathrel{\text{:=}} \mathop{\sup }\limits_{{x \in D}}\left| {{f}^{\prime }\left( x\right) }\right| < 1 \]\n\nThen the equation \( f\left( x\right) = x \) ... | Proof. Equipped with the norm \( \parallel \cdot \parallel = \left| \cdot \right| \) the space \( \mathbb{R} \) is complete. By the mean value theorem, for \( x, y \in D \) with \( x < y \), we have that\n\n\[ f\left( x\right) - f\left( y\right) = {f}^{\prime }\left( \xi \right) \left( {x - y}\right) \]\n\nfor some int... | Yes |
Theorem 6.2 Let \( x \) be a fixed point of a continuously differentiable function \( f \) such that \( \left| {{f}^{\prime }\left( x\right) }\right| < 1 \) . Then the method of successive approximations \( {x}_{\nu + 1} \mathrel{\text{:=}} f\left( {x}_{\nu }\right) \) is locally convergent; i.e., there exists a neighb... | Proof. Since \( {f}^{\prime } \) is continuous and \( \left| {{f}^{\prime }\left( x\right) }\right| < 1 \), there exist constants \( 0 < q < 1 \) and \( \delta > 0 \) such that \( \left| {{f}^{\prime }\left( y\right) }\right| \leq q \) for all \( y \in B \mathrel{\text{:=}} \left\lbrack {x - \delta, x + \delta }\right\... | Yes |
In order to describe a division by iteration, for \( a > 0 \) we consider the function \( f : \mathbb{R} \rightarrow \mathbb{R} \) given by \( f\left( x\right) \mathrel{\text{:=}} {2x} - a{x}^{2} \) . The graph of this function is a parabola with maximum value \( 1/a \) attained at \( 1/a \) . By solving the quadratic ... | From the the property \( x < f\left( x\right) < 1/a \), which is valid for \( 0 < x < 1/a \) , it follows that the sequence \( {x}_{\nu + 1} \mathrel{\text{:=}} 2{x}_{\nu } - a{x}_{\nu }^{2} \) is monotonicly increasing and bounded. Hence, the successive approximations converge to the fixed point \( x = 1/a \) for arbi... | Yes |
For computing the square root of a positive real number \( a \) by an iterative method we consider the function \( f : \left( {0,\infty }\right) \rightarrow \left( {0,\infty }\right) \) given by\n\n\[ f\left( x\right) \mathrel{\text{:=}} \frac{1}{2}\left( {x + \frac{a}{x}}\right) . \] | By solving the quadratic equation \( f\left( x\right) = x \) it can be seen that \( f \) has the fixed point \( x = \sqrt{a} \) . By the arithmetic geometric mean inequality we have that \( f\left( x\right) > \sqrt{a} \) for \( x > 0 \) ; i.e., \( f \) maps the open interval \( \left( {0,\infty }\right) \) into \( \lbr... | Yes |
Example 6.5 Consider the function \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) given by\n\n\[ f\left( x\right) \mathrel{\text{:=}} \cos x. \]\n\nHere we have\n\n\[ q = \mathop{\sup }\limits_{{0 \leq x \leq 1}}\left| {{f}^{\prime }\left( x\right) }\right| = \sin 1 < 1 \] | and Theorem 6.1 implies that the successive approximations \( {x}_{\nu + 1} \mathrel{\text{:=}} \cos {x}_{\nu } \) converge to the unique solution \( x \) of \( \cos x = x \) for each \( {x}_{0} \in \left\lbrack {0,1}\right\rbrack \) . Table 6.1 illustrates the convergence, which is notably slower than in the two previ... | Yes |
Example 6.6 The function \( h : \left( {0,1}\right) \rightarrow \left( {-\infty ,\infty }\right) \) given by \( h\left( x\right) \mathrel{\text{:=}} x + \ln x \) is strictly monotonically increasing with limits \( \mathop{\lim }\limits_{{x \rightarrow 0}}h\left( x\right) = - \infty \) and \( \mathop{\lim }\limits_{{x \... | \[ \left| {{f}^{\prime }\left( x\right) }\right| = \frac{1}{x} > 1 \] implies that \( f \) is not contracting in a neighborhood of the fixed point. However, we can still design a convergent scheme because \( x = - \ln x \) is equivalent to \( {e}^{-x} = x \) . We consider the inverse function \[ g\left( x\right) \mathr... | Yes |
Theorem 6.8 Let \( D \subset {\mathbb{R}}^{n} \) be closed and convex (with a nonempty interior) and let \( f : D \rightarrow D \) be a continuous mapping. Assume further that \( f \) is continuously differentiable in the interior of \( D \) and that its Jacobian can be continuously extended to all of \( D \) such that... | Proof. By the mean value Theorem 6.7 the mapping \( f : D \rightarrow D \) is a contraction.\n\nBy Theorem 3.26 we have that each of the conditions\n\n\[ \mathop{\sup }\limits_{{x \in D}}\mathop{\max }\limits_{{j = 1,\ldots, n}}\mathop{\sum }\limits_{{k = 1}}^{n}\left| {\frac{\partial {f}_{j}}{\partial {x}_{k}}\left( x... | Yes |
Theorem 6.14 Let \( D \subset {\mathbb{R}}^{n} \) be open and convex and let \( f : D \rightarrow {\mathbb{R}}^{n} \) be continuously differentiable. Assume that for some norm \( \parallel \cdot \parallel \) on \( {\mathbb{R}}^{n} \) and some \( {x}_{0} \in D \) the following conditions hold:\n\n(a) \( f \) satisfies\n... | Proof. 1. Let \( x, y, z \in D \). From the proof of Theorem 6.7 we know that\n\n\[ f\left( y\right) - f\left( x\right) = {\int }_{0}^{1}{f}^{\prime }\left\lbrack {{\lambda x} + \left( {1 - \lambda }\right) y}\right\rbrack \left( {y - x}\right) {d\lambda }.\]\n\nHence\n\n\[ f\left( y\right) - f\left( x\right) - {f}^{\p... | Yes |
Corollary 6.15 Let \( D \subset {\mathbb{R}}^{n} \) be open and let \( f : D \rightarrow {\mathbb{R}}^{n} \) be twice continuously differentiable, and assume that \( {x}^{ * } \) is a zero of \( f \) such that the Jacobian \( {f}^{\prime }\left( {x}^{ * }\right) \) is nonsingular. Then Newton’s method is locally conver... | Proof. Since \( f \) is twice continuously differentiable, by the mean value Theorem 6.7 applied to the components of \( {f}^{\prime } \) there exists \( \gamma > 0 \) such that\n\n\[ \begin{Vmatrix}{{f}^{\prime }\left( x\right) - {f}^{\prime }\left( y\right) }\end{Vmatrix} \leq \gamma \parallel x - y\parallel \]\n\nfo... | Yes |
Corollary 6.16 Let \( f : \left( {a, b}\right) \rightarrow \mathbb{R} \) be twice continuously differentiable and assume that \( {x}^{ * } \) is a simple zero of \( f \) . Then Newton’s method is locally convergent. | Proof. For simple zeros we have \( {f}^{\prime }\left( {x}^{ * }\right) \neq 0 \). | No |
Theorem 6.20 Under the assumptions of Theorem 6.14 Newton's method converges quadratically. | Proof. Using condition (b) of Theorem 6.14 and the inequality (6.5) we can estimate\n\n\[ \n\begin{Vmatrix}{{x}^{ * } - {x}_{\nu + 1}}\end{Vmatrix} = \begin{Vmatrix}{{x}^{ * } - {x}_{\nu } + {\left\lbrack {f}^{\prime }\left( {x}_{\nu }\right) \right\rbrack }^{-1}f\left( {x}_{\nu }\right) }\end{Vmatrix}\n\]\n\n\[ \n\leq... | Yes |
Theorem 6.21 Under the assumptions of Theorem 6.14 the simplified Newton method converges linearly to the unique zero of \( f \) in \( B\left\lbrack {{x}_{0}, r}\right\rbrack \) . | Proof. Recall that the function\n\n\[ g\left( x\right) \mathrel{\text{:=}} x - {\left\lbrack {f}^{\prime }\left( {x}_{0}\right) \right\rbrack }^{-1}f\left( x\right) \]\n\ndefined in the proof of Theorem 6.14 is a contraction. We show that \( g \) maps \( B\left\lbrack {{x}_{0}, r}\right\rbrack \) into itself. For this ... | Yes |
For the polynomial \( p\left( x\right) \mathrel{\text{:=}} {x}^{3} - {x}^{2} + {3x} - 5 \) the Horner scheme | <table><thead><tr><th>\( z \)</th><th>1</th><th>-1</th><th>3</th><th>- 5</th></tr></thead><tr><td>2</td><td>1</td><td>1</td><td>5</td><td>5</td></tr><tr><td>2</td><td>1</td><td>3</td><td>11</td><td></td></tr><tr><td>2</td><td>1</td><td>5</td><td></td><td></td></tr><tr><td>2</td><td>1</td><td></td><td></td><td></td></tr... | Yes |
Theorem 7.3 (Rayleigh) Let \( A \) be a Hermitian \( n \times n \) matrix with eigenvalues\n\n\[ \n{\lambda }_{1} \geq {\lambda }_{2} \geq \cdots \geq {\lambda }_{n}\n\]\n\n(where multiple eigenvalues occur according to their multiplicity) and corresponding orthonormal eigenvectors \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \)... | Proof. Let \( x \in {V}_{j} \) with \( x \neq 0 \) . Then\n\n\[ \nx = \mathop{\sum }\limits_{{k = j}}^{n}\left( {x,{x}_{k}}\right) {x}_{k}\;\text{ and }\;\mathop{\sum }\limits_{{k = j}}^{n}{\left| \left( x,{x}_{k}\right) \right| }^{2} = \left( {x, x}\right) .\n\]\n\nHence\n\n\[ \n{Ax} = \mathop{\sum }\limits_{{k = j}}^... | Yes |
Theorem 7.4 (Courant) Let \( A \) be a Hermitian \( n \times n \) matrix with eigenvalues\n\n\[ \n{\lambda }_{1} \geq {\lambda }_{2} \geq \cdots \geq {\lambda }_{n} \n\]\n\n(where multiple eigenvalues occur according to their multiplicity). Then\n\n\[ \n{\lambda }_{j} = \mathop{\min }\limits_{{{U}_{j} \in {M}_{j}}}\mat... | Proof. First we note that because of\n\n\[ \n\mathop{\sup }\limits_{\substack{{x \in {U}_{j}} \\ {x \neq 0} }}\frac{\left( Ax, x\right) }{\left( x, x\right) } = \mathop{\sup }\limits_{\substack{{x \in {U}_{j}} \\ {\left( {x, x}\right) = 1} }}\left( {{Ax}, x}\right) \n\]\n\nand the continuity of the function \( x \mapst... | Yes |
Corollary 7.5 Let \( A \) and \( B \) be two Hermitian \( n \times n \) matrices with eigenvalues \( {\lambda }_{1}\left( A\right) \geq {\lambda }_{2}\left( A\right) \geq \cdots \geq {\lambda }_{n}\left( A\right) \) and \( {\lambda }_{1}\left( B\right) \geq {\lambda }_{2}\left( B\right) \geq \cdots \geq {\lambda }_{n}\... | Proof. From the Cauchy-Schwarz inequality we have that\n\n\[ \n\left( {{Ax} - {Bx}, x}\right) \leq \parallel \left( {A - B}\right) x{\parallel }_{2}\parallel x{\parallel }_{2} \leq \parallel A - B{\parallel }_{2}\parallel x{\parallel }_{2}^{2} \n\] \n\nand hence\n\n\[ \n\left( {{Ax}, x}\right) \leq \left( {{Bx}, x}\rig... | Yes |
Corollary 7.6 For the eigenvalues \( {\lambda }_{1} \geq {\lambda }_{2} \geq \cdots \geq {\lambda }_{n} \) of a Hermitian \( n \times n \) matrix \( A = \left( {a}_{jk}\right) \) we have that\n\n\[ \n{\left| {\lambda }_{i} - {a}_{ii}^{\prime }\right| }^{2} \leq \mathop{\sum }\limits_{\substack{{j, k = 1} \\ {j \neq k} ... | Proof. Use \( B = \operatorname{diag}\left( {a}_{jj}^{\prime }\right) \) and \( \parallel \cdot \parallel = \parallel \cdot {\parallel }_{2} \) in the preceding corollary. | No |
Theorem 7.7 (Gerschgorin) Let \( A = \left( {a}_{jk}\right) \) be a complex \( n \times n \) matrix and define the disks\n\n\[ \n{G}_{j} \mathrel{\text{:=}} \left\{ {\lambda \in \mathbb{C} : \left| {\lambda - {a}_{jj}}\right| \leq \mathop{\sum }\limits_{\substack{{k = 1} \\ {k \neq j} }}^{n}\left| {a}_{jk}\right| }\rig... | Proof. Assume that \( {Ax} = {\lambda x} \) and \( \parallel x{\parallel }_{\infty } = 1 \), and for \( x = {\left( {x}_{1},\ldots ,{x}_{n}\right) }^{T} \) choose \( j \) such that \( \left| {x}_{j}\right| = \parallel x{\parallel }_{\infty } = 1 \) . Then \n\n\[ \n\left| {\lambda - {a}_{jj}}\right| = \left| {\left( {\l... | Yes |
Lemma 7.8 The Frobenius norm\n\n\\[ \n\\parallel A{\\parallel }_{F} \\mathrel{\\text{:=}} {\\left( \\mathop{\\sum }\\limits_{{j, k = 1}}^{n}{\\left| {a}_{jk}\\right| }^{2}\\right) }^{1/2}\n\\]\n\nof an \\( n \\times n \\) matrix \\( A = \\left( {a}_{jk}\\right) \\) is invariant with respect to unitary transformations. | Proof. The trace\n\n\\[ \n\\operatorname{tr}A \\mathrel{\\text{:=}} \\mathop{\\sum }\\limits_{{j = 1}}^{n}{a}_{jj}\n\\]\n\nof a matrix \\( A \\) is commutative; i.e., \\( \\operatorname{tr}{AB} = \\operatorname{tr}{BA} \\) . This follows from\n\n\\[ \n\\mathop{\\sum }\\limits_{{j = 1}}^{n}{\\left( AB\\right) }_{jj} = \... | Yes |
Corollary 7.9 The eigenvalues of an \( n \times n \) matrix \( A \) (counted repeatedly according to their algebraic multiplicity) satisfy Schur's inequality\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {\lambda }_{j}\right| }^{2} \leq \parallel A{\parallel }_{F}^{2} \]\n\nEquality holds if and only if the matrix \... | Proof. By Theorem 3.27 there exists a unitary matrix \( Q \) such that \( R \mathrel{\text{:=}} {Q}^{ * }{AQ} \) is an upper triangular matrix. Hence\n\n\[ \parallel A{\parallel }_{F}^{2} = \parallel R{\parallel }_{F}^{2} = \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {\lambda }_{j}\right| }^{2} + \mathop{\sum }\limits_{... | Yes |
Lemma 7.10 Normal matrices \( A \) satisfy\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {\lambda }_{j}\right| }^{2} = \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {a}_{jj}\right| }^{2} + {\left\lbrack N\left( A\right) \right\rbrack }^{2}. \] | Proof. This follows from Corollary 7.9. | No |
Lemma 7.11 For each pair \( j < k \) and each \( \varphi \in \mathbb{R} \) the matrix\n\n\[ U = \left( \begin{matrix} 1 & & & & & \\ & \cdot & & & & \\ & & \cos \varphi & & - \sin \varphi & \\ & & & \cdot & & \\ & & \sin \varphi & & \cos \varphi & \\ & & & & & \cdot \\ & & & & & 1 \end{matrix}\right) ,\]\n\nwhich coinc... | Proof. This follows from\n\n\[ \left( \begin{matrix} \cos \varphi & - \sin \varphi \\ \sin \varphi & \cos \varphi \end{matrix}\right) \left( \begin{matrix} \cos \varphi & \sin \varphi \\ - \sin \varphi & \cos \varphi \end{matrix}\right) = \left( \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right) \]\n\nand\n\n\[ \left(... | Yes |
Lemma 7.12 Let \( A \) be a real symmetric matrix and let \( U \) be the unitary matrix of Lemma 7.11. Then \( B = {U}^{ * }{AU} \) is also real and symmetric and has the entries\n\n\[ \n{b}_{jj} = {a}_{jj}{\cos }^{2}\varphi + {a}_{jk}\sin {2\varphi } + {a}_{kk}{\sin }^{2}\varphi \]\n\n\[ \n{b}_{kk} = {a}_{jj}{\sin }^{... | Proof. The matrix \( B \) is real, since \( A \) and \( U \) are real, and it is symmetric, since the unitary transformation of a Hermitian matrix is again Hermitian. Elementary calculations show that\n\n\[ \n\left( \begin{matrix} \cos \varphi & \sin \varphi \\ - \sin \varphi & \cos \varphi \end{matrix}\right) \left( \... | Yes |
Lemma 7.13 For\n\n\[ \tan {2\varphi } = \frac{2{a}_{jk}}{{a}_{jj} - {a}_{kk}},\;{a}_{jj} \neq {a}_{kk}, \]\n\n\[ \varphi = \frac{\pi }{4},\;{a}_{jj} = {a}_{kk}, \]\n\nthe transformation of Lemma 7.12 annihilates the elements\n\n\[ {b}_{jk} = {b}_{kj} = 0 \]\n\nand reduces the off-diagonal elements according to\n\n\[ {\... | Proof. \( {b}_{jk} = {b}_{kj} = 0 \) follows immediately from Lemma 7.12. Applying Lemma 7.8 to the matrices\n\n\[ \left( \begin{matrix} {a}_{jj} & {a}_{jk} \\ {a}_{kj} & {a}_{kk} \end{matrix}\right) \;\mathrm{{and}}\;\left( \begin{matrix} {b}_{jj} & {b}_{jk} \\ {b}_{kj} & {b}_{kk} \end{matrix}\right) \]\n\nyields\n\n\... | Yes |
Theorem 7.14 The classical Jacobi method converges; i.e., the sequence \( \left( {A}_{\nu }\right) \) converges to a diagonal matrix with the eigenvalues of \( A \) as diagonal elements. | Proof. For one step of the Jacobi method, from\n\n\[ \n{\left\lbrack N\left( A\right) \right\rbrack }^{2} \leq \left( {{n}^{2} - n}\right) \mathop{\max }\limits_{\substack{{i, l = 1,\ldots, n} \\ {i \neq l} }}{a}_{il}^{2} \n\]\n\nwe obtain that\n\n\[ \n{a}_{jk}^{2} \geq \frac{{\left\lbrack N\left( A\right) \right\rbrac... | Yes |
For the matrix\n\n\[ A = \left( \begin{array}{rrr} 2 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array}\right) \]\n\nthe first six transformed matrices for the classical Jacobi method are given by | \[ {A}_{1} = \left( \begin{array}{rrr} {1.0000} & {0.0000} & - {0.7071} \\ {0.0000} & {3.0000} & - {0.7071} \\ - {0.7071} & - {0.7071} & {2.0000} \end{array}\right) \]\n\n\[ {A}_{2} = \left( \begin{array}{rrr} {0.6340} & - {0.3251} & {0.0000} \\ - {0.3251} & {3.0000} & - {0.6280} \\ {0.0000} & - {0.6280} & {2.3660} \en... | Yes |
An \( n \times n \) matrix \( A \) is diagonalizable if and only if it has \( n \) linearly independent eigenvectors. | Assume that \( {C}^{-1}{AC} = D \), where \( D = \operatorname{diag}\left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \), is diagonal. Then \( D{e}_{j} = {\lambda }_{j}{e}_{j}, j = 1,\ldots, n \), with the canonical orthonormal basis \( {e}_{1},\ldots ,{e}_{n} \) of \( {\mathbb{C}}^{n} \) . This implies that the ve... | Yes |
Theorem 7.19 Assume that \( A \) is a diagonalizable \( n \times n \) matrix with eigenvalues\n\n\[ \n\\left| {\\lambda }_{1}\\right| > \\left| {\\lambda }_{2}\\right| > \\cdots > \\left| {\\lambda }_{n}\\right| \n\]\n\nand corresponding eigenvectors \( {x}_{1},{x}_{2},\\ldots ,{x}_{n} \), and set\n\n\[ \n{T}_{m} \\mat... | Proof. 1. Without loss of generality we may assume that \( {\\begin{Vmatrix}{x}_{j}\\end{Vmatrix}}_{2} = 1 \) for \( j = 1,\\ldots, n \) . From Lemma 7.18 it follows that\n\n\[ \n{\\begin{Vmatrix}{P}_{{A}^{\\nu }{S}_{m}} - {P}_{{T}_{m}}\\end{Vmatrix}}_{2} \\leq M{r}^{\\nu },\\;m = 1,\\ldots, n - 1,\\;\\nu \\in \\mathbb... | Yes |
Theorem 7.20 (QR algorithm) Let \( A \) be a diagonalizable matrix with eigenvalues\n\n\[ \left| {\lambda }_{1}\right| > \left| {\lambda }_{2}\right| > \cdots > \left| {\lambda }_{n}\right| \]\n\nand corresponding eigenvectors \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \), and assume that\n\n\[ \operatorname{span}\left\{ {{e}_... | Proof. This is just a special case of Theorem 7.19. | Yes |
Example 7.23 Let\n\n\[ A = \\left( \\begin{matrix} {a}_{1} & {c}_{2} & & & & \\\\ {c}_{2} & {a}_{2} & {c}_{3} & & & \\\\ & {c}_{3} & {a}_{3} & {c}_{4} & & \\\\ & & \\cdot & \\cdot & \\cdot & \\\\ & & & {c}_{n - 1} & {a}_{n - 1} & {c}_{n} \\\\ & & & & {c}_{n} & {a}_{n} \\end{matrix}\\right) \]\n\nbe a symmetric tridiago... | Proof. The recursion (7.25) follows by expanding \( \\det \\left( {{A}_{k} - {\\lambda I}}\\right) \) with respect to the last column, and (7.26) is obtained by differentiating (7.25). | Yes |
Theorem 8.1 For \( n \in \mathbb{N} \cup \{ 0\} \), each polynomial in \( {P}_{n} \) that has more than \( n \) (complex) zeros, where each zero is counted repeatedly according to its multiplicity, must vanish identically; i.e., all its coefficients must be equal to zero. | Proof. Obviously, the statement is true for \( n = 0 \) . Assume that it has been proven for some \( n \geq 0 \) . By using the binomial formula for \( {x}^{k} = {\left\lbrack \left( x - z\right) + z\right\rbrack }^{k} \) we can rewrite the polynomial \( p \in {P}_{n + 1} \) in the form\n\n\[ p\left( x\right) = \mathop... | Yes |
Theorem 8.2 The monomials \( {u}_{k}\left( x\right) \mathrel{\text{:=}} {x}^{k}, k = 0,\ldots, n \), are linearly independent. | Proof. In order to prove this, assume that\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}{u}_{k} = 0 \]\n\nthat is,\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}{x}^{k} = 0,\;x \in \left\lbrack {a, b}\right\rbrack \]\n\nThen the polynomial with coefficients \( {a}_{0},{a}_{1},\ldots ,{a}_{n} \) has more than \( n ... | Yes |
Theorem 8.3 Given \( n + 1 \) distinct points \( {x}_{0},\ldots ,{x}_{n} \in \left\lbrack {a, b}\right\rbrack \) and \( n + 1 \) values \( {y}_{0},\ldots ,{y}_{n} \in \mathbb{R} \), there exists a unique polynomial \( {p}_{n} \in {P}_{n} \) with the property\n\n\[ \n{p}_{n}\left( {x}_{j}\right) = {y}_{j},\;j = 0,\ldots... | Proof. We note that \( {\ell }_{k} \in {P}_{n} \) for \( k = 0,\ldots, n \) and that the equations\n\n\[ \n{\ell }_{k}\left( {x}_{j}\right) = {\delta }_{jk},\;j, k = 0,\ldots, n\n\]\n\nhold, where \( {\delta }_{jk} = 1 \) for \( k = j \), and \( {\delta }_{jk} = 0 \) for \( k \neq j \) . It follows that \( {p}_{n} \) g... | Yes |
Lemma 8.6 The divided differences satisfy the relation\n\n\[ \n{D}_{j}^{k} = \mathop{\sum }\limits_{{m = j}}^{{j + k}}{y}_{m}\mathop{\prod }\limits_{\substack{{i = j} \\ {i \neq m} }}^{{j + k}}\frac{1}{{x}_{m} - {x}_{i}},\;j = 0,\ldots, n - k,\;k = 1,\ldots, n. \n\]\n\n(8.4) | Proof. We proceed by induction with respect to the order \( k \) . Trivially,(8.4) holds for \( k = 1 \) . We assume that (8.4) has been proven for order \( k - 1 \) for some \( k \geq 2 \) . Then, using Definition 8.4, the induction assumption, and the identity\n\n\[ \n\frac{1}{{x}_{j + k} - {x}_{j}}\left\{ {\frac{1}{... | Yes |
Theorem 8.7 In the Newton representation, for \( n \geq 1 \) the uniquely determined interpolation polynomial \( {p}_{n} \) of Theorem 8.3 is given by\n\n\[ \n{p}_{n}\left( x\right) = {y}_{0} + \mathop{\sum }\limits_{{k = 1}}^{n}{D}_{0}^{k}\mathop{\prod }\limits_{{i = 0}}^{{k - 1}}\left( {x - {x}_{i}}\right) .\n\]\n\n(... | Proof. We denote the right-hand side of (8.5) by \( {\widetilde{p}}_{n} \) and establish \( {p}_{n} = {\widetilde{p}}_{n} \) by induction with respect to the degree \( n \) . For \( n = 1 \) the representation (8.5) is correct. We assume that (8.5) has been proven for degree \( n - 1 \) for some \( n \geq 2 \) and cons... | Yes |
Given \( n + 1 \) distinct points \( {x}_{0},\ldots ,{x}_{n} \in \left\lbrack {a, b}\right\rbrack \) and \( n + 1 \) values \( {y}_{0},\ldots ,{y}_{n} \in \mathbb{R} \), the uniquely determined interpolation polynomials \( {p}_{i}^{k} \in {P}_{k}, i = 0,\ldots, n - k, k = 0,\ldots, n \), with the interpolation property... | Proof. We again proceed by induction with respect to the degree \( k \) . Obviously, the statement is true for \( k = 1 \) . Assume that the assertion has been proven for degree \( k - 1 \) for some \( k \geq 2 \) . Then the right-hand side of (8.6) describes a polynomial \( p \in {P}_{k} \), and by the induction assum... | Yes |
Theorem 8.9 Given \( n + 1 \) distinct points \( {x}_{0},\ldots ,{x}_{n} \in \left\lbrack {a, b}\right\rbrack \) and \( n + 1 \) values \( {y}_{0},\ldots ,{y}_{n} \in \mathbb{R} \), the uniquely determined interpolation polynomials \( {p}_{i}^{k} \in {P}_{k}, i = 0,\ldots, n - k, k = 0,\ldots, n \), with the interpolat... | Proof. We again proceed by induction with respect to the degree \( k \) . Obviously, the statement is true for \( k = 1 \) . Assume that the assertion has been proven for degree \( k - 1 \) for some \( k \geq 2 \) . Then the right-hand side of (8.6) describes a polynomial \( p \in {P}_{k} \), and by the induction assum... | Yes |
Theorem 8.10 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be \( \left( {n + 1}\right) \) -times continuously differentiable. Then the remainder \( {R}_{n}f \mathrel{\text{:=}} f - {L}_{n}f \) for polynomial interpolation with \( n + 1 \) distinct points \( {x}_{0},\ldots ,{x}_{n} \in \left\lbra... | Proof. Since (8.8) is trivially satisfied if \( x \) coincides with one of the interpolation points \( {x}_{0},\ldots ,{x}_{n} \), we need be concerned only with the case where \( x \) does not coincide with one of the interpolation points. We define\n\n\[ {q}_{n + 1}\left( x\right) \mathrel{\text{:=}} \mathop{\prod }\... | Yes |
The linear interpolation is given by\n\n\\[ \n\\left( {{L}_{1}f}\\right) \\left( x\\right) = \\frac{1}{h}\\left\\lbrack {f\\left( {x}_{0}\\right) \\left( {{x}_{1} - x}\\right) + f\\left( {x}_{1}\\right) \\left( {x - {x}_{0}}\\right) }\\right\\rbrack \n\\]\n\nwith the step width \\( h = {x}_{1} - {x}_{0} \\) . For the p... | Therefore, by Corollary 8.11, the error occurring in linear interpolation of a twice continuously differentiable function \\( f \\) can be estimated by\n\n\\[ \n\\left| {\\left( {{R}_{1}f}\\right) \\left( x\\right) }\\right| \\leq \\frac{{h}^{2}}{8}\\mathop{\\max }\\limits_{{y \\in \\left\\lbrack {{x}_{0},{x}_{1}}\\rig... | Yes |
Example 8.13 Let \( f\left( x\right) \mathrel{\text{:=}} \sin x \) and let \( {x}_{0},\ldots ,{x}_{n} \in \left\lbrack {0,\pi }\right\rbrack \) be \( n + 1 \) distinct points. Since | \[ \left| {{f}^{\left( n + 1\right) }\left( x\right) }\right| \leq 1,\;x \in \left\lbrack {0,\pi }\right\rbrack \] and \[ \left| {{q}_{n + 1}\left( x\right) }\right| \leq {\pi }^{n + 1},\;x \in \left\lbrack {0,\pi }\right\rbrack \] by Corollary 8.11, we have the estimate \[ \left| {\left( {{R}_{n}f}\right) \left( x\rig... | Yes |
Theorem 8.16 (Marcinkiewicz) For each function \( f \in C\left\lbrack {a, b}\right\rbrack \) there exists a sequence of interpolation points \( \left( {x}_{j}^{\left( n\right) }\right), j = 0,\ldots, n, n = 0,1,\ldots \) , such that the sequence \( \left( {{L}_{n}f}\right) \) of interpolation polynomials \( {L}_{n}f \i... | Proof. The proof relies on the Weierstrass approximation theorem and the Chebyshev alternation theorem. The Weierstrass approximation theorem (see [16]) ensures that for each \( f \in C\left\lbrack {a, b}\right\rbrack \) there exists a sequence of polynomials \( {p}_{n} \in {P}_{n} \) such that \( {\begin{Vmatrix}{p}_{... | Yes |
Theorem 8.17 (Faber) For each sequence of interpolation points \( \left( {x}_{j}^{\left( n\right) }\right) \) there exists a function \( f \in C\left\lbrack {a, b}\right\rbrack \) such that the sequence \( \left( {{L}_{n}f}\right) \) of interpolation polynomials \( {L}_{n}f \in {P}_{n} \) does not converge to \( f \) u... | Proof. This is a consequence of the uniform boundedness principle, Theorem 12.7. It implies that from the convergence of the sequence \( \left( {{L}_{n}f}\right) \) for all \( f \in C\left\lbrack {a, b}\right\rbrack \) it follows that there must exist a constant \( C > 0 \) such that \( {\begin{Vmatrix}{L}_{n}\end{Vmat... | Yes |
Theorem 8.18 Given \( n + 1 \) distinct points \( {x}_{0},\ldots ,{x}_{n} \in \left\lbrack {a, b}\right\rbrack \) and \( {2n} + 2 \) values \( {y}_{0},\ldots ,{y}_{n} \in \mathbb{R} \) and \( {y}_{0}^{\prime },\ldots ,{y}_{n}^{\prime } \in \mathbb{R} \), there exists a unique polynomial \( {p}_{{2n} + 1} \in {P}_{{2n} ... | This Hermite interpolation polynomial is given by\n\n\[ \n{p}_{{2n} + 1} = \mathop{\sum }\limits_{{k = 0}}^{n}\left\lbrack {{y}_{k}{H}_{k}^{0} + {y}_{k}^{\prime }{H}_{k}^{1}}\right\rbrack\n\]\n\n(8.11)\n\nwith the Hermite factors\n\n\[ \n{H}_{k}^{0}\left( x\right) \mathrel{\text{:=}} \left\lbrack {1 - 2{\ell }_{k}^{\pr... | Yes |
Theorem 8.21 A trigonometric polynomial in \( {T}_{n} \) that has more than \( {2n} \) distinct zeros in the periodicity interval \( \lbrack 0,{2\pi }) \) must vanish identically; i.e., all its coefficients must be equal to zero. | Proof. We consider a trigonometric polynomial \( q \in {T}_{n} \) of the form\n\n\[ q\left( t\right) = \frac{{a}_{0}}{2} + \mathop{\sum }\limits_{{k = 1}}^{n}\left\lbrack {{a}_{k}\cos {kt} + {b}_{k}\sin {kt}}\right\rbrack \]\n\n(8.14)\n\nSetting \( {b}_{0} = 0 \) ,\n\n\[ {\gamma }_{k} \mathrel{\text{:=}} \frac{1}{2}\le... | Yes |
Theorem 8.22 The cosine functions \( {c}_{k}\left( t\right) \mathrel{\text{:=}} \cos {kt}, k = 0,1,\ldots, n \), and the sine functions \( {s}_{k}\left( t\right) \mathrel{\text{:=}} \sin {kt}, k = 1,\ldots, n \), are linearly independent in the function space \( C\left\lbrack {0,{2\pi }}\right\rbrack \) . | Proof. To prove this, assume that\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}{c}_{k} + \mathop{\sum }\limits_{{k = 1}}^{n}{b}_{k}{s}_{k} = 0 \]\n\nthat is,\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}\cos {kt} + \mathop{\sum }\limits_{{k = 1}}^{n}{b}_{k}\sin {kt} = 0,\;t \in \left\lbrack {0,{2\pi }}\right\rbra... | Yes |
Theorem 8.23 Given \( {2n} + 1 \) distinct points \( {t}_{0},\ldots ,{t}_{2n} \in \lbrack 0,{2\pi }) \) and \( {2n} + 1 \) values \( {y}_{0},\ldots ,{y}_{2n} \in \mathbb{R} \), there exists a uniquely determined trigonometric polynomial \( {q}_{n} \in {T}_{n} \) with the property\n\n\[ \n{q}_{n}\left( {t}_{j}\right) = ... | Proof. The function \( {q}_{n} \) belongs to \( {T}_{n} \), since the Lagrange factors are trigonometric polynomials of degree \( n \) . The latter is a consequence of\n\n\[ \n\sin \frac{t - {t}_{0}}{2}\sin \frac{t - {t}_{1}}{2} = \frac{1}{2}\cos \frac{{t}_{1} - {t}_{0}}{2} - \frac{1}{2}\cos \left( {t - \frac{{t}_{1} +... | Yes |
Theorem 8.24 There exists a unique trigonometric polynomial\n\n\[ \n{q}_{n}\left( t\right) = \frac{{a}_{0}}{2} + \mathop{\sum }\limits_{{k = 1}}^{n}\left\lbrack {{a}_{k}\cos {kt} + {b}_{k}\sin {kt}}\right\rbrack \n\]\n\nsatisfying the interpolation property\n\n\[ \n{q}_{n}\left( \frac{2\pi j}{{2n} + 1}\right) = {y}_{j}... | Its coefficients are given by\n\n\[ \n{a}_{k} = \frac{2}{{2n} + 1}\mathop{\sum }\limits_{{j = 0}}^{{2n}}{y}_{j}\cos \frac{2\pi jk}{{2n} + 1},\;k = 0,\ldots, n, \n\]\n\n\[ \n{b}_{k} = \frac{2}{{2n} + 1}\mathop{\sum }\limits_{{j = 0}}^{{2n}}{y}_{j}\sin \frac{2\pi jk}{{2n} + 1},\;k = 1,\ldots, n. \n\] | Yes |
Theorem 8.25 There exists a unique trigonometric polynomial\n\n\[ \n{q}_{n}\left( t\right) = \frac{{a}_{0}}{2} + \mathop{\sum }\limits_{{k = 1}}^{{n - 1}}\left\lbrack {{a}_{k}\cos {kt} + {b}_{k}\sin {kt}}\right\rbrack + \frac{{a}_{n}}{2}\cos {nt} \n\]\n\nsatisfying the interpolation property\n\n\[ \n{q}_{n}\left( \frac... | Its coefficients are given by\n\n\[ \n{a}_{k} = \frac{1}{n}\mathop{\sum }\limits_{{j = 0}}^{{{2n} - 1}}{y}_{j}\cos \frac{\pi jk}{n},\;k = 0,\ldots, n \n\]\n\n\[ \n{b}_{k} = \frac{1}{n}\mathop{\sum }\limits_{{j = 0}}^{{{2n} - 1}}{y}_{j}\sin \frac{\pi jk}{n},\;k = 1,\ldots, n - 1. \n\] | Yes |
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