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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 6.9 Let \( x \) be a fixed point of a continuously differentiable function \( f \) such that \( \begin{Vmatrix}{{f}^{\prime }\left( x\right) }\end{Vmatrix} < 1 \) in some norm \( \parallel \cdot \parallel \) on \( {\mathbb{R}}^{n} \) . Then the method of successive approximations \( {x}_{\nu + 1} \mathrel{\text... | Null | No |
Example 6.10 For the system\n\n\[ \n{x}_{1} = {0.5}\cos {x}_{1} - {0.5}\sin {x}_{2} \n\]\n\n\[ \n{x}_{2} = {0.5}\sin {x}_{1} + {0.5}\cos {x}_{2} \n\]\n\nwe have\n\[ \n{f}^{\prime }\left( x\right) = \left( \begin{array}{rr} - {0.5}\sin {x}_{1} & - {0.5}\cos {x}_{2} \\ {0.5}\cos {x}_{1} & - {0.5}\sin {x}_{2} \end{array}\... | Null | No |
Example 6.12 For the function\n\n\\[ \nf\\left( x\\right) \\mathrel{\\text{:=}} a - \\frac{1}{x} \n\\]\n\nwhere \\( a > 0 \\), the Newton iteration is given by\n\n\\[ \n{x}_{\\nu + 1} \\mathrel{\\text{:=}} 2{x}_{\\nu } - a{x}_{\\nu }^{2} \n\\]\n\nBy Example 6.3 we have convergence for all \\( {x}_{0} \\in \\left( {0,2/... | Null | No |
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 |
For the function \( f\left( x\right) \mathrel{\text{:=}} x - \cos x \) the Newton iteration reads\n\n\[ \n{x}_{\nu + 1} \mathrel{\text{:=}} {x}_{\nu } - \frac{{x}_{\nu } - \cos {x}_{\nu }}{1 + \sin {x}_{\nu }} \n\] | Null | No |
For the function \( f\left( x\right) \mathrel{\text{:=}} x - {e}^{-x} \) the Newton iteration reads\n\n\[ {x}_{\nu + 1} \mathrel{\text{:=}} {x}_{\nu } - \frac{{x}_{\nu } - {e}^{-{x}_{\nu }}}{1 + {e}^{-{x}_{\nu }}} \] | Null | 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 |
Theorem 6.22 Let\n\n\\[ p\\left( x\\right) = {a}_{0}{x}^{n} + {a}_{1}{x}^{n - 1} + {a}_{2}{x}^{n - 2} + \\cdots + {a}_{n - 1}x + {a}_{n} \\]\n\nbe a polynomial of degree \\( n \\) . For \\( z \\in \\mathbb{C} \\) the complete Horner scheme\n\n<table><thead><tr><th></th><th>\\( {a}_{0} \\)</th><th>\\( {a}_{1} \\)</th><t... | Null | No |
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 |
Example 6.24 The polynomial\n\n\[ p\\left( x\\right) \\mathrel{\\text{:=}} \\left( {x - 1}\\right) \\left( {x - 2}\\right) \\cdots \\left( {x - {10}}\\right) = {x}^{10} - {55}{x}^{9} + \\cdots + {10}! \]\n\nhas the zeros \( 1,2,\\ldots ,{10} \), which are well separated from each other. We perturb the coefficient of \(... | Null | No |
The vibrations of a string are modeled by the so-called wave equation\n\n\\[ \n\\frac{{\\partial }^{2}w}{\\partial {x}^{2}} = \\frac{1}{{c}^{2}}\\frac{{\\partial }^{2}w}{\\partial {t}^{2}}\n\\]\n\nwhere \\( w = w\\left( {x, t}\\right) \\) denotes the vertical elongation and \\( c \\) is the speed of sound in the string... | Null | No |
Consider the eigenvalue problem\n\n\[ \n{\int }_{0}^{1}K\left( {x, y}\right) \varphi \left( y\right) {dy} = {\lambda \varphi }\left( x\right) ,\;x \in \left\lbrack {0,1}\right\rbrack \n\] \n\nfor a linear integral operator with continuous kernel \( K \) . For the numerical approximation we proceed as in Example 2.3 and... | Null | No |
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. | No |
Theorem 7.22 To each \( n \times n \) matrix \( A \) there exist \( n - 2 \) Householder matrices \( {H}_{1},\ldots ,{H}_{n - 2} \) such for \( Q = {H}_{n - 2}\cdots {H}_{1} \) the matrix\n\n\[ B = {Q}^{ * }{AQ} \]\n\n## is a Hessenberg matrix. | Null | No |
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 |
The \( n \times n \) tridiagonal matrix\n\n\[ A = \left( \begin{array}{rrrrrr} 2 & - 1 & & & & \\ - 1 & 2 & - 1 & & & \\ & - 1 & 2 & - 1 & & \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ & & & - 1 & 2 & - 1 \\ & & & & - 1 & 2 \end{array}\right) \]\n\nhas the eigenvalues\n\n\[ {\lambda }_{j} = 4{\sin }^{2}\frac{j\... | Null | No |
Theorem 7.25 Let \( B = \left( {b}_{jk}\right) \) be an irreducible Hessenberg matrix and let \( \lambda \in \mathbb{C} \) . Starting from \( {\xi }_{n} = 1,{\eta }_{n} = 0 \), compute recursively\n\n\[ \n{\xi }_{n - k} = \frac{1}{{b}_{n - k + 1, n - k}}\left\{ {\lambda {\xi }_{n - k + 1} - \mathop{\sum }\limits_{{j = ... | Null | No |
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 |
Corollary 8.11 Under the assumptions of Theorem 8.10 we have the error estimate \[ {\begin{Vmatrix}{R}_{n}f\end{Vmatrix}}_{\infty } \leq \frac{1}{\left( {n + 1}\right) !}{\begin{Vmatrix}{q}_{n + 1}\end{Vmatrix}}_{\infty }{\begin{Vmatrix}{f}^{\left( n + 1\right) }\end{Vmatrix}}_{\infty }.\] | Null | No |
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 |
Example 8.14 A first detailed example of the insufficiency of polynomial interpolation even for analytic functions was investigated by Runge in 1901. He considered the simple function\n\n\[ f\left( x\right) = \frac{1}{1 + {25}{x}^{2}} \]\n\non the interval \( \left\lbrack {-1,1}\right\rbrack \) with equidistant interpo... | Null | No |
Example 8.15 Consider the continuous function\n\n\\[ \nf\\left( x\\right) \\mathrel{\\text{:=}} \\left\\{ \\begin{array}{ll} x\\sin \\frac{\\pi }{x}, & x \\in (0,1\\rbrack \\\\ 0, & x = 0 \\end{array}\\right. \n\\]\n\nWith the interpolation points chosen as\n\n\\[ \n{x}_{j} = \\frac{1}{j + 1},\\;j = 0,\\ldots, n \n\\]\... | Null | No |
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 8.19 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be \( \left( {{2n} + 2}\right) \) -times continuously differentiable. Then the remainder \( {R}_{n}f \mathrel{\text{:=}} f - {H}_{n}f \) for Hermite interpolation with \( n + 1 \) distinct points \( {x}_{0},\ldots ,{x}_{n} \in \left\lbra... | Null | No |
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 |
Lemma 8.28 Let \( m = 2\ell - 1 \) with \( \ell \in \mathbb{N} \) and \( \ell \geq 2 \), and let \( f \in {C}^{\ell }\left\lbrack {a, b}\right\rbrack \) . Assume that the spline \( s \in {S}_{m}^{n} \) interpolates \( f \), i.e., \[ s\left( {x}_{j}\right) = f\left( {x}_{j}\right) ,\;j = 0,\ldots, n, \] (8.26) and that ... | Proof. We have that \[ {\int }_{a}^{b}{\left\lbrack {f}^{\left( \ell \right) }\left( x\right) - {s}^{\left( \ell \right) }\left( x\right) \right\rbrack }^{2}{dx} = {\int }_{a}^{b}{\left\lbrack {f}^{\left( \ell \right) }\left( x\right) \right\rbrack }^{2}{dx} - {\int }_{a}^{b}{\left\lbrack {s}^{\left( \ell \right) }\lef... | Yes |
Lemma 8.29 Under the assumptions of Lemma 8.28 let \( f = 0 \) . Then \( s = 0 \) . | Proof. For \( f = 0 \), from (8.28) it follows that\n\n\[ \n{\int }_{a}^{b}{\left\lbrack {s}^{\left( \ell \right) }\left( x\right) \right\rbrack }^{2}{dx} = 0 \n\]\n\nThis implies that \( {s}^{\left( \ell \right) } = 0 \), and therefore \( s \in {P}_{\ell - 1} \) on \( \left\lbrack {a, b}\right\rbrack \) . Now the boun... | Yes |
Theorem 8.30 Let \( m = 2\ell - 1 \) with \( \ell \in \mathbb{N} \) and \( \ell \geq 2 \) . Then, given \( n + 1 \) values \( {y}_{0},\ldots ,{y}_{n} \) and \( m - 1 \) boundary data \( {a}_{1},\ldots ,{a}_{\ell - 1} \) and \( {b}_{1},\ldots ,{b}_{\ell - 1} \) , there exists a unique spline \( s \in {S}_{m}^{n} \) sati... | Proof. Representing the spline in the form (8.25), i.e.,\n\n\[ s\left( x\right) = \mathop{\sum }\limits_{{k = 0}}^{m}{\alpha }_{k}{u}_{k} + \mathop{\sum }\limits_{{k = 1}}^{{n - 1}}{\beta }_{k}{v}_{k} \]\n\n(8.31)\n\n\nit follows that the interpolation conditions (8.29) and boundary conditions (8.30) are satisfied if a... | Yes |
Theorem 8.31 For \( m \in \mathbb{N} \cup \{ 0\} \) the B-splines\n\n\[ \n{B}_{m}\left( {\cdot - k}\right) ,\;k = 0,\ldots, m \n\]\n\n(8.37)\n\nare linearly independent on the interval \( {I}_{m} \mathrel{\text{:=}} \left\lbrack {\frac{m - 1}{2},\frac{m + 1}{2}}\right\rbrack \) . | Proof. This is trivial for \( m = 0 \), and we assume that it has been proven for degree \( m - 1 \) for some \( m \geq 1 \) . Let\n\n\[ \n\mathop{\sum }\limits_{{k = 0}}^{m}{\alpha }_{k}{B}_{m}\left( {x - k}\right) = 0,\;x \in {I}_{m} \n\]\n\n(8.38)\n\nThen, with the aid of (8.33), differentiating (8.38) yields\n\n\[ ... | Yes |
Corollary 8.32 Let \( {x}_{k} = a + {hk}, k = 0,\ldots, n \), be an equidistant subdivision of the interval \( \left\lbrack {a, b}\right\rbrack \) of step size \( h = \left( {b - a}\right) /n \) with \( n \geq 2 \), and let \( m = 2\ell - 1 \) with \( \ell \in \mathbb{N} \) . Then the B-splines\n\n\[ \n{B}_{m, k}\left(... | Proof. The \( n + m \) splines (8.39) belong to \( {S}_{m}^{n} \), and by the preceding Theorem 8.31 they can be shown to be linearly independent on \( \left\lbrack {a, b}\right\rbrack \) . Hence, the statement follows from Theorem 8.27. | Yes |
Theorem 8.33 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be twice continuously differentiable and let \( s \in {S}_{3}^{n} \) be the uniquely determined cubic spline satisfying the interpolation and boundary conditions of Lemma 8.28. Then\n\n\[ \parallel f - s{\parallel }_{\infty } \leq \frac{... | Proof. The error function \( r \mathrel{\text{:=}} f - s \) has \( n + 1 \) zeros \( {x}_{0},\ldots ,{x}_{n} \) . Hence, the distance between two consecutive zeros of \( r \) is less than or equal to \( h \) . By Rolle’s theorem, the derivative \( {r}^{\prime } \) has \( n \) zeros with distance less than or equal to \... | Yes |
Theorem 8.34 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be four-times continuously differentiable and let \( s \in {S}_{3}^{n} \) be the uniquely determined cubic spline satisfying the interpolation and boundary conditions of Lemma 8.28 for an equidistant subdivision with step width \( h \) .... | Proof. By \( {L}_{1} : C\left\lbrack {a, b}\right\rbrack \rightarrow {S}_{1}^{n} \) we denote the interpolation operator mapping \( g \in C\left\lbrack {a, b}\right\rbrack \) onto its uniquely determined piecewise linear interpolation. From Example 8.12 we obtain that\n\n\[ \parallel r{\parallel }_{\infty } = {\begin{V... | Yes |
Theorem 8.37 The Bernstein polynomials are nonnegative on \( \\left\\lbrack {0,1}\\right\\rbrack \) and provide a partition of unity; i.e., \[ {B}_{k}^{n}\\left( t\\right) \\geq 0,\\;t \\in \\left\\lbrack {0,1}\\right\\rbrack \] and \[ \\mathop{\\sum }\\limits_{{k = 0}}^{n}{B}_{k}^{n}\\left( t\\right) = 1,\\;t \\in \\m... | Proof. The first five properties are obvious. The statement on the maximum of \( {B}_{k}^{n} \) is a consequence of \[ \\frac{d}{dt}{B}_{k}^{n}\\left( t\\right) = \\left( \\begin{array}{l} n \\\\ k \\end{array}\\right) {t}^{k - 1}{\\left( 1 - t\\right) }^{n - k - 1}\\left( {k - {nt}}\\right) ,\\;k = 0,\\ldots n. \] The... | No |
Theorem 8.39 Let\n\n\\[ \np\\left( t\\right) = \\mathop{\\sum }\\limits_{{k = 0}}^{n}{b}_{k}{B}_{k}^{n}\\left( t\\right) ,\\;t \\in \\left\\lbrack {0,1}\\right\\rbrack \n\\]\n\nbe a Bézier polynomial on \\( \\left\\lbrack {0,1}\\right\\rbrack \\) . Then\n\n\\[ \n{p}^{\\left( j\\right) }\\left( t\\right) = \\frac{n!}{\\... | Proof. Obviously, the statement is true for \\( j = 0 \\) . We assume that it has been proven for some \\( 0 \\leq j < n \\) . Then with the aid of (8.54) we obtain\n\n\\[ \n{p}^{\\left( j + 1\\right) }\\left( t\\right) = \\frac{n!}{\\left( {n - j}\\right) !}\\mathop{\\sum }\\limits_{{k = 0}}^{{n - j}}{\\bigtriangleup ... | Yes |
Corollary 8.40 The polynomial from Theorem 8.39 has the derivatives\n\n\\[ \n{p}^{\\left( j\\right) }\\left( 0\\right) = \\frac{n!}{\\left( {n - j}\\right) !}{\\bigtriangleup }^{j}{b}_{0},\\;{p}^{\\left( j\\right) }\\left( 1\\right) = \\frac{n!}{\\left( {n - j}\\right) !}{\\bigtriangleup }^{j}{b}_{n - j} \n\\]\n\n at t... | From Corollary 8.40 we note that \\( {p}^{\\left( j\\right) }\\left( 0\\right) \\) depends only on \\( {b}_{0},\\ldots ,{b}_{j} \\) and that \\( {p}^{\\left( j\\right) }\\left( 1\\right) \\) depends only on \\( {b}_{n - j},\\ldots ,{b}_{n} \\) . In particular, we have that\n\n\\[ \n{p}^{\\prime }\\left( 0\\right) = n\\... | Yes |
Theorem 8.41 The subpolynomials \( {b}_{i}^{k} \) of a Bézier polynomial \( p \) of degree \( n \) satisfy the recursion formulae\n\n\[ \n{b}_{i}^{k}\left( t\right) = \left( {1 - t}\right) {b}_{i}^{k - 1}\left( t\right) + t{b}_{i + 1}^{k - 1}\left( t\right) \n\]\n\n(8.57)\n\nfor \( i = 0,\ldots, n - k \) and \( k = 1,\... | Proof. We insert the recursion formulae (8.51) and (8.52) for the Bernstein polynomials into the definition (8.56) for the subpolynomials and obtain\n\n\[ \n{b}_{i}^{k}\left( t\right) = {b}_{i}{B}_{0}^{k}\left( t\right) + \mathop{\sum }\limits_{{j = 1}}^{{k - 1}}{b}_{i + j}{B}_{j}^{k}\left( t\right) + {b}_{i + k}{B}_{k... | Yes |
Theorem 8.42 The Bézier polynomials\n\n\\[ \n{p}_{1}\\left( x\\right) \\mathrel{\\text{:=}} \\mathop{\\sum }\\limits_{{k = 0}}^{n}{b}_{0}^{k}\\left( t\\right) {B}_{k}^{n}\\left( {x;0, t}\\right) \\;\\text{ and }\\;{p}_{2}\\left( x\\right) \\mathrel{\\text{:=}} \\mathop{\\sum }\\limits_{{k = 0}}^{n}{b}_{k}^{n - k}\\left... | Proof. Inserting the equivalent definition (8.56) of the subpolynomials and reordering the summation, we find that\n\n\\[ \n{p}_{1}\\left( x\\right) = \\mathop{\\sum }\\limits_{{k = 0}}^{n}\\mathop{\\sum }\\limits_{{j = 0}}^{k}{b}_{j}{B}_{j}^{k}\\left( t\\right) {B}_{k}^{n}\\left( {x;0, t}\\right) = \\mathop{\\sum }\\l... | Yes |
Theorem 9.1 The polynomial interpolatory quadrature of order \( n \) defined by\n\n\[ \n{Q}_{n}\left( f\right) \mathrel{\text{:=}} {\int }_{a}^{b}\left( {{L}_{n}f}\right) \left( x\right) {dx} \n\]\n\n(9.3)\n\nis of the form (9.2) with the weights given by\n\n\[ \n{a}_{k} = \frac{1}{{q}_{n + 1}^{\prime }\left( {x}_{k}\r... | Proof. From (8.2) we obtain\n\n\[ \n{\int }_{a}^{b}\left( {{L}_{n}f}\right) \left( x\right) {dx} = \mathop{\sum }\limits_{{k = 0}}^{n}f\left( {x}_{k}\right) {\int }_{a}^{b}{\ell }_{k}\left( x\right) {dx} \n\]\n\nwith\n\[ \n{a}_{k} = {\int }_{a}^{b}{\ell }_{k}\left( x\right) {dx} = {\int }_{a}^{b}\mathop{\prod }\limits_... | Yes |
Theorem 9.2 Given \( n + 1 \) distinct quadrature points \( {x}_{0},\ldots ,{x}_{n} \in \left\lbrack {a, b}\right\rbrack \) , the interpolatory quadrature (9.3) of order \( n \) is uniquely determined by its property of integrating all polynomials \( p \in {P}_{n} \) exactly, i.e., by the property\n\n\[ \n\mathop{\sum ... | Proof. From (9.3) and \( {L}_{n}p = p \) for all \( p \in {P}_{n} \) it follows that\n\n\[ \n\mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}p\left( {x}_{k}\right) = {\int }_{a}^{b}\left( {{L}_{n}p}\right) \left( x\right) {dx} = {\int }_{a}^{b}p\left( x\right) {dx} \n\]\n\ni.e., the quadrature is exact for all \( p \in {P}_{... | Yes |
Theorem 9.3 The polynomial interpolatory quadrature of order \( n \) with equidistant quadrature points\n\n\[ \n{x}_{k} = a + {kh},\;k = 0,\ldots, n, \]\n\nand step width \( h = \left( {b - a}\right) /n \) is called the Newton-Cotes quadrature formula of order \( n \) . Its weights are given by\n\n\[ \n{a}_{k} = h\frac... | Proof. The weights are obtained from (9.4) by substituting \( x = {x}_{0} + {hz} \) and observing that\n\n\[ \n{q}_{n + 1}\left( x\right) = {h}^{n + 1}\mathop{\prod }\limits_{{j = 0}}^{n}\left( {z - j}\right) \]\n\nand\n\n\[ \n{q}_{n + 1}^{\prime }\left( {x}_{k}\right) = {\left( -1\right) }^{n - k}k!\left( {n - k}\righ... | Yes |
Theorem 9.4 Let \( f : C\left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be twice continuously differentiable. Then the error for the trapezoidal rule can be represented in the form\n\n\[{\int }_{a}^{b}f\left( x\right) {dx} - \frac{b - a}{2}\left\lbrack {f\left( a\right) + f\left( b\right) }\right\rbrack = - ... | Proof. Let \( {L}_{1}f \) denote the linear interpolation of \( f \) at the interpolation points \( {x}_{0} = a \) and \( {x}_{1} = b \) . By construction of the trapezoidal rule we have that the error\n\n\[{E}_{1}\left( f\right) \mathrel{\text{:=}} {\int }_{a}^{b}f\left( x\right) {dx} - \frac{b - a}{2}\left\lbrack {f\... | Yes |
Theorem 9.5 Let \( f : C\left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be four-times continuously differentiable. Then the error for Simpson's rule can be represented in the form\n\n\[{\int }_{a}^{b}f\left( x\right) {dx} - \frac{b - a}{6}\left\lbrack {f\left( a\right) + {4f}\left( \frac{a + b}{2}\right) + f... | Proof. Let \( {L}_{2}f \) denote the quadratic interpolation polynomial for \( f \) at the interpolation points \( {x}_{0} = a,{x}_{1} = \left( {a + b}\right) /2 \), and \( {x}_{2} = b \) . By construction of Simpson's rule we have that the error\n\n\[{E}_{2}\left( f\right) \mathrel{\text{:=}} {\int }_{a}^{b}f\left( x\... | Yes |
Example 9.6 The approximation of\n\n\\[ \ln 2 = {\\int }_{0}^{1}\\frac{dx}{1 + x} \\]\n\nby the trapezoidal rule yields\n\n\\[ \ln 2 \\approx \\frac{1}{2}\\left\\lbrack {1 + \\frac{1}{2}}\\right\\rbrack = {0.75} \\] | For \\( f\\left( x\\right) \\mathrel{\\text{:=}} 1/\\left( {1 + x}\\right) \\) we have\n\n\\[ \\frac{{h}^{3}}{12}{\\begin{Vmatrix}{f}^{\\prime \\prime }\\end{Vmatrix}}_{\\infty } = \\frac{1}{6} \\]\n\nand hence, from Theorem 9.4, we obtain the estimate \\( \\left| {\\ln 2 - {0.75}}\\right| \\leq {0.167} \\) as compared... | Yes |
Theorem 9.7 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be twice continuously differentiable. Then the error for the composite trapezoidal rule is given by\n\n\[{\int }_{a}^{b}f\left( x\right) {dx} - {T}_{h}\left( f\right) = - \frac{b - a}{12}{h}^{2}{f}^{\prime \prime }\left( \xi \right)\]\n\n... | Proof. By Theorem 9.4 we have that\n\n\[{\int }_{a}^{b}f\left( x\right) {dx} - {T}_{h}\left( f\right) = - \frac{{h}^{3}}{12}\mathop{\sum }\limits_{{k = 1}}^{n}{f}^{\prime \prime }\left( {\xi }_{k}\right)\]\n\nwhere \( a \leq {\xi }_{1} \leq {\xi }_{2} \leq \cdots \leq {\xi }_{n} \leq b \). From\n\n\[n\mathop{\min }\lim... | Yes |
Theorem 9.10 (Szegö) Let\n\n\[ \n{Q}_{n}\left( f\right) = \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}^{\left( n\right) }f\left( {x}_{k}^{\left( n\right) }\right) \n\]\n\nbe a sequence of quadrature formulae that converges for all polynomials, i.e,\n\n\[ \n\mathop{\lim }\limits_{{n \rightarrow \infty }}{Q}_{n}\left( p\ri... | Proof. Let \( f \in C\left\lbrack {a, b}\right\rbrack \) and \( \varepsilon > 0 \) be arbitrary. By the Weierstrass approximation theorem (see [16]) there exists a polynomial \( p \) such that\n\n\[ \n\parallel f - p{\parallel }_{\infty } \leq \frac{\varepsilon }{2\left( {C + b - a}\right) }.\n\]\n\nThen, since by (9.1... | Yes |
Corollary 9.11 (Steklov) Assume that the sequence \( \left( {Q}_{n}\right) \) of quadrature formulae converges for all polynomials and that all the weights are nonnegative. Then the sequence \( \left( {Q}_{n}\right) \) is convergent. | Proof. This follows from\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}\left| {a}_{k}^{\left( n\right) }\right| = \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}^{\left( n\right) } = {Q}_{n}\left( 1\right) \rightarrow {\int }_{a}^{b}{dx} = b - a,\;n \rightarrow \infty ,\]\n\nand the preceding Theorem 9.10. | Yes |
Lemma 9.13 Let \( {x}_{0},\ldots ,{x}_{n} \) be the \( n + 1 \) distinct quadrature points of a Gaussian quadrature formula. Then\n\n\[ \n{\int }_{a}^{b}w\left( x\right) {q}_{n + 1}\left( x\right) q\left( x\right) {dx} = 0 \n\]\n\n(9.16)\n\nfor \( {q}_{n + 1}\left( x\right) \mathrel{\text{:=}} \left( {x - {x}_{0}}\righ... | Proof. Since \( {q}_{n + 1}q \in {P}_{{2n} + 1} \) and \( {q}_{n + 1}\left( {x}_{k}\right) = 0 \), we have that\n\n\[ \n{\int }_{a}^{b}w\left( x\right) {q}_{n + 1}\left( x\right) q\left( x\right) {dx} = \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}{q}_{n + 1}\left( {x}_{k}\right) q\left( {x}_{k}\right) = 0 \n\]\n\nfor all... | Yes |
Lemma 9.14 Let \( {x}_{0},\ldots ,{x}_{n} \) be \( n + 1 \) distinct points satisfying the condition (9.16). Then the corresponding polynomial interpolatory quadrature is a Gaussian quadrature formula. | Proof. Let \( {L}_{n} \) denote the polynomial interpolation operator for the interpolation points \( {x}_{0},\ldots ,{x}_{n} \) . By construction, for the interpolatory quadrature we have\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}f\left( {x}_{k}\right) = {\int }_{a}^{b}w\left( x\right) \left( {{L}_{n}f}\right) \... | Yes |
Lemma 9.15 There exists a unique sequence \( \left( {q}_{n}\right) \) of polynomials of the form \( {q}_{0} = 1 \) and\n\n\[ \n{q}_{n}\left( x\right) = {x}^{n} + {r}_{n - 1}\left( x\right) ,\;n = 1,2,\ldots ,\n\]\n\nwith \( {r}_{n - 1} \in {P}_{n - 1} \) satisfying the orthogonality relation\n\n\[ \n{\int }_{a}^{b}w\le... | Proof. This follows by the Gram-Schmidt orthogonalization procedure from Theorem 3.18 applied to the linearly independent functions \( {u}_{n}\left( x\right) \mathrel{\text{:=}} {x}^{n} \) for \( n = 0,1,\ldots \) and the scalar product\n\n\[ \n\left( {f, g}\right) \mathrel{\text{:=}} {\int }_{a}^{b}w\left( x\right) f\... | Yes |
Lemma 9.16 Each of the orthogonal polynomials \( {q}_{n} \) from Lemma 9.15 has \( n \) simple zeros in \( \left( {a, b}\right) \) . | Proof. For \( m = 0 \), from (9.18) we have that\n\n\[ \n{\int }_{a}^{b}w\left( x\right) {q}_{n}\left( x\right) {dx} = 0 \n\]\n\nfor \( n > 0 \) . Hence, since \( w \) is positive on \( \left( {a, b}\right) \), the polynomial \( {q}_{n} \) must have at least one zero in \( \left( {a, b}\right) \) where the sign of \( {... | Yes |
Theorem 9.17 For each \( n = 0,1,\ldots \) there exists a unique Gaussian quadrature formula of order \( n \) . Its quadrature points are given by the zeros of the orthogonal polynomial \( {q}_{n + 1} \) of degree \( n + 1 \) . | Proof. This is a consequence of Lemmas 9.13-9.16. | Yes |
Theorem 9.18 The weights of the Gaussian quadrature formulae are all positive. | Proof. Define\n\n\[ \n{f}_{k}\left( x\right) \mathrel{\text{:=}} {\left\lbrack \frac{{q}_{n + 1}\left( x\right) }{x - {x}_{k}}\right\rbrack }^{2},\;k = 0,\ldots, n. \]\n\nThen\n\n\[ \n{a}_{k}{\left\lbrack {q}_{n + 1}^{\prime }\left( {x}_{k}\right) \right\rbrack }^{2} = \mathop{\sum }\limits_{{j = 0}}^{n}{a}_{j}{f}_{k}\... | Yes |
Corollary 9.19 The sequence of Gaussian quadrature formulae is convergent. | Proof. For each polynomial \( p \) we have\n\n\[ \n{Q}_{n}\left( p\right) = {\int }_{a}^{b}w\left( x\right) p\left( x\right) {dx} \]\n\nprovided that \( {2n} + 1 \) is greater than or equal to the degree of \( p \) . From their proofs it is obvious that Theorem 9.10 and its Corollary 9.11 remain valid for the integral ... | Yes |
Theorem 9.20 Let \( f \in {C}^{{2n} + 2}\left\lbrack {a, b}\right\rbrack \) . Then the error for the Gaussian quadrature formula of order \( n \) is given by\n\n\[{\int }_{a}^{b}w\left( x\right) f\left( x\right) {dx} - \mathop{\sum }\limits_{{k = 0}}^{n}{a}_{k}f\left( {x}_{k}\right) = \frac{{f}^{\left( 2n + 2\right) }\... | Proof. Recall the Hermite interpolation polynomial \( {H}_{n}f \in {P}_{{2n} + 1} \) for \( f \) from Theorem 8.18. Since \( \left( {{H}_{n}f}\right) \left( {x}_{k}\right) = f\left( {x}_{k}\right), k = 0,\ldots, n \), for the error\n\n\[{E}_{n}\left( f\right) \mathrel{\text{:=}} {\int }_{a}^{b}w\left( x\right) f\left( ... | Yes |
Example 9.21 We consider the Gaussian quadrature formulae for the weight function\n\n\[ \nw\left( x\right) = \frac{1}{\sqrt{1 - {x}^{2}}},\;x \in \left\lbrack {-1,1}\right\rbrack .\n\] | The Chebyshev polynomial \( {T}_{n} \) of degree \( n \) is defined by\n\n\[ \n{T}_{n}\left( x\right) \mathrel{\text{:=}} \cos \left( {n\arccos x}\right) ,\; - 1 \leq x \leq 1.\n\]\n\nObviously \( {T}_{0}\left( x\right) = 1 \) and \( {T}_{1}\left( x\right) = x \) . From the addition theorem for the cosine function, \( ... | Yes |
The Legendre polynomial \( {L}_{n} \) of degree \( n \) is defined by \[ {L}_{n}\left( x\right) \mathrel{\text{:=}} \frac{1}{{2}^{n}n!}\frac{{d}^{n}}{d{x}^{n}}{\left( {x}^{2} - 1\right) }^{n}. \] | Obviously, \( {L}_{n} \in {P}_{n} \) . If \( m < n \), by repeated partial integration we see that \[ {\int }_{-1}^{1}{x}^{m}\frac{{d}^{n}}{d{x}^{n}}{\left( {x}^{2} - 1\right) }^{n}{dx} = 0 \] since \( {\left( {x}^{2} - 1\right) }^{n} \) has zeros of order \( n \) at the endpoints -1 and 1 . Therefore, \[ {\int }_{-1}^... | Yes |
Lemma 9.24 The Bernoulli polynomials have the symmetry property\n\n\\[ \n{B}_{n}\\left( x\\right) = {\\left( -1\\right) }^{n}{B}_{n}\\left( {1 - x}\\right) ,\\;x \\in \\mathbb{R},\\;n = 0,1,\\ldots \n\\]\n\n(9.24) | Proof. Obviously (9.24) holds for \\( n = 0 \\) . Assume that (9.24) has been proven for some \\( n \\geq 0 \\) . Then, integrating (9.24), we obtain\n\n\\[ \n{B}_{n + 1}\\left( x\\right) = {\\left( -1\\right) }^{n + 1}{B}_{n + 1}\\left( {1 - x}\\right) + {\\beta }_{n + 1} \n\\]\n\nfor some constant \\( {\\beta }_{n + ... | No |
Lemma 9.25 The Bernoulli polynomials \( {B}_{{2m} + 1}, m = 1,2,\ldots \), of odd degree have exactly three zeros in \( \left\lbrack {0,1}\right\rbrack \), and these zeros are at the points \( 0,1/2 \), and 1 . The Bernoulli polynomials \( {B}_{2m}, m = 0,1,\ldots \), of even degree satisfy \( {B}_{2m}\left( 0\right) \... | Proof. From (9.23) and (9.24) we conclude that \( {B}_{{2m} + 1} \) vanishes at the points \( 0,1/2 \), and 1 . We prove by induction that these are the only zeros of \( {B}_{{2m} + 1} \) in \( \left\lbrack {0,1}\right\rbrack \) . This is true for \( m = 1 \), since \( {B}_{3} \) is a polynomial of degree three. Assume... | Yes |
Theorem 9.26 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be \( m \) times continuously differentiable for \( m \geq 2 \) . Then we have the Euler-Maclaurin expansion\n\n\[{\int }_{a}^{b}f\left( x\right) {dx} = {T}_{h}\left( f\right) - \mathop{\sum }\limits_{{j = 1}}^{\left\lbrack \frac{m}{2}\r... | Proof. Let \( g \in {C}^{m}\left\lbrack {0,1}\right\rbrack \) . Then, by \( m - 1 \) partial integrations and using (9.23) we find that\n\n\[{\int }_{0}^{1}{B}_{1}\left( z\right) {g}^{\prime }\left( z\right) {dz} = \mathop{\sum }\limits_{{j = 2}}^{m}{\left( -1\right) }^{j}{B}_{j}\left( 0\right) \left\lbrack {{g}^{\left... | Yes |
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