Calculators, Power Series and Chebyshev Polynomials - Versionsgeschichte

Graeme Cohen: /* Approximation by Chebyshev Polynomials */

2010-07-04T03:00:42Z

‎Approximation by Chebyshev Polynomials

Graeme Cohen am 4. Juli 2010 um 02:52 Uhr

2010-07-04T02:52:47Z

Graeme Cohen: /* Conclusion: The Point of the Story */

2010-07-04T02:51:59Z

‎Conclusion: The Point of the Story

Graeme Cohen: /* Conclusion --- The point of the story */

2010-07-04T02:48:03Z

‎Conclusion --- The point of the story

Graeme Cohen: /* The power series for the sine function */

2010-07-04T02:47:04Z

‎The power series for the sine function

Graeme Cohen: /* Using Chebyshev polynomials */

2010-07-04T02:46:21Z

‎Using Chebyshev polynomials

Änderungen zeigen

Graeme Cohen: /* The power series for the sine function */

2010-07-04T02:42:42Z

‎The power series for the sine function

Graeme Cohen: /* The power series for the sine function */

2010-07-04T02:40:47Z

‎The power series for the sine function

Graeme Cohen: /* The power series for the sine function */

2010-07-04T02:40:11Z

‎The power series for the sine function

Graeme Cohen: /* The power series for the sine function */

2010-07-04T02:39:36Z

‎The power series for the sine function

@@ Zeile 201: / Zeile 201: @@
 are closer to values of <math>\,\sin\,x</math>, for <math>|x|\le2</math>, than are values of the cubic function
 <math>x-\frac16x^3</math>, which is obtained from the power series representation of <math>\ \sin\,x</math>.
 ==Conclusion: The Point of the Story==

← Nächstältere Version		Version vom 4. Juli 2010, 02:52 Uhr
Zeile 1:		Zeile 1:
	Originating author: Graeme Cohen		Originating author: Graeme Cohen
−
−	~~(THIS ARTICLE IS CURRENTLY BEING REVISED.)~~
−
−	~~Originating author: Graeme Cohen~~

	Of all the familiar functions, such as trigonometric, exponential and logarithmic functions, surely the simplest to evaluate are		Of all the familiar functions, such as trigonometric, exponential and logarithmic functions, surely the simplest to evaluate are

@@ Zeile 209: / Zeile 209: @@
 ==Conclusion: The Point of the Story==
-The effectiveness of Chebyshev polynomials for our purpose here arises in part from a property of the cosine function. It implies
+The effectiveness of Chebyshev polynomials for our purpose here arises in part from a property of the cosine function. It implies that for all integers <math>k\ge0\,</math> and for <math>|x|\le1\,</math>, we have <math>|T_k(x)|\le 1</math>. Pafnuty Lvovich Chebyshev, whose surname is often given alternatively as Tchebycheff, was Russian; he introduced these polynomials in a paper in 1854.  They are denoted by <math>T_k\,</math> because of ''Tchebycheff''.
 that for all integers <math>k\ge0\,</math> and for <math>|x|\le1\,</math>, we have <math>|T_k(x)|\le 1</math>. Pafnuty Lvovich
 Chebyshev, whose surname is often given alternatively as Tchebycheff, was Russian; he introduced these polynomials in a paper in
 .  They are denoted by <math>T_k\,</math> because of ''Tchebycheff''.
-The procedure outlined above is known as ''economization'' of power series and is studied in the branch of mathematics known as
+The procedure outlined above is known as ''economization'' of power series and is studied in the branch of mathematics known as [http://en.wikipedia.org/wiki/numerical_analysis numerical analysis].
-[http://en.wikipedia.org/wiki/numerical_analysis numerical analysis]. Economization is not always necessary for evaluating the
+Economization is not always necessary for evaluating the sine function. Because <math>\sin\,x</math> is approximately <math>x\,</math> when <math>|x|\,</math> is small, this is often good enough! See Chuck Allison, [http://uvu.freshsources.com/page1/page8/page16/files/sine.pdf Machine Computation of the Sine Function], for more on this. We mentioned at the beginning that the Cordic algorithm is often better still, but for evaluating other functions on a calculator, particularly <math>\tan^{-1}\,x</math>, which has the ''slowly converging'' power series expansion found in (3), economization is considered to be essential.
 sine function. Because <math>\sin\,x</math> is approximately <math>x\,</math> when <math>|x|\,</math> is small, this is often
 good enough! See Chuck Allison, [http://uvu.freshsources.com/page1/page8/page16/files/sine.pdf Machine Computation of the Sine
 Function], for more on this. We mentioned at the beginning that the Cordic algorithm is often better still, but for evaluating
 other functions on a calculator, particularly <math>\tan^{-1}\,x</math>, which has the ''slowly converging'' power series
 expansion found in (3), economization is considered to be essential.
 Perhaps the point of the story is that the obvious can very often be improved upon.

@@ Zeile 207: / Zeile 207: @@
 near <math>x=\pm2</math>.
-==Conclusion --- The point of the story==
+==Conclusion: The Point of the Story==
-The effectiveness of Chebyshev polynomials for such a purpose arises from properties of the cosine function. They tell us in
+The effectiveness of Chebyshev polynomials for our purpose here arises in part from a property of the cosine function. It implies
-particular that for all integers <math>k\ge0</math> and for <math>|x|\le1</math>, we have <math>|T_k(x)|\le 1</math>. Pafnuty
+that for all integers <math>k\ge0\,</math> and for <math>|x|\le1\,</math>, we have <math>|T_k(x)|\le 1</math>. Pafnuty Lvovich
-Lvovich Chebyshev was Russian; he introduced these polynomials in a paper in 1854. His surname is often given alternatively as
+Chebyshev, whose surname is often given alternatively as Tchebycheff, was Russian; he introduced these polynomials in a paper in
-Tchebycheff, and this is why the polynomials are denoted with <math>T\,</math>'s.
+.  They are denoted by <math>T_k\,</math> because of ''Tchebycheff''.
-The procedure above is called ''economization of power series''. As mentioned in the Introduction, there are other methods of
+The procedure outlined above is known as ''economization'' of power series and is studied in the branch of mathematics known as
-speeding up the calculation of values of the trigonometric functions, such as the Cordic algorithm for the sine function, but
+[http://en.wikipedia.org/wiki/numerical_analysis numerical analysis]. Economization is not always necessary for evaluating the
-economization is considered to be essential for ''slowly converging'' series such as that for <math>\tan^{-1}x\,</math>
+sine function. Because <math>\sin\,x</math> is approximately <math>x\,</math> when <math>|x|\,</math> is small, this is often
-established in (3), above. Perhaps the point of this story is that, very often, the obvious approach is not necessarily best.
+good enough! See Chuck Allison, [http://uvu.freshsources.com/page1/page8/page16/files/sine.pdf Machine Computation of the Sine

@@ Zeile 105: / Zeile 105: @@
 values of <math>x\,</math>. Then partial sums of the series, obtained by stopping after some finite number of terms, should give
 polynomial functions that can be used to find approximate values of the sine function, such as you find in tables of
-trigonometric functions or as output on a calculator.7x^2+\cdots, </math>
+trigonometric functions or as output on a calculator.
-−
-so <math> a_5=1/(5\cdot4\cdot3\cdot2)=1/5! </math>.
 ==Approximation by Chebyshev Polynomials==

@@ Zeile 83: / Zeile 83: @@
 </math>
-<math> \sin\,x = 4\cdot3\cdot2a_4+5\cdot4\cdot3\cdot2a_5x+6\cdot5\cdot4\cdot3a_6x^2+\cdots, \qquad\mathrm{so}\ a_4=0, </math>
+<math>
 <math>
-\cos\,x = 5\cdot4\cdot3\cdot2a_5+6\cdot5\cdot4\cdot3\cdot2a_6x+7\cdot6\cdot5\cdot4\cdot3a_7x^2+\cdots, \qquad\mathrm{so}\a_5=\frac1{5\cdot4\cdot3\cdot2}=\frac1{5!}.
+\cos\,x = 5\cdot4\cdot3\cdot2a_5+6\cdot5\cdot4\cdot3\cdot2a_6x+7\cdot6\cdot5\cdot4\cdot3a_7x^2+\cdots, \qquad\mathrm{so}\ a_5=\frac1{5\cdot4\cdot3\cdot2}=\frac1{5!}.
 </math>
 In this way, we can find a formula for all the coefficients <math>a_0,\,a_1,\,a_2,\,\ldots</math>, namely,

@@ Zeile 82: / Zeile 82: @@
 -\cos\,x = 3\cdot2a_3+4\cdot3\cdot2a_4x+5\cdot4\cdot3a_5x^2+6\cdot5\cdot4a_6x^3+\cdots, \qquad\mathrm{so}\ a_3=\frac{-1}{3\cdot2}=\frac{-1}{3!},
 </math>
 <math> \sin\,x = 4\cdot3\cdot2a_4+5\cdot4\cdot3\cdot2a_5x+6\cdot5\cdot4\cdot3a_6x^2+\cdots, \qquad\mathrm{so}\ a_4=0, </math>
-<math> \cos\,x = 5\cdot4\cdot3\cdot2a_5+6\cdot5\cdot4\cdot3\cdot2a_6x+7\cdot6\cdot5\cdot4\cdot3a_7x^2+\cdots, \qquad\mathrm{so}\
+<math>
-a_5=\frac1{5\cdot4\cdot3\cdot2}=\frac1{5!}. </math>
+\cos\,x = 5\cdot4\cdot3\cdot2a_5+6\cdot5\cdot4\cdot3\cdot2a_6x+7\cdot6\cdot5\cdot4\cdot3a_7x^2+\cdots, \qquad\mathrm{so}\
 In this way, we can find a formula for all the coefficients <math>a_0,\,a_1,\,a_2,\,\ldots</math>, namely,

@@ Zeile 79: / Zeile 79: @@
 <math> -\sin\,x = 2a_2+3\cdot2a_3x+4\cdot3a_4x^2+5\cdot4a_5x^3+\cdots, \qquad\mathrm{so}\ a_2=0, </math>
-<math> -\cos\,x = 3\cdot2a_3+4\cdot3\cdot2a_4x+5\cdot4\cdot3a_5x^2+6\cdot5\cdot4a_6x^3+\cdots, \qquad\mathrm{so}\
+<math>
-a_3=\frac{-1}{3\cdot2}=\frac{-1}{3!}, </math>
+-\cos\,x = 3\cdot2a_3+4\cdot3\cdot2a_4x+5\cdot4\cdot3a_5x^2+6\cdot5\cdot4a_6x^3+\cdots, \qquad\mathrm{so}\ a_3=\frac{-1}{3\cdot2}=\frac{-1}{3!},
 <math> \sin\,x = 4\cdot3\cdot2a_4+5\cdot4\cdot3\cdot2a_5x+6\cdot5\cdot4\cdot3a_6x^2+\cdots, \qquad\mathrm{so}\ a_4=0, </math>