光的世界︰派生科學計算三

2016-07-26 懸鉤子

從『聯立方程式』 Simultaneous equations 、『線性方程組』 System of linear equations 、『矩陣』 Matrix 、『行列式』 Determinant …… 觀之，『高斯消去法』 Gaussian Elimination 歷史

History

The method of Gaussian elimination appears in the Chinese mathematical text Chapter Eight Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations. The first reference to the book by this title is dated to 179 CE, but parts of it were written as early as approximately 150 BCE.^[1]^[2] It was commented on by Liu Hui in the 3rd century.

The method in Europe stems from the notes of Isaac Newton.^[3]^[4] In 1670, he wrote that all the algebra books known to him lacked a lesson for solving simultaneous equations, which Newton then supplied. Cambridge University eventually published the notes as Arithmetica Universalis in 1707 long after Newton left academic life. The notes were widely imitated, which made (what is now called) Gaussian elimination a standard lesson in algebra textbooks by the end of the 18th century. Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems.^[5] The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject.^[6]

Some authors use the term Gaussian elimination to refer only to the procedure until the matrix is in echelon form, and use the term Gauss-Jordan elimination to refer to the procedure which ends in reduced echelon form. The name is used because it is a variation of Gaussian elimination as described by Wilhelm Jordan in 1888. However, the method also appears in an article by Clasen published in the same year. Jordan and Clasen probably discovered Gauss–Jordan elimination independently.^[7]

久遠矣。或許很多人不曉此法漢代《九章算術》中已有，甚或不知作注的『Liu Hui』劉徽是何許人也！！而後歷經數百代凝鑄成

線性代數

線性代數是關於向量空間和線性映射的一個數學分支。它包括對線、面和子空間的研究，同時也涉及到所有的向量空間的一般性質。

坐標滿足滿足線性方程的點集形成 n 維空間中的一個超平面。n 個超平面相交於一點的條件是線性代數研究的一個重要焦點。此項研究源於包含多個未知數的線性方程組。這樣的方程組可以很自然地表示為矩陣和向量的形式。^[1]^[2]

線性代數既是純數學也是應用數學的核心。例如，放寬向量空間的公理就產生了抽象代數，也就出現了若干推廣。泛函分析研究無窮維情形的向量空間理論。線性代數與微積分結合，使得微分方程線性系統的求解更加便利。線性代數的理論已被泛化為算子理論。

線性代數的方法還用在解析幾何、工程、物理、自然科學、計算機科學、計算機動畫和社會科學（尤其是經濟學）中。由於線性代數是一套完善的理論，非線性數學模型通常可以被近似為線性模型。

三維歐氏空間R³是一個向量空間，而通過原點的線及平面是R³的向量子空間

歷史

線性代數的研究最初出現於對行列式的研究上。行列式當時被用來求解線性方程組。萊布尼茨在1693年使用了行列式。隨後，加布里爾·克拉默在1750年推導出求解線性方程組的克萊姆法則。然後，高斯利用高斯消元法發展出求解線性系統的理論。這也被列為大地測量學的一項進展。^[3]^[4]

現代線性代數的歷史可以上溯到19世紀中期的英國。1843年，哈密頓發現了四元數。1844年，赫爾曼·格拉斯曼發表了他的著作《線性外代數》（Die lineare Ausdehnungslehre），包括了今日線性代數的一些主題。1848年，詹姆斯·西爾維斯特引入了矩陣（matrix），該詞是「子宮」的拉丁語。阿瑟·凱萊在研究線性變換時引入了矩陣乘法和轉置的概念。很重要的是，凱萊使用了一個字母來代表一個矩陣，因此將矩陣當做了聚合對象。他也意識到矩陣和行列式之間的聯繫。^[3]

不過除了這些早期的文獻以外．線性代數主要是在二十世紀發展的。在抽象代數的環論開發之前，矩陣只有模糊不清的定義。隨著狹義相對論的到來，很多開拓者發現了線性代數的微妙。進一步的，解偏微分方程的克萊姆法則的例行應用導致了大學的標準教育中包括了線性代數。例如，E.T. Copson寫到：

“

當我在1922年到愛丁堡做年輕的講師的時候，我驚奇的發現了不同於牛津的課程。這裡包括了我根本就不知道的主題如勒貝格積分、矩陣論、數值分析、黎曼幾何…

”

——E.T. Copson，《偏微分方程》前言, 1973

1882年，Hüseyin Tevfik Pasha寫了一本書，名為《線性代數》。^[5]^[6]第一次現代化精確定義向量空間是在1888年，由朱塞佩·皮亞諾提出。在1888年，弗蘭西斯·高爾頓還發起了相關係數的應用。經常有多於一個隨機變量出現並且它們可以互相關。在多變元隨機變量的統計分析中，相關矩陣是自然的工具。所以這種隨機向量的統計研究幫助了矩陣用途的開發。到1900年，一種有限維向量空間的線性變換理論被提出。在20世紀上半葉，許多前幾世紀的想法和方法被總結成抽象代數，線性代數第一次有了它的現代形式。矩陣在量子力學、狹義相對論和統計學上的應用幫助線性代數的主題超越了純數學的範疇。計算機的發展導致更多地研究致力於有關高斯消元法和矩陣分解的有效算法上。線性代數成為了數字模擬和模型的基本工具。^[3]

其論述能不厚重乎？其內容能不範圍廣闊耶！系統符號混雜、概念層出不窮、各種分支頻起旁徵博引，豈不目不暇給望洋興嘆呀！！

此處藉著『高斯消去法』例子為範︰

Gaussian elimination

Example of the algorithm

Suppose the goal is to find and describe the set of solutions to the following system of linear equations:

{\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&8&\qquad (L_{1})\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&-11&\qquad (L_{2})\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&-3&\qquad (L_{3})\end{alignedat}}

The table below is the row reduction process applied simultaneously to the system of equations, and its associated augmented matrix. In practice, one does not usually deal with the systems in terms of equations but instead makes use of the augmented matrix, which is more suitable for computer manipulations. The row reduction procedure may be summarized as follows: eliminate x from all equations below $L_{1}$ , and then eliminate y from all equations below $L_{2}$ . This will put the system into triangular form. Then, using back-substitution, each unknown can be solved for.

System of equations	Row operations	Augmented matrix
${\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&8&\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&-11&\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&-3&\end{alignedat}}$		$\left[{\begin{array}{ccc\|c}2&1&-1&8\\-3&-1&2&-11\\-2&1&2&-3\end{array}}\right]$
${\begin{alignedat}{7}2x&&\;+&&y&&\;-&&\;z&&\;=\;&&8&\\&&&&{\frac {1}{2}}y&&\;+&&\;{\frac {1}{2}}z&&\;=\;&&1&\\&&&&2y&&\;+&&\;z&&\;=\;&&5&\end{alignedat}}$	$L_{2}+{\frac {3}{2}}L_{1}\rightarrow L_{2}$ $L_{3}+L_{1}\rightarrow L_{3}$	$\left[{\begin{array}{ccc\|c}2&1&-1&8\\0&1/2&1/2&1\\0&2&1&5\end{array}}\right]$
${\begin{alignedat}{7}2x&&\;+&&y\;&&-&&\;z\;&&=\;&&8&\\&&&&{\frac {1}{2}}y\;&&+&&\;{\frac {1}{2}}z\;&&=\;&&1&\\&&&&&&&&\;-z\;&&\;=\;&&1&\end{alignedat}}$	$L_{3}+-4L_{2}\rightarrow L_{3}$	$\left[{\begin{array}{ccc\|c}2&1&-1&8\\0&1/2&1/2&1\\0&0&-1&1\end{array}}\right]$
The matrix is now in echelon form (also called triangular form)
${\begin{alignedat}{7}2x&&\;+&&y\;&&&&\;\;&&=\;&&7&\\&&&&{\frac {1}{2}}y\;&&&&\;\;&&=\;&&{\frac {3}{2}}&\\&&&&&&&&\;-z\;&&\;=\;&&1&\end{alignedat}}$	$L_{2}+{\frac {1}{2}}L_{3}\rightarrow L_{2}$ $L_{1}-L_{3}\rightarrow L_{1}$	$\left[{\begin{array}{ccc\|c}2&1&0&7\\0&1/2&0&3/2\\0&0&-1&1\end{array}}\right]$
${\begin{alignedat}{7}2x&&\;+&&y\;&&&&\;\;&&=\;&&7&\\&&&&y\;&&&&\;\;&&=\;&&3&\\&&&&&&&&\;z\;&&\;=\;&&-1&\end{alignedat}}$	$2L_{2}\rightarrow L_{2}$ $-L_{3}\rightarrow L_{3}$	$\left[{\begin{array}{ccc\|c}2&1&0&7\\0&1&0&3\\0&0&1&-1\end{array}}\right]$
${\begin{alignedat}{7}x&&\;&&\;&&&&\;\;&&=\;&&2&\\&&&&y\;&&&&\;\;&&=\;&&3&\\&&&&&&&&\;z\;&&\;=\;&&-1&\end{alignedat}}$	$L_{1}-L_{2}\rightarrow L_{1}$ ${\frac {1}{2}}L_{1}\rightarrow L_{1}$	$\left[{\begin{array}{ccc\|c}1&0&0&2\\0&1&0&3\\0&0&1&-1\end{array}}\right]$

The second column describes which row operations have just been performed. So for the first step, the x is eliminated from $L_{2}$ by adding ${\begin{matrix}{\frac {3}{2}}\end{matrix}}L_{1}$ to $L_{2}$ . Next x is eliminated from $L_{3}$ by adding $L_{1}$ to $L_{3}$ . These row operations are labelled in the table as

L_{2}+{\frac {3}{2}}L_{1}\rightarrow L_{2}

L_{3}+L_{1}\rightarrow L_{3}.

Once y is also eliminated from the third row, the result is a system of linear equations in triangular form, and so the first part of the algorithm is complete. From a computational point of view, it is faster to solve the variables in reverse order, a process known as back-substitution. One sees the solution is z = -1, y = 3, and x = 2. So there is a unique solution to the original system of equations.

Instead of stopping once the matrix is in echelon form, one could continue until the matrix is in reduced row echelon form, as it is done in the table. The process of row reducing until the matrix is reduced is sometimes referred to as Gauss-Jordan elimination, to distinguish it from stopping after reaching echelon form.

翻古出新，談談如何用『SymPy』之

‧ Matrices

‧ Matrices (linear algebra)

模組為『工具』，操作

初等矩陣

線性代數中，初等矩陣（又稱為基本矩陣^[1]）是一個與單位矩陣只有微小區別的矩陣。具體來說，一個n階單位矩陣E經過一次初等行變換或一次初等列變換所得矩陣稱為n階初等矩陣。^[2]

操作

初等矩陣分為3種類型，分別對應著3種不同的行/列變換。

兩行（列）互換：: $R_i \leftrightarrow R_j$

把某行（列）乘以一非零常數：: $kR_i \rightarrow R_i,\$ 其中 $k \neq 0$

把第i行（列）加上第j行（列）的k倍：: $R_i + kR_j \rightarrow R_i$

初等矩陣即是將上述3種初等變換應用於一單位矩陣的結果。以下只討論對某行的變換，列變換可以類推。

行互換

這一變換T_ij，將一單位矩陣的第i行的所有元素與第j行互換。

T_{i,j} = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 0 & & 1 & & \\ & & & \ddots & & & & \\ & & 1 & & 0 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix}\quad

性質

逆矩陣即自身： $T_{ij}^{-1} = T_{ij}$ 。
因為單位矩陣的行列式為1，故 $|T_{ij}|=-1$ 。與其他相同大小的方陣A亦有一下性質： $|T_{ij}A|=-|A|$ 。

把某行乘以一非零常數

這一變換T_i（m），將第i行的所有元素乘以一非零常數m。

T_i (m) = \begin{bmatrix} 1 & & & & & & \\ & \ddots & & & & & \\ & & 1 & & & & \\ & & & m & & & \\ & & & & 1 & & \\ & & & & & \ddots & \\ & & & & & & 1\end{bmatrix}\quad

性質

逆矩陣為 $T_{i}(m)^{-1} = T_{i}(\frac{1}{m})$ 。
此矩陣及其逆矩陣均為對角矩陣。
其行列式 $|T_{i}(m)|=m$ 。故對於一等大方陣A有 $|T_{i}(m)A|=m|A|$ 。

把第i行加上第j行的m倍

這一變換T_ij（m），將第i行加上第j行的m倍。

T_{i,j}(m) = \begin{bmatrix} 1 & & & & & & & \\ & \ddots & & & & & & \\ & & 1 & & & & & \\ & & & \ddots & & & & \\ & & m & & 1 & & \\ & & & & & & \ddots & \\ & & & & & & & 1\end{bmatrix}

性質

逆矩陣具有性質 $T_{ij}(m)^{-1}=T_{ij}(-m)$ 。
此矩陣及其逆矩陣均為三角矩陣。
$|T_{ij}(m)|=1$ 。故對於一等大方陣A有： $|T_{ij}(m)A| = |A|$ 。

親自『實證』消去法的精神，體驗用『工具』來『學習』之樂趣吧！！

省思這個『初等』當真可『模擬』紙筆運算嗎？？

pi@raspberrypi:~ $ python3
Python 3.4.2 (default, Oct 19 2014, 13:31:11) 
[GCC 4.9.1] on linux
Type "help", "copyright", "credits" or "license" for more information.

>>> from sympy import *
>>> init_printing()
>>> a, b, c, d, e, f, g, h, i, m = symbols('a, b, c, d, e, f, g, h, i, m')
>>> M = Matrix([[a, b, c], [d, e, f], [g, h, i]])
>>> M
⎡a  b  c⎤
⎢       ⎥
⎢d  e  f⎥
⎢       ⎥
⎣g  h  i⎦

>>> 二三列交換 = Matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])
>>> 二三列交換
⎡1  0  0⎤
⎢       ⎥
⎢0  0  1⎥
⎢       ⎥
⎣0  1  0⎦
>>> 二三列交換 * M
⎡a  b  c⎤
⎢       ⎥
⎢g  h  i⎥
⎢       ⎥
⎣d  e  f⎦

>>> 第二列乘m = Matrix([[1, 0, 0], [0, m, 0], [0, 0, 1]])
>>> 第二列乘m
⎡1  0  0⎤
⎢       ⎥
⎢0  m  0⎥
⎢       ⎥
⎣0  0  1⎦
>>> 第二列乘m * M
⎡ a    b    c ⎤
⎢             ⎥
⎢d⋅m  e⋅m  f⋅m⎥
⎢             ⎥
⎣ g    h    i ⎦

>>> 第三列加第二列乘m = Matrix([[1, 0, 0], [0, 1, 0], [0, m, 1]])
>>> 第三列加第二列乘m
⎡1  0  0⎤
⎢       ⎥
⎢0  1  0⎥
⎢       ⎥
⎣0  m  1⎦
>>> 第三列加第二列乘m * M
⎡   a        b        c   ⎤
⎢                         ⎥
⎢   d        e        f   ⎥
⎢                         ⎥
⎣d⋅m + g  e⋅m + h  f⋅m + i⎦
>>>

或可得『自學』之法哩！！？？

>>> AM = Matrix([[2, 1, -1, 8], [-3, -1, 2, -11], [-2, 1, 2, -3]])
>>> AM
⎡2   1   -1   8 ⎤
⎢               ⎥
⎢-3  -1  2   -11⎥
⎢               ⎥
⎣-2  1   2   -3 ⎦

>>> L2加二分之三L1 = Matrix([[1, 0, 0], [3/2, 1, 0], [0, 0, 1]])
>>> L2加二分之三L1
⎡ 1   0  0⎤
⎢         ⎥
⎢1.5  1  0⎥
⎢         ⎥
⎣ 0   0  1⎦

>>> L2加二分之三L1 * AM
⎡2    1   -1    8 ⎤
⎢                 ⎥
⎢0   0.5  0.5  1.0⎥
⎢                 ⎥
⎣-2   1    2   -3 ⎦

>>> L3加上L1 = Matrix([[1, 0, 0], [0, 1, 0], [1, 0, 1]]) 
>>> L3加上L1
⎡1  0  0⎤
⎢       ⎥
⎢0  1  0⎥
⎢       ⎥
⎣1  0  1⎦

>>> L3加上L1 * AM
⎡2   1   -1   8 ⎤
⎢               ⎥
⎢-3  -1  2   -11⎥
⎢               ⎥
⎣0   2   1    5 ⎦

>>> L3加上L1 * L2加二分之三L1 * AM
⎡2   1   -1    8 ⎤
⎢                ⎥
⎢0  0.5  0.5  1.0⎥
⎢                ⎥
⎣0   2    1    5 ⎦

>>> L2加二分之三L1 * L3加上L1 * AM
⎡2   1   -1    8 ⎤
⎢                ⎥
⎢0  0.5  0.5  1.0⎥
⎢                ⎥
⎣0   2    1    5 ⎦
# ……

樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰派生科學計算二‧下

2016-07-25 懸鉤子

如果我們藉著維基百科『三次函數』詞條，來個拉格朗日求解之旅，自會發現用『SymPy』作符號運算的好處︰

Cubic function

In algebra, a cubic function is a function of the form

f(x)=ax^{3}+bx^{2}+cx+d,\,

where $a$ is nonzero.

Setting $f (x) = 0$ produces a cubic equation of the form:

ax^{3}+bx^{2}+cx+d=0.\,

The solutions of this equation are called roots of the polynomial $f (x)$ . If all of the coefficients $a$ , $b$ , $c$ , and $d$ of the cubic equation are real numbers then there will be at least one real root (this is true for all odd degree polynomials) and if the coefficients are complex numbers then there will be at least one complex root (this is true for all non-constant polynomials). All of the roots of the cubic equation can be found algebraically. (This is also true of a quadratic or quartic (fourth degree) equation, but no higher-degree equation, by the Abel–Ruffini theorem). The roots can also be found trigonometrically. Alternatively, numerical approximations of the roots can be found using root-finding algorithms like Newton’s method.

The coefficients do not need to be complex numbers. Much of what is covered below is valid for coefficients of any field with characteristic $0$ or greater than $3$ . The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are non-rational (and even non-real) complex numbers.

Lagrange’s method

In his paper Réflexions sur la résolution algébrique des équations (“Thoughts on the algebraic solving of equations”),^[35] Joseph Louis Lagrange introduced a new method to solve equations of low degree.

This method works well for cubic and quartic equations, but Lagrange did not succeed in applying it to a quintic equation, because it requires solving a resolvent polynomial of degree at least six.^[36]^[37]^[38] This is explained by the Abel–Ruffini theorem, which proves that such polynomials cannot be solved by radicals. Nevertheless, the modern methods for solving solvable quintic equations are mainly based on Lagrange’s method.^[38]

In the case of cubic equations, Lagrange’s method gives the same solution as Cardano’s. By drawing attention to a geometrical problem that involves two cubes of different size Cardano explains in his book Ars Magna how he arrived at the idea of considering the unknown of the cubic equation as a sum of two other quantities. Lagrange’s method may also be applied directly to the general cubic equation (1), $ax 3 + bx 2 + cx + d = 0$ , without using the reduction to the depressed cubic equation (2), $t 3 + pt + q = 0$ . Nevertheless, the computation is much easier with this reduced equation.

Suppose that $x 0$ , $x 1$ and $x 2$ are the roots of equation (1) or (2), and define $ξ = - 1 / 2 + 1 / 2 \sqrt3 i$ (a complex cube root of $1$ , i.e. a primitive third root of unity) which satisfies the relation $ξ 2 + ξ + 1 = 0$ . We now set

s_{0}=x_{0}+x_{1}+x_{2},\,

s_{1}=x_{0}+\zeta x_{1}+\zeta ^{2}x_{2},\,

s_{2}=x_{0}+\zeta ^{2}x_{1}+\zeta x_{2}.\,

This is the discrete Fourier transform of the roots: observe that while the coefficients of the polynomial are symmetric in the roots, in this formula an order has been chosen on the roots, so these are not symmetric in the roots. The roots may then be recovered from the three $s i$ by inverting the above linear transformation via the inverse discrete Fourier transform, giving

x_{0}={\tfrac {1}{3}}(s_{0}+s_{1}+s_{2}),\,

x_{1}={\tfrac {1}{3}}(s_{0}+\zeta ^{2}s_{1}+\zeta s_{2}),\,

x_{2}={\tfrac {1}{3}}(s_{0}+\zeta s_{1}+\zeta ^{2}s_{2}).\,

The polynomial $s 0$ is equal, by Vieta’s formulas, to $- b / a$ in case of equation (1) and to $0$ in case of equation (2), so we only need to seek values for the other two.

即使已知 $s_0, s_1, s_2$ 與 $x_0, x_1, x_2$ 的線性關係，也知 ${\xi}^2 + \xi +1 =0$ ，將 $x_0, x_1, x_2$ 以 $s_0, s_1, s_2$ 來表示亦需要許多計算！因此人們往往不願從頭來過，只想知道結果就好！！不過這樣就少了許多探索的樂趣了。難道使用『SymPy』能有什麼不同的嗎？終究也只是得到『答案』而已乎？？

pi@raspberrypi:~ {\xi}^2 + \xi +1 =0 python3
Python 3.4.2 (default, Oct 19 2014, 13:31:11) 
[GCC 4.9.1] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> from sympy import *
>>> x0, x1, x2, ζ = symbols('x0, x1, x2, ζ')
>>> s0 = x0 + x1 + x2
>>> s1 = x0 + ζ*x1 + ζ**2 * x2
>>> s2 = x0 + ζ**2 * x1 + ζ*x2

>>> (s1**2).expand()
x0**2 + 2*x0*x1*ζ + 2*x0*x2*ζ**2 + x1**2*ζ**2 + 2*x1*x2*ζ**3 + x2**2*ζ**4

>>> collect(x0**2 + 2*x0*x1*ζ + 2*x0*x2*ζ**2 + x1**2*ζ**2 + 2*x1*x2 + x2**2*ζ, ζ)

x0**2 + 2*x1*x2 + ζ**2*(2*x0*x2 + x1**2) + ζ*(2*x0*x1 + x2**2)

>>> (s1**3).expand()
x0**3 + 3*x0**2*x1*ζ + 3*x0**2*x2*ζ**2 + 3*x0*x1**2*ζ**2 + 6*x0*x1*x2*ζ**3 + 3*x0*x2**2*ζ**4 + x1**3*ζ**3 + 3*x1**2*x2*ζ**4 + 3*x1*x2**2*ζ**5 + x2**3*ζ**6

>>> collect(x0**3 + 3*x0**2*x1*ζ + 3*x0**2*x2*ζ**2 + 3*x0*x1**2*ζ**2 + 6*x0*x1*x2 + 3*x0*x2**2*ζ + x1**3 + 3*x1**2*x2*ζ + 3*x1*x2**2*ζ**2 + x2**3, ζ)

x0**3 + 6*x0*x1*x2 + x1**3 + x2**3 + ζ**2*(3*x0**2*x2 + 3*x0*x1**2 + 3*x1*x2**2) + ζ*(3*x0**2*x1 + 3*x0*x2**2 + 3*x1**2*x2)

>>> (s1*s2).expand()
x0**2 + x0*x1*ζ**2 + x0*x1*ζ + x0*x2*ζ**2 + x0*x2*ζ + x1**2*ζ**3 + x1*x2*ζ**4 + x1*x2*ζ**2 + x2**2*ζ**3

>>> collect(x0**2 + x0*x1*ζ**2 + x0*x1*ζ + x0*x2*ζ**2 + x0*x2*ζ + x1**2 + x1*x2*ζ + x1*x2*ζ**2 + x2**2, ζ)

x0**2 + x1**2 + x2**2 + ζ**2*(x0*x1 + x0*x2 + x1*x2) + ζ*(x0*x1 + x0*x2 + x1*x2)
>>>

如是在清楚通熟『四次函數』

Quartic function

In algebra, a quartic function is a function of the form

f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,

where a is nonzero, which is defined by a polynomial of degree four, called quartic polynomial.

Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form

f(x)=ax^{4}+cx^{2}+e.

A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form

ax^{4}+bx^{3}+cx^{2}+dx+e=0,

where a ≠ 0.

The derivative of a quartic function is a cubic function.

Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if a is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum.

The degree four (quartic case) is the highest degree such that every polynomial equation can be solved by radicals.

Solving by Lagrange resolvent

The symmetric group $S 4$ on four elements has the Klein four-group as a normal subgroup. This suggests using a resolvent cubic whose roots may be variously described as a discrete Fourier transform or a Hadamard matrix transform of the roots; see Lagrange resolvents for the general method. Denote by $x i$ , for $i$ from $0$ to $3$ , the four roots of $x 4 + bx 3 + cx 2 + dx + e$ . If we set

{\begin{aligned}s_{0}&={\tfrac {1}{2}}(x_{0}+x_{1}+x_{2}+x_{3}),\\s_{1}&={\tfrac {1}{2}}(x_{0}-x_{1}+x_{2}-x_{3}),\\s_{2}&={\tfrac {1}{2}}(x_{0}+x_{1}-x_{2}-x_{3}),\\s_{3}&={\tfrac {1}{2}}(x_{0}-x_{1}-x_{2}+x_{3}),\end{aligned}}

then since the transformation is an involution we may express the roots in terms of the four $s i$ in exactly the same way. Since we know the value $s 0 = - b / 2$ , we only need the values for $s 1$ , $s 2$ and $s 3$ . These are the roots of the polynomial

(s^{2}-{s_{1}}^{2})(s^{2}-{s_{2}}^{2})(s^{2}-{s_{3}}^{2}).

Substituting the $s i$ by their values in term of the $x i$ , this polynomial may be expanded in a polynomial in $s$ whose coefficients are symmetric polynomials in the $x i$ . By the fundamental theorem of symmetric polynomials, these coefficients may be expressed as polynomials in the coefficients of the monic quartic. If, for simplification, we suppose that the quartic is depressed, that is $b = 0$ , this results in the polynomial

{s}^{6}+2c\,{s}^{4}+({c}^{2}-4e)\,{s}^{2}-{d}^{2}

(3)

This polynomial is of degree six, but only of degree three in $s 2$ , and so the corresponding equation is solvable by the method described in the article about cubic function. By substituting the roots in the expression of the $x i$ in terms of the $s i$ , we obtain expression for the roots. In fact we obtain, apparently, several expressions, depending on the numbering of the roots of the cubic polynomial and of the signs given to their square roots. All these different expressions may be deduced from one of them by simply changing the numbering of the $x i$ .

These expressions are unnecessarily complicated, involving the cubic roots of unity, which can be avoided as follows. If $s$ is any non-zero root of (3), and if we set

{\begin{aligned}F_{1}(x)&={x}^{2}+sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}-{\frac {d}{2s}}\\F_{2}(x)&={x}^{2}-sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}+{\frac {d}{2s}}\end{aligned}}

then

F_{1}(x)\times F_{2}(x)=x^{4}+cx^{2}+dx+e.

We therefore can solve the quartic by solving for $s$ and then solving for the roots of the two factors using the quadratic formula.

Note that this gives exactly the same formula for the roots as the one provided by Descartes’ method.

之後，或能欣賞阿貝爾和伽羅瓦優美的

可解群

在數學的歷史中，群論原本起源於對五次方程及更高次方程無一般的公式解之證明的找尋，最終隨著伽羅瓦理論的提出而確立。可解群的概念產生於描述其根可以只用根式(平方根、立方根等等及其和與積)表示的多項式所對應的自同構群所擁有的性質。

一個群被稱為可解的，若它擁有一個其商群皆為阿貝爾群的正規列。或者等價地說，若其降正規列

G\triangleright G^{{(1)}}\triangleright G^{{(2)}}\triangleright \cdots ,

之中，每一個子群都會是前一個的導群，且最後一個為G的當然子群{1}。上述兩個定義是等價的，對一個群H及H的正規子群N，其商群H/N為可交換的若且唯若N包含著H⁽¹⁾。

對於有限群，有一個等價的定義為：一可解群為一有著其商群皆為質數階的循環群之合成列的群。此一定義會等價是因為每一個簡單阿貝爾群都是有質數階的循環群。若爾當-赫爾德定理表示若一個合成列有此性質，則其循環群即會對應到某個體上的n個根。但此一定義的等價性並不必然於無限群中亦會成立：例如，因為每一個在加法下的整數群Z的非當然子群皆同構於Z本身，它不會有合成列，但是其有著唯一同構於Z的商群之正規列{0,Z}，證明了其確實是可解的。

和喬治·波里亞的格言「若有一個你無法算出的問題，則會有的你可以算出的較簡單的問題」相一致的，可解群通常在簡化有關一複雜的群的推測至一系列有著簡單結構－阿貝爾群的群的推測有著很有用的功用。

理論吧！！！

樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰派生科學計算二‧中

2016-07-24 懸鉤子

照說一元二次方程式 $x^2 + px + q = 0$ 用『配方法』來求解，應是自然順理成章之事。若從『發現』之『邏輯』來考察這一『思路』莫非源於 ${(x - \Box)}^2 = {\bigcirc}^2$ 的『形式』之『啟發』耶？如果由二次式函數 $f(x) = x^2 + px +q$ 之觀點來看，難到不能平移『座標系』 $x = y + \alpha$ ，使之簡化為 $y \times (y - \Box)$ 之『形式』乎？？一個簡單的例子 $x^2 - 4x + 3 = 0$ 或可說明，如此尋找 $\alpha$ 以求簡化方程式的求解，實無攸利也！那個 $\alpha$ 還得滿足原方程式 ${\alpha}^2 - 4 \alpha + 3 = 0$ 矣！！

Figure q_0

Fig 1 : $f(x) = x^2 - 4x +3 = (x - 1) \times (x - 3)$

Figure q_1

Fig 2 : $f(y) = y^2 - 1 = (y - 1) \times (y + 1)$

Figure q_2

Fig 3 : $f(y) = y^2 - 2y = y \times (y - 2)$

pi@raspberrypi:~ x^2 = 1x = \pm 1x^2 + px + q = (x - \alpha)(x -\beta)r_1 = \alpha + \betar_2 = \alpha - \betaNr_2 = \alpha - \beta \beta - \alpha- r_2\alpha - \beta{(\alpha - \beta)}^2 = {(\alpha + \beta)}^2 - 4 \alpha \betar_2\alpha = \frac{r_1 + r_2}{2}\beta = \frac{r_1 - r_2}{2}\sqrt[4]{-1}\sqrt[3]{-1}\pm \sqrt{x}\sqrt[3]{x}\sqrt[n]{x}P(x) = \sum \limits_{i=0}^{i=n} c_i \cdot x^i = 0 \ =?= \prod \limits_{k=1}^{k=n} x - x_kc_ix_k\sqrt{2}\sqrt{2}\frac{p}{q}pqQ\sqrt{Q}\sqrt{Q} = \frac{p}{q}p^2 = q^2 \cdot Qp, qp = k \cdot Qk, qq^2 = k^2 \cdot Qq = m QpqQ\sqrt{Q}P(x) = x^2 -2 = 0 = (x + \sqrt{2})(x - \sqrt{2})$ 時，這裡所說的『多項式』與『方程式』是一樣的嗎？它們的內在聯繫又是什麼的呢？

── 摘自《【Sonic π】電路學之補充《四》無窮小算術‧中下下‧上》

樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰派生科學計算二‧上

2016-07-23 懸鉤子

為什麼『IPython』那麼強調『互動性』 interactive 呢？因為人們的『認知』是積累的，偶或『靈感』乍現，當下若不『捕捉』，恐怕稍後不復『記得』矣！更何況今日之『計算』極其『複雜』，若是每每紙筆『推導』，或是一定得先寫好『程式』，總覺不切實際，終將誰願為的耶？？反觀則朗朗親見『互動性』伴隨『符號運算』的好處？？！！

口說無憑，且以拉格朗日之『預解形‧式』再探伽羅瓦『方程式』論的淵源，及於『SymPy』之省勞少錯方便乎！！？？

就讓我們略窺一下『伽羅瓦』的思考法吧。假使 $x_1,x_2,\cdots, x_n$ 是『多項式』 $P(x) = \sum \limits_{k=0}^{k=n} c_k x^k = 0$ 的『解』，此處係數 $c_k$ 都是『有理數』。如果我們建構一個『對稱函數』

$f(x_1,x_2,x_3,\cdots,x_n) = (x-x_1)(x-x_2)(x-x_3)\cdots(x-x_n)$

，將它展開後 $c_n f(x_1,x_2,x_3,\cdots,x_n)$ 應該就是 $P(x)$ 的吧。如果將這些『解』 $x_1,x_2,\cdots, x_n$ ，作任意的『排列』permutation $\begin{pmatrix} x_1 & x_2 & x_3 & \cdots & x_n \\ x_2 & x_n & x_4 & \cdots & x_1\end{pmatrix}$ ，此處是說上一排的『解』的『位置』用下一排的『解』來『置換』，由於 $f$ 函數的特殊『形式』，我們會得到 $f(x_1,x_2,x_3, \cdots,x_n)=f(x_2,x_n,x_4,\cdots,x_1)$ 。事實上對於任意的『置換』，都會有 $f(x_1,x_2,...,x_n)=f(x_2,x_1,\cdots,x_n)=f(x_3,x_1,\cdots,x_n,x_{n−1})$ 。所以函數 $f$ 稱之為『對稱函數』，這個『置換』的『不變性』就是『伽羅瓦』主要研究的對象。舉例來說，考慮一個二次方程式 $x^2 + A x + B = 0$ 有兩個根 $\lambda_1, \lambda_2$ ， $F(\lambda_1, \lambda_2) = (x - \lambda_1)(x - \lambda_2)$
$= x^2 -(\lambda_1 + \lambda_2) x + \lambda_1 \cdot \lambda_2$
$= x^2 + A x + B$
，比對後得到
$\lambda_1 + \lambda_2 = -A$ ，和
$\lambda_1 \cdot \lambda_2 = B$
。這兩個『代數式』對於 $\lambda_1, \lambda_2$ 來講，也是『對稱的』，如果將它們看成兩個變數的『聯立方程組』，化簡後所得到的也定然就是『對等的』二次方程式 ${\lambda_1}^2 + A \lambda_1 + B = 0$ 與 ${\lambda_2}^2 + A \lambda_2 + B = 0$ 。

這產生了很重要的結果，假使 $\lambda_1 = a + b \sqrt{Q}$ 是方程式的一個根，假設另一個根是 $\lambda_2 = c + d \sqrt{Q}$ ，由於
$\lambda_1 + \lambda_2 = -A$
$\Longrightarrow (a + b \sqrt{Q}) + (c + d \sqrt{Q}) = -A$
$\Longrightarrow (a + c +A) + (b + d) \sqrt{Q} = 0$
$\therefore a + c + A = 0, \ b + d =0$
，再由
$\lambda_1 \cdot \lambda_2 = B$
$\Longrightarrow (a + b \sqrt{Q}) \cdot (c + d \sqrt{Q}) = B$
$\Longrightarrow (a c + b d Q - B) + (a d + b c) \sqrt{Q}) =0$
$\therefore a c + b d Q = B, \ a d + b c =0$
。因此得到 $d = -b, \ c = a$ 。而且 $a = - \frac{A}{2}, \ b \sqrt{Q} = \frac{\sqrt{A^2 - 4B}}{2}$ 。也就是說這兩個根是熟悉的 $\frac{- A + \sqrt{A^2 - 4B}}{2}$ 與 $\frac{- A - \sqrt{A^2 - 4B}}{2}$ 。於是一個『對稱函數』 $f(x_1,x_2,x_3,\cdots,x_n) = (x-x_1)(x-x_2)(x-x_3)\cdots(x-x_n)$ 如果某一個根 $x_k$ 是 $a + b \sqrt{Q}$ 的形式﹐那麼必然有另一個根 $x_j$ 是 $a - b \sqrt{Q}$ 的形式，這就是由於那個『多項式』的係數是『有理數』的原故，它的『二次方根』的解，總是『成對』出現的啊！於是『二次方根』解的個數也必然是『偶數』的了！！

── 摘自《【Sonic π】電路學之補充《四》無窮小算術‧中下下‧中》

Resolvent (Galois theory)

In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are

$X^{2}-\Delta$ where $\Delta$ is the discriminant, which is a resolvent for the alternating group. In the case of a cubic equation, this resolvent is sometimes called the quadratic resolvent; its roots appear explicitly in the formulas for the roots of a cubic equation.
The cubic resolvent of a quartic equation, which is a resolvent for the dihedral group of 8 elements.
The Cayley resolvent is a resolvent for the maximal resoluble Galois group in degree five. It is a polynomial of degree 6.

These three resolvents have the property of being always separable, which means that, if they have a multiple root, then the polynomial p is not irreducible. It is not known if there is an always separable resolvent for every group of permutations.

For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble.

───

即使祇以文獻上記載的歷史來說，

配方法

配方法是一種代數的計算技巧，可以用來解二次方程式、判別解析幾何中某些方程式的圖形，或者用來計算微積分中的某些積分型式。配方法最主要的目的就是將一個一元二次方程式或多項式化為一個一次式的完全平方，以便簡化計算。

將下方左邊的二次式化成右邊的形式，就是配方法的目標：

ax^{2}+bx+c=a(x-h)^{2}+k

，其中h和k是常數

其由來亦古早矣︰

求根公式的由來

中亞細亞的花拉子米

花拉子米全名是阿布·阿卜杜拉·穆罕默德·伊本·穆薩·花拉子米^[1]（Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī，約780年－約850年^[2]），他是一位波斯數學家、天文學家及地理學家，也是巴格達智慧之家的學者。

他的《代數學》（Kitab al-Jabr wa-l-Muqabala）是第一本解決一次方程及一元二次方程的系統著作，他因而被稱為代數的創造者^[3]，與丟番圖享名。十二世紀，花拉子米在印度數字方面的著作被翻譯成拉丁文，十進制因此傳入西方世界^[4]。此外，他修訂了托勒密的《地理》，並著有天文學及占星學方面的書籍。

從一些詞就可以看出他對數學的重要貢獻，「代數」（algebra）一詞出自阿拉伯文拉丁轉寫「al-jabr」^[5]，「al-jabr」是用以解決一元二次方程的兩個辦法之一。算法（Algorism、Algorithm）出自「Algoritmi」，這是花拉子米（al-Khwārizmī）的拉丁文譯名^[6]，而西班牙語「guarismo」及葡萄牙語「algarismo」亦是由此名字而來，這兩個詞語都解作數字^[7]。

……

(約780-約850) 在公元820年左右出版了《代數學》。書中給出了一元二次方程的求根公式，並把方程的未知數叫做「根」，其後譯成拉丁文radix。

我們通常把 $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$ 稱之為 $ax^{2}+bx+c=0\,$ 的求根公式：

${\begin{aligned}ax^{2}+bx+c&=0\\x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}&=0\\x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}-\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}&=0\\\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a^{2}}}+{\frac {c}{a}}&=0\\\left(x+{\frac {b}{2a}}\right)^{2}&={\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}\\\left(x+{\frac {b}{2a}}\right)^{2}&={\frac {b^{2}-4ac}{4a^{2}}}\\x+{\frac {b}{2a}}&={\frac {\pm {\sqrt {b^{2}-4ac}}}{2a}}\\x&={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\end{aligned}}$

或不將 $x^{2}$ 係數化為1：

${\begin{aligned}ax^{2}+bx+c&=0\\ax^{2}+bx+\left({\frac {b}{2{\sqrt {a}}}}\right)^{2}&=\left({\frac {b}{2{\sqrt {a}}}}\right)^{2}-c\\\left(x{\sqrt {a}}+{\frac {b}{2{\sqrt {a}}}}\right)^{2}&=\left({\frac {b}{2{\sqrt {a}}}}\right)^{2}-c\\x{\sqrt {a}}+{\frac {b}{2{\sqrt {a}}}}&=\pm {\sqrt {\left({\frac {b}{2{\sqrt {a}}}}\right)^{2}-c}}\\x{\sqrt {a}}+{\frac {b}{2{\sqrt {a}}}}&=\pm {\sqrt {{\frac {b^{2}}{4a}}-c}}\\x+{\frac {b}{2a}}&=\pm {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}\\x+{\frac {b}{2a}}&=\pm {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {4ac}{4a^{2}}}}}\\x&=-{\frac {b}{2a}}\pm {\sqrt {{\frac {b^{2}-4ac}{4a^{2}}}}}\\x&={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\end{aligned}}$

至於說用『單位圓』之

單位根

數學上， $n\,$ 次單位根是 } $n\,$ 次冪為1的複數。它們位於複平面的單位圓上，構成正n邊形的頂點，其中一個頂點是1。

定義

z^{n}=1\qquad (n=1,2,3,\cdots )

這方程的複數根 $z \,$ 為 $n\,$ 次單位根。

單位的 $n\,$ 次根有 $n\,$ 個：

e^{\frac {2\pi k{i}}{n}}\qquad (k=0,1,2,\cdots ,n-1)

。

本原根

單位的 $n\,$ 次根以乘法構成 $n$ 階循環群。它的生成元是 $n\,$ 次本原單位根。 $n\,$ 次本原單位根是 $e^{\frac {2\pi k{i}}{n}}$ ，其中 $k\,$ 和 $n\,$ 互質。 $n\,$ 次本原單位根數目為歐拉函數 $\varphi (n)$ 。

例子

一次單位根有一個 $1\,$ 。

二次單位根有兩個： $+1\,$ 和 $-1\,$ ，只有 $-1\,$ 是本原根。

三次單位根是

\left\{1,{\frac {-1+{\sqrt {3}}{i}}{2}},{\frac {-1-{\sqrt {3}}{i}}{2}}\right\},

其中 ${{\mathrm {i}}}\,$ 是虛數單位；除 $1\,$ 外都是本原根。

四次單位根是

\left\{1,+{i},-1,-{i}\right\},

其中 $+{{\mathrm {i}}}\,$ 和 $-{{\mathrm {i}}}\,$ 是本原根。

來『系統化』研究『一元多次方程式』之求解卻是拉格朗日的直覺『先見』哩！！此處僅擷維基百科之文本一段︰

Quadratic formula

In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way.

The general quadratic equation is

ax^{2}+bx+c=0.

Here $x$ represents an unknown, while $a$ , $b$ , and $c$ are constants with $a$ not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation, by inserting the former into the latter. Each of the solutions given by the quadratic formula is called a root of the quadratic equation.

Geometrically, these roots represent the $x$ values at which any parabola, explicitly given as $y = ax 2 + bx + c$ , crosses the $x$ -axis. As well as being a formula that will yield the zeros of any parabola, the quadratic equation will give the axis of symmetry of the parabola, and it can be used to immediately determine how many zeros it has.

……

By Lagrange resolvents

For more details on this topic, see Lagrange resolvents.

An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents,^[24] which is an early part of Galois theory.^[25] This method can be generalized to give the roots of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group.

This approach focuses on the roots more than on rearranging the original equation. Given a monic quadratic polynomial

x^2+px+q,

assume that it factors as

x^2+px+q=(x-\alpha)(x-\beta),

Expanding yields

x^2+px+q=x^2-(\alpha+\beta)x+\alpha \beta,

where $p = -(α + β)$ and $q = αβ$ .

Since the order of multiplication does not matter, one can switch $α$ and $β$ and the values of $p$ and $q$ will not change: one can say that $p$ and $q$ are symmetric polynomials in $α$ and $β$ . In fact, they are the elementary symmetric polynomials – any symmetric polynomial in $α$ and $β$ can be expressed in terms of $α + β$ and $αβ$ The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one “break the symmetry” and recover the roots? Thus solving a polynomial of degree $n$ is related to the ways of rearranging (“permuting”) $n$ terms, which is called the symmetric group on $n$ letters, and denoted $S n$ . For the quadratic polynomial, the only way to rearrange two terms is to swap them (“transpose” them), and thus solving a quadratic polynomial is simple.

To find the roots $α$ and $β$ , consider their sum and difference:

\text{[math]}

These are called the Lagrange resolvents of the polynomial; notice that one of these depends on the order of the roots, which is the key point. One can recover the roots from the resolvents by inverting the above equations:

\text{[math]}

Thus, solving for the resolvents gives the original roots.

Now $r 1 = α + β$ is a symmetric function in $α$ and $β$ , so it can be expressed in terms of $p$ and $q$ , and in fact $r 1 = - p$ as noted above. But $r 2 = α - β$ is not symmetric, since switching $α$ and $β$ yields $- r 2 = β - α$ (formally, this is termed a group action of the symmetric group of the roots). Since $r 2$ is not symmetric, it cannot be expressed in terms of the coefficients $p$ and $q$ , as these are symmetric in the roots and thus so is any polynomial expression involving them. Changing the order of the roots only changes $r 2$ by a factor of −1, and thus the square $r 22 = (α - β) 2$ is symmetric in the roots, and thus expressible in terms of $p$ and $q$ . Using the equation

(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta\!

yields

r_2^2 = p^2 - 4q\!

and thus

r_2 = \pm \sqrt{p^2 - 4q}.\!

If one takes the positive root, breaking symmetry, one obtains:

\text{[math]}

and thus

\text{[math]}

Thus the roots are

\textstyle{\frac{1}{2}}\left(-p \pm \sqrt{p^2 - 4q}\right)

which is the quadratic formula. Substituting $p = b / a, q = c / a$ yields the usual form for when a quadratic is not monic. The resolvents can be recognized as $r 1 / 2 = - p / 2 = - b / 2 a$ being the vertex, and $r 22 = p 2 - 4 q$ is the discriminant (of a monic polynomial).

A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation (the “resolving polynomial”) relating $r 2$ and $r 3$ , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved.^[24] The same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and in fact solutions to quintic equations in general cannot be expressed using only roots.

───

不知是否能得『配方法』之根本焉？？？

樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰派生科學計算一

2016-07-22 懸鉤子

學習『工具』最簡單的方法，就是『使用』它︰

pi@raspberrypi:~ $ ipython3
Python 3.4.2 (default, Oct 19 2014, 13:31:11) 
Type "copyright", "credits" or "license" for more information.

IPython 2.3.0 -- An enhanced Interactive Python.
?         -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help      -> Python's own help system.
object?   -> Details about 'object', use 'object??' for extra details.

In [1]:

不過對此『工具』是什麼□○都尚且不知，又將如何『使用』呢？通常『工具』的『使用』始於閱讀『概觀』︰

Introduction

Overview

One of Python’s most useful features is its interactive interpreter. It allows for very fast testing of ideas without the overhead of creating test files as is typical in most programming languages. However, the interpreter supplied with the standard Python distribution is somewhat limited for extended interactive use.

The goal of IPython is to create a comprehensive environment for interactive and exploratory computing. To support this goal, IPython has three main components:

An enhanced interactive Python shell.
A decoupled two-process communication model, which allows for multiple clients to connect to a computation kernel, most notably the web-based notebook
An architecture for interactive parallel computing.

All of IPython is open source (released under the revised BSD license).

………

從『簡介』尋找，發現『上手』之處︰

Introducing IPython

You don’t need to know anything beyond Python to start using IPython – just type commands as you would at the standard Python prompt. But IPython can do much more than the standard prompt. Some key features are described here. For more information, check the tips page, or look at examples in the IPython cookbook.

If you’ve never used Python before, you might want to look at the official tutorial or an alternative, Dive into Python.

The four most helpful commands

The four most helpful commands, as well as their brief description, is shown to you in a banner, every time you start IPython:

command	description
?	Introduction and overview of IPython’s features.
%quickref	Quick reference.
help	Python’s own help system.
object?	Details about ‘object’, use ‘object??’ for extra details.

Tab completion

Tab completion, especially for attributes, is a convenient way to explore the structure of any object you’re dealing with. Simply type object_name.<TAB> to view the object’s attributes (see the readline section for more). Besides Python objects and keywords, tab completion also works on file and directory names.

………

要是還能知道『參考文件』在哪裡固然很好︰

IPython Documentation

Release:	3.2.1
Date:	September 25, 2015

Welcome to the official IPython documentation.

───

往往根本什麼說明也沒有哩！？此時亦無礙徒手上陣，嘗試錯誤，將之 TRY 出耶？！

依稀記得作者當初下的第一個指令就是『?』︰

? -> Introduction and overview of IPython’s features.

匆匆瀏覽後，隨即上路︰

IPython — An enhanced Interactive Python
=========================================

IPython offers a combination of convenient shell features, special commands and a history mechanism for both input (command history) and output (results caching, similar to Mathematica). It is intended to be a fully compatible replacement for the standard Python interpreter, while offering vastly improved functionality and flexibility.

At your system command line, type ‘ipython -h’ to see the command line options available. This document only describes interactive features.

MAIN FEATURES
————-

* Access to the standard Python help. As of Python 2.1, a help system is available with access to object docstrings and the Python manuals. Simply type ‘help’ (no quotes) to access it.

* Magic commands: type %magic for information on the magic subsystem.

* System command aliases, via the %alias command or the configuration file(s).

* Dynamic object information:

Typing ?word or word? prints detailed information about an object. If
certain strings in the object are too long (docstrings, code, etc.) they get snipped in the center for brevity.

Typing ??word or word?? gives access to the full information without
snipping long strings. Long strings are sent to the screen through the less pager if longer than the screen, printed otherwise.

The ?/?? system gives access to the full source code for any object (if
available), shows function prototypes and other useful information.

If you just want to see an object’s docstring, type ‘%pdoc object’ (without quotes, and without % if you have automagic on).

Both %pdoc and ?/?? give you access to documentation even on things which are not explicitely defined. Try for example typing {}.get? or after import os, type os.path.abspath??. The magic functions %pdef, %source and %file operate similarly.

* Completion in the local namespace, by typing TAB at the prompt.

At any time, hitting tab will complete any available python commands or variable names, and show you a list of the possible completions if there’s no unambiguous one. It will also complete filenames in the current directory.

This feature requires the readline and rlcomplete modules, so it won’t work if your Python lacks readline support (such as under Windows).
* Search previous command history in two ways (also requires readline):

– Start typing, and then use Ctrl-p (previous,up) and Ctrl-n (next,down) to search through only the history items that match what you’ve typed so far. If you use Ctrl-p/Ctrl-n at a blank prompt, they just behave like normal arrow keys.

– Hit Ctrl-r: opens a search prompt. Begin typing and the system searches your history for lines that match what you’ve typed so far, completing as much as it can.

– %hist: search history by index (this does *not* require readline).

* Persistent command history across sessions.

* Logging of input with the ability to save and restore a working session.

* System escape with !. Typing !ls will run ‘ls’ in the current directory.

* The reload command does a ‘deep’ reload of a module: changes made to the module since you imported will actually be available without having to exit.

* Verbose and colored exception traceback printouts. See the magic xmode and xcolor functions for details (just type %magic).

* Input caching system:

IPython offers numbered prompts (In/Out) with input and output caching. All input is saved and can be retrieved as variables (besides the usual arrow key recall).

The following GLOBAL variables always exist (so don’t overwrite them!):
_i: stores previous input.
_ii: next previous.
_iii: next-next previous.
_ih : a list of all input _ih[n] is the input from line n.

Additionally, global variables named _i<n> are dynamically created (<n> being the prompt counter), such that _i<n> == _ih[<n>]

For example, what you typed at prompt 14 is available as _i14 and _ih[14].

You can create macros which contain multiple input lines from this history, for later re-execution, with the %macro function.

The history function %hist allows you to see any part of your input history by printing a range of the _i variables. Note that inputs which contain magic functions (%) appear in the history with a prepended comment. This is because they aren’t really valid Python code, so you can’t exec them.

* Output caching system:
For output that is returned from actions, a system similar to the input
cache exists but using _ instead of _i. Only actions that produce a result (NOT assignments, for example) are cached. If you are familiar with Mathematica, IPython’s _ variables behave exactly like Mathematica’s % variables.

The following GLOBAL variables always exist (so don’t overwrite them!):
_ (one underscore): previous output.
__ (two underscores): next previous.
___ (three underscores): next-next previous.

Global variables named _<n> are dynamically created (<n> being the prompt counter), such that the result of output <n> is always available as _<n>.

Finally, a global dictionary named _oh exists with entries for all lines
which generated output.

* Directory history:

Your history of visited directories is kept in the global list _dh, and the
magic %cd command can be used to go to any entry in that list.

* Auto-parentheses and auto-quotes (adapted from Nathan Gray’s LazyPython)

1. Auto-parentheses

Callable objects (i.e. functions, methods, etc) can be invoked like
this (notice the commas between the arguments)::

In [1]: callable_ob arg1, arg2, arg3

and the input will be translated to this::

callable_ob(arg1, arg2, arg3)

This feature is off by default (in rare cases it can produce
undesirable side-effects), but you can activate it at the command-line
by starting IPython with `–autocall 1`, set it permanently in your
configuration file, or turn on at runtime with `%autocall 1`.

You can force auto-parentheses by using ‘/’ as the first character
of a line. For example::

In [1]: /globals # becomes ‘globals()’

Note that the ‘/’ MUST be the first character on the line! This
won’t work::

In [2]: print /globals # syntax error

In most cases the automatic algorithm should work, so you should
rarely need to explicitly invoke /. One notable exception is if you
are trying to call a function with a list of tuples as arguments (the
callable_ob(arg1, arg2, arg3)

You can force auto-parentheses by using ‘/’ as the first character
of a line. For example::

In [1]: /globals # becomes ‘globals()’

Note that the ‘/’ MUST be the first character on the line! This
won’t work::

In [2]: print /globals # syntax error

In [1]: zip (1,2,3),(4,5,6) # won’t work

but this will work::

In [2]: /zip (1,2,3),(4,5,6)
——> zip ((1,2,3),(4,5,6))
Out[2]= [(1, 4), (2, 5), (3, 6)]

IPython tells you that it has altered your command line by
displaying the new command line preceded by –>. e.g.::

In [18]: callable list
——-> callable (list)

2. Auto-Quoting

You can force auto-quoting of a function’s arguments by using ‘,’ as
the first character of a line. For example::

In [1]: ,my_function /home/me # becomes my_function(“/home/me”)

If you use ‘;’ instead, the whole argument is quoted as a single
string (while ‘,’ splits on whitespace)::

In [2]: ,my_function a b c # becomes my_function(“a”,”b”,”c”)
In [3]: ;my_function a b c # becomes my_function(“a b c”)

Note that the ‘,’ MUST be the first character on the line! This
won’t work::

In [4]: x = ,my_function /home/me # syntax error

一旦需要什麼 □□ ，或想要怎樣 ○○ ，再查詢、翻閱、谷歌……文件，仔細精讀文本一番乎！！？？果然常用之故，有時大腦恐已忘記，誰知指頭竟能自書，當真『熟能生巧』隨心所欲哩？？！！

因此『派生互動筆記』界面，作者也作如是觀矣！！！

The IPython Notebook

Introduction

The notebook extends the console-based approach to interactive computing in a qualitatively new direction, providing a web-based application suitable for capturing the whole computation process: developing, documenting, and executing code, as well as communicating the results. The IPython notebook combines two components:

A web application: a browser-based tool for interactive authoring of documents which combine explanatory text, mathematics, computations and their rich media output.

Notebook documents: a representation of all content visible in the web application, including inputs and outputs of the computations, explanatory text, mathematics, images, and rich media representations of objects.

Main features of the web application

In-browser editing for code, with automatic syntax highlighting, indentation, and tab completion/introspection.
The ability to execute code from the browser, with the results of computations attached to the code which generated them.
Displaying the result of computation using rich media representations, such as HTML, LaTeX, PNG, SVG, etc. For example, publication-quality figures rendered by the matplotlib library, can be included inline.
In-browser editing for rich text using the Markdown markup language, which can provide commentary for the code, is not limited to plain text.
The ability to easily include mathematical notation within markdown cells using LaTeX, and rendered natively by MathJax.

Notebook documents

Notebook documents contains the inputs and outputs of a interactive session as well as additional text that accompanies the code but is not meant for execution. In this way, notebook files can serve as a complete computational record of a session, interleaving executable code with explanatory text, mathematics, and rich representations of resulting objects. These documents are internally JSON files and are saved with the .ipynb extension. Since JSON is a plain text format, they can be version-controlled and shared with colleagues.

Notebooks may be exported to a range of static formats, including HTML (for example, for blog posts), reStructuredText, LaTeX, PDF, and slide shows, via the new nbconvert command.

Furthermore, any .ipynb notebook document available from a public URL can be shared via the IPython Notebook Viewer (nbviewer). This service loads the notebook document from the URL and renders it as a static web page. The results may thus be shared with a colleague, or as a public blog post, without other users needing to install IPython themselves. In effect, nbviewer is simply nbconvert as a web service, so you can do your own static conversions with nbconvert, without relying on nbviewer.

2016 年 7 月
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歷史

Example of the algorithm

操作

行互換

性質

把某行乘以一非零常數

性質

把第i行加上第j行的m倍

性質

Lagrange’s method

Solving by Lagrange resolvent

定義

本原根

例子

By Lagrange resolvents

Overview

The four most helpful commands

Tab completion

Contents

Introduction

Main features of the web application

Notebook documents

輕。鬆。學。部落客