===== General polynomial square root solution =====
===== General polynomial square root solution =====
The polynomial square root algorithm above may be summarized and generalized into standard math syntax for use extracting the square roots from any size of squared polynomial, and is easily translated into computer language for use in rapid computations. If the squared polynomial highest order term is not 1, the polynomial must first be preprocessed by dividing it with the value of the highest order term, and then the extracted polynomial factors must be post processed by multiplying them with the square root of the same value.
The polynomial square root algorithm above may be summarized and generalized into standard math syntax for use extracting the square roots from any size of squared polynomial, and is easily translated into computer language for use in rapid computations. If the squared polynomial highest order term is not 1, the polynomial must first be preprocessed by dividing it with the value of the highest order term, and then the extracted polynomial factors must be post processed by multiplying them with the square root of the same value.
<math>\begin{align}
<math>\begin{align}
\end{align}</math>
\end{align}</math>
Note that once <math>R_{m-i}</math> has been calculated for <math>i=m</math>, the R polynomial has been completed and the following <math>T_{n-i}</math> and <math>T_{n-i-k}</math> computations may be neglected in that the results are no longer used after that point, but if executed, the results may be used as a validity check to insure the S polynomial is a squared polynomial and the algorithm executed correctly by insuring the final values of the T and D vectors are identical, as seen in the table..
Note that once <math>R_{m-i}</math> has been calculated for <math>i=m</math>, the R polynomial has been completed and the following <math>T_{n-i}</math> and <math>T_{n-i-k}</math> computations may be neglected in that the results are no longer used after that point, but if executed, the results may be used as a validity check to insure the S polynomial is a squared polynomial and the algorithm executed correctly by insuring the final values of the T and D vectors are identical, as seen in the table..
Polynomial square root
Square-free polynomial#Yun’s algorithm
Factoring a squared polynomial into square roots
[edit]
In general, most polynomials do not have square roots. However, some applications, such as the electrical engineers function of obtaining the Y parameters from a driving point impedance of a two port network[1], do utilize squared polynomials that must be factored into two identical square root polynomials, . The algorithm below will factor a squared polynomial, x 6 − 6 x 5 + 17 x 4 − 36 x 3 + 52 x 2 + 48 x + 36 {\displaystyle {\sqrt {x^{6}-6x^{5}+17x^{4}-36x^{3}+52x^{2}+48x+36}}} , into two identical polynomial roots, R = X 3 − 3 x 2 + 4 x − 6 {\displaystyle R=X^{3}-3x^{2}+4x-6} , using the example from Mathematics Stack Exchange.[2][3]
Q R R ( 2 Q + R ) 0 x 3 x 6 x 3 − 3 x 2 6 x 5 − 9 x 4 x 3 − 3 x 2 4 x 8 x 4 − 24 x 3 + 16 x 2 x 3 − 3 x 2 + 4 x − 6 − 12 x 3 + 36 x 2 − 48 x + 36 R x 3 − 3 x 2 4 x − 6 1 x 6 − 6 x 5 17 x 4 − 36 x 3 52 x 2 48 x 36 x 6 2 − 6 x 5 17 x 4 − 6 x 5 9 x 4 3 8 x 4 − 36 x 3 52 x 2 8 x 4 − 24 x 3 16 x 2 4 − 12 x 3 36 x 2 − 48 x 36 − 12 x 3 36 x 2 − 48 x 36 {\displaystyle {\begin{aligned}&{\begin{array}{|c|c|c|}\hline Q&R&R(2Q+R)\\\hline \\0&x^{3}&x^{6}\\\hline \\x^{3}&-3x^{2}&6x^{5}-9x^{4}\\\hline \\x^{3}-3x^{2}&4x&8x^{4}-24x^{3}+16x^{2}\\\hline \\x^{3}-3x^{2}+4x&-6&-12x^{3}+36x^{2}-48x+36\\\hline \end{array}}&{\begin{array}{|c|c|c|c|c|c|c|c|}\hline R&x^{3}&&-3x^{2}&&4x&&-6\\\hline 1&x^{6}&-6x^{5}&17x^{4}&-36x^{3}&52x^{2}&48x&36\\&x^{6}&&&&&&\\\hline 2&-6x^{5}&17x^{4}&&&&\\&-6x^{5}&9x^{4}&&&&\\\hline 3&&&8x^{4}&-36x^{3}&52x^{2}&&\\&&&8x^{4}&-24x^{3}&16x^{2}&&\\\hline 4&&&&-12x^{3}&36x^{2}&-48x&36\\&&&&-12x^{3}&36x^{2}&-48x&36\\\hline \end{array}}\end{aligned}}}
Steps:
Step1: Compute the square root of the leading term, x 6 {\displaystyle x^{6}} , and place it, x 3 {\displaystyle x^{3}} , in the leading term of the solution R polynomial solution row on top, and place the x 6 {\displaystyle x^{6}} term in row 1 just below the polynomial to be factored, as shown.
Step2: Subtract the newly placed x 6 {\displaystyle x^{6}} from the polynomial to be factored, and bring down the next two terms into row 2.
Step3: Double the current state of the solution R polynomial, then add a new term, Q, such that R(2Q+R) negates the leading term of row 2, and place the negative of R(2Q+R) in the lower space of row 2.
Step4: Subtract the two numbers in row 2, place the results in row 3, and bring down the next two terms of row 1 into row 3..
Step 5: Repeat for all remaining rows and columns until complete.
When complete, the solution R polynomial will show up in the R column in the left side table and the R row of the right side table.
General polynomial square root solution
[edit]
The polynomial square root algorithm above may be summarized and generalized into standard math syntax for use in extracting the square roots from any size of squared polynomial, and is easily translated into computer language for use in rapid computations. If the squared polynomial highest order term is not 1, the polynomial must first be preprocessed by dividing it with the value of the highest order term, and then the extracted polynomial factors must be post processed by multiplying them with the square root of the same value. The generalized math summary is:
n = order of the squared polynomial being factored m = order of the extracted square root polynomial = n / 2 S is the squared polynomial, indexed in powers of x, and normalized to the highest order term value of 1 R is the square root polynomial, indexed in powers of x, and with the highest order term initialized to 1 T and D are vectors of length n, with all entries initialized to 0 ∑ i = 1 m [ ( ∑ k = i 0 D n − i − k = { S n − i − k , if k ≥ i − 1 D n − i − k − T n − i − k , if k < i − 1 ) ; R m − i = D n − i 2 ; T n − i = D n − i ; ( ∑ k = 1 i T n − i − k = { R m − k D n − i , if k < i R m − k R m − i , if k ≥ i ) ] {\displaystyle {\begin{aligned}n&={\text{order of the squared polynomial being factored}}\\m&={\text{order of the extracted square root polynomial}}=n/2\\S&{\text{ is the squared polynomial, indexed in powers of x, and normalized to the highest order term value of 1}}\\R&{\text{ is the square root polynomial, indexed in powers of x, and with the highest order term initialized to 1}}\\T&{\text{ and }}D{\text{ are vectors of length n, with all entries initialized to 0}}\\\\&\sum _{i=1}^{m}{{\Bigg [}{\Big (}\sum _{k=i}^{0}}D_{n-i-k}={\begin{cases}S_{n-i-k},&{\text{if }}k\geq i-1\\D_{n-i-k}-T_{n-i-k},&{\text{if }}k<i-1\end{cases}}{\Big )}{\text{ ;}}\quad R_{m-i}={\frac {D_{n-i}}{2}}{\text{ ;}}\quad T_{n-i}=D_{n-i}{\text{ ;}}\quad {\Big (}\sum _{k=1}^{i}{T_{n-i-k}={\begin{cases}R_{m-k}D_{n-i},&{\text{if }}k<i\\R_{m-k}R_{m-i},&{\text{if }}k\geq i\end{cases}}{\Big )}{\Bigg ]}}\end{aligned}}} .
![{\displaystyle {\begin{aligned}n&={\text{order of the squared polynomial being factored}}\\m&={\text{order of the extracted square root polynomial}}=n/2\\S&{\text{ is the squared polynomial, indexed in powers of x, and normalized to the highest order term value of 1}}\\R&{\text{ is the square root polynomial, indexed in powers of x, and with the highest order term initialized to 1}}\\T&{\text{ and }}D{\text{ are vectors of length n, with all entries initialized to 0}}\\\\&\sum _{i=1}^{m}{{\Bigg [}{\Big (}\sum _{k=i}^{0}}D_{n-i-k}={\begin{cases}S_{n-i-k},&{\text{if }}k\geq i-1\\D_{n-i-k}-T_{n-i-k},&{\text{if }}k<i-1\end{cases}}{\Big )}{\text{ ;}}\quad R_{m-i}={\frac {D_{n-i}}{2}}{\text{ ;}}\quad T_{n-i}=D_{n-i}{\text{ ;}}\quad {\Big (}\sum _{k=1}^{i}{T_{n-i-k}={\begin{cases}R_{m-k}D_{n-i},&{\text{if }}k<i\\R_{m-k}R_{m-i},&{\text{if }}k\geq i\end{cases}}{\Big )}{\Bigg ]}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38dbcc96ba4b97053b8c3f407379ebc43e230621)
Note that once R m − i {\displaystyle R_{m-i}} has been calculated for i = m {\displaystyle i=m} , the R polynomial has been completed and the following T n − i {\displaystyle T_{n-i}} and T n − i − k {\displaystyle T_{n-i-k}} computations may be neglected in that the results are no longer used after that point, but if executed, the results may be used as a validity check to insure the S polynomial is a squared polynomial and the algorithm executed correctly by insuring the final values of the T and D vectors are identical, as seen in the table..
Overcoming undefinedness
[edit]
To potentially overcome undefinedness, consider the following equation to be solved,
0 = c o s h ( x ) − l n 2 ( x ) then: x n + 1 = x n − c o s h ( x n ) − l n 2 ( x n ) s i n h ( x n ) − 2 l n ( x n ) / x n {\displaystyle {\begin{aligned}0&=cosh(x)-ln^{2}(x)\\{\text{then:}}&\\x_{n+1}&=x_{n}-{\frac {cosh(x_{n})-ln^{2}(x_{n})}{sinh(x_{n})-2ln(x_{n})/x_{n}}}\\\end{aligned}}}
if x 0 = 1 {\displaystyle x_{0}=1} is used to initialize the method, then x 1 = − .313035 {\displaystyle x_{1}=-.313035} , x 2 {\displaystyle x_{2}} cannot be calculated, Undefinedness is encountered, and Newton’s method cannot continue in the set of real numbers. However, if the following replacement function is solved:
0 = c o s h ( m a x ( x , 10 − 10 ) ) − l n 2 ( m a x ( x , 10 − 10 ) ) {\displaystyle {\begin{aligned}0&=cosh{\big (}max(x,10^{-10}){\big )}-ln^{2}{\big (}max(x,10^{-10}){\big )}\\\end{aligned}}}
then the argument of the l n ( ) {\displaystyle ln()} function is forced to be positive at all times and the equation is no longer undefined when x < 0 {\displaystyle x<0} . If x 0 = 1 {\displaystyle x_{0}=1} is again chosen as the initial point, the method converges to 0.35643617 after 20 iterations.
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Multivariate interpolation
[edit]
V D = I N T E R P ( M n , V I n ) V D n = I N T E R P ( M n − 1 , V I ) {\displaystyle {\begin{aligned}V_{D}&=INTERP(Mn,V_{In})\\V_{D}n&=INTERP(M_{n-1},V_{I})\\\end{aligned}}}
- ^ Kinayman, Noyan; Aksun, M. I. (2005). Modern Microwave Circuits. 685 Canton Street, Norwood, MA, US: Artech House. pp. 130–131, 510. ISBN 1-58053-725-1.
{{cite book}}: CS1 maint: location (link) - ^ Steven Alexis Gregory (https://math.stackexchange.com/users/75410/steven-alexis-gregory), Algorithm for finding the square root of a polynomial…, URL (version: 2018-07-10): https://math.stackexchange.com/q/1854191
- ^ Steven Alexis Gregory (https://math.stackexchange.com/users/75410/steven-alexis-gregory), How to find the Square Root of a Polynomial, URL (version: 2021-05-21): https://math.stackexchange.com/q/4146459
