Continuous phase modulation: Difference between revisions

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In numerical analysis, the Clenshaw algorithm^[1] is a recursive method to evaluate a linear combination of Chebyshev polynomials. It is a generalization of Horner's method for evaluating a linear combination of monomials.

It generalizes to more than just Chebyshev polynomials; it applies to any class of functions that can be defined by a three-term recurrence relation.^[2]

Clenshaw algorithm

Suppose that $ϕ_{k}, k = 0, 1, \dots$ is a sequence of functions that satisfy the linear recurrence relation

ϕ_{k + 1} (x) = α_{k} (x) ϕ_{k} (x) + β_{k} (x) ϕ_{k - 1} (x),

where the coefficients $α_{k}$ and $β_{k}$ are known in advance. Note that in the most common applications, $α (x)$ does not depend on $k$ , and $β$ is a constant that depends on neither $x$ nor $k$ .

Our goal is to evaluate a weighted sum of these functions

S (x) = \sum_{k = 0}^{n} a_{k} ϕ_{k} (x)

Given the coefficients $a_{0}, \dots, a_{n}$ , compute the values $b_{k} (x)$ by the "reverse" recurrence formula:

\begin{aligned} b_{n + 1} (x) & = b_{n + 2} (x) = 0, \\ b_{k} (x) & = a_{k} + α_{k} (x) b_{k + 1} (x) + β_{k + 1} (x) b_{k + 2} (x) . \end{aligned}

The linear combination of the $ϕ_{k}$ satisfies:

S (x) = ϕ_{0} (x) a_{0} + ϕ_{1} (x) b_{1} (x) + β_{1} (x) ϕ_{0} (x) b_{2} (x) .

See Fox and Parker^[3] for more information and stability analyses.

Horner as a special case of Clenshaw

A particularly simple case occurs when evaluating a polynomial of the form

S (x) = \sum_{k = 0}^{n} a_{k} x^{k}

.

The functions are simply

\begin{aligned} ϕ_{0} (x) & = 1, \\ ϕ_{k} (x) & = x^{k} = x ϕ_{k - 1} (x) \end{aligned}

and are produced by the recurrence coefficients $α (x) = x$ and $β = 0$ .

In this case, the recurrence formula to compute the sum is

b_{k} (x) = a_{k} + x b_{k + 1} (x)

and, in this case, the sum is simply

S (x) = a_{0} + x b_{1} (x) = b_{0} (x)

,

which is exactly the usual Horner's method.

Special case for Chebyshev series

Consider a truncated Chebyshev series

p_{n} (x) = a_{0} + a_{1} T_{1} (x) + a_{2} T_{2} (x) + \dots + a_{n} T_{n} (x) .

The coefficients in the recursion relation for the Chebyshev polynomials are

α (x) = 2 x, β = - 1,

with the initial conditions

T_{0} (x) = 1, T_{1} (x) = x .

Thus, the recurrence is

b_{k} (x) = a_{k} + 2 x b_{k + 1} (x) - b_{k + 2} (x)

and the final sum is

p_{n} (x) = a_{0} + x b_{1} (x) - b_{2} (x) .

One way to evaluate this is to continue the recurrence one more step, and compute

b_{0} (x) = 2 a_{0} + 2 x b_{1} (x) - b_{2} (x),

(note the doubled a₀ coefficient) followed by

p_{n} (x) = b_{0} (x) - x b_{1} (x) - a_{0} = \frac{1}{2} [b_{0} (x) - b_{2} (x)] .

Geodetic applications

Clenshaw's algorithm is extensively used in geodetic applications where it is usually referred to as Clenshaw summation.^[4] A simple application is summing the trigonometric series to compute the meridian arc. These have the form

m (θ) = C_{0} θ + C_{1} \sin θ + C_{2} \sin 2 θ + \dots + C_{n} \sin n θ .

Leaving off the initial $C_{0} θ$ term, the remainder is a summation of the appropriate form. There is no leading term because $ϕ_{0} (θ) = \sin 0 θ = \sin 0 = 0$ .

The recurrence relation for $\sin k θ$ is

\sin k θ = 2 \cos θ \sin (k - 1) θ - \sin (k - 2) θ

,

making the coefficients in the recursion relation

α_{k} (θ) = 2 \cos θ, β_{k} = - 1 .

and the evaluation of the series is given by

\begin{aligned} b_{n + 1} (θ) & = b_{n + 2} (θ) = 0, \\ b_{k} (θ) & = C_{k} + 2 b_{k + 1} (θ) \cos θ - b_{k + 2} (θ) (n \geq k \geq 1) . \end{aligned}

The final step is made particularly simple because $ϕ_{0} (θ) = \sin 0 = 0$ , so the end of the recurrence is simply $b_{1} (θ) \sin (θ)$ ; the $C_{0} θ$ term is added separately:

m (θ) = C_{0} θ + b_{1} (θ) \sin θ .

Note that the algorithm requires only the evaluation of two trigonometric quantities $\cos θ$ and $\sin θ$ .

References

↑ Template:Cite doi Note that this paper is written in terms of the Shifted Chebyshev polynomials of the first kind $T_{n}^{*} (x) = T_{n} (2 x - 1)$ .
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[Clenshaw55-1] Template:Cite doi Note that this paper is written in terms of the Shifted Chebyshev polynomials of the first kind $T_{n}^{*} (x) = T_{n} (2 x - 1)$ .

[2] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[FoxParker-3] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[4] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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+In [[numerical analysis]], the '''Clenshaw algorithm'''<ref name="Clenshaw55">{{Cite doi|10.1090/S0025-5718-1955-0071856-0}}  Note that this paper is written in terms of the ''Shifted'' Chebyshev polynomials of the first kind <math>T^*_n(x) = T_n(2x-1)</math>.</ref> is a [[Recursion|recursive]] method to evaluate a linear combination of [[Chebyshev polynomials]]. It is a generalization of [[Horner's method]] for evaluating a linear combination of [[monomial]]s.
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+It generalizes to more than just Chebyshev polynomials; it applies to any class of functions that can be defined by a three-term [[recurrence relation]].<ref>{{Citation |last1=Press |first1=WH |last2=Teukolsky |first2=SA |last3=Vetterling |first3=WT |last4=Flannery |first4=BP |year=2007 |title=Numerical Recipes: The Art of Scientific Computing |edition=3rd |publisher=Cambridge University Press |publication-place=New York |isbn=978-0-521-88068-8 |chapter=Section 5.4.2. Clenshaw's Recurrence Formula |chapter-url=http://apps.nrbook.com/empanel/index.html?pg=222}}</ref>
+==Clenshaw algorithm==
+Suppose that <math>\phi_k,\; k=0, 1, \ldots</math> is a sequence of functions that satisfy the linear recurrence relation
+:<math>\phi_{k+1}(x) = \alpha_k(x)\,\phi_k(x) + \beta_k(x)\,\phi_{k-1}(x),</math>
+where the coefficients <math>\alpha_k</math> and <math>\beta_k</math> are known in advance.  Note that in the most common applications, <math>\alpha(x)</math> does not depend on <math>k</math>, and <math>\beta</math> is a constant that depends on neither <math>x</math> nor <math>k</math>.
+Our goal is to evaluate a weighted sum of these functions
+:<math>S(x) = \sum_{k=0}^n a_k \phi_k(x)</math>
+Given the coefficients <math>a_0, \ldots, a_n</math>, compute the values <math>b_k(x)</math> by the "reverse" recurrence formula:
+:<math>\begin{align}
+  b_{n+1}(x) &= b_{n+2}(x) = 0, \\[.5em]
+  b_{k}(x) &= a_k + \alpha_k(x)\,b_{k+1}(x) + \beta_{k+1}(x)\,b_{k+2}(x).
+\end{align}</math>
+The linear combination of the <math>\phi_k</math> satisfies:
+:<math>S(x) = \phi_0(x)a_0 + \phi_1(x)b_1(x) + \beta_1(x)\phi_0(x)b_2(x).</math>
+See Fox and Parker<ref name="FoxParker">{{Citation |author1=L. Fox |author2=I. B. Parker |title=Chebyshev Polynomials in Numerical Analysis |publisher=Oxford University Press |year=1968 |isbn=0-19-859614-6}}</ref> for more information and stability analyses.
+===Horner as a special case of Clenshaw===
+A particularly simple case occurs when evaluating a polynomial of the form
+:<math>S(x) = \sum_{k=0}^n a_k x^k</math>.
+The functions are simply
+:<math>\begin{align}
+  \phi_0(x) &= 1, \\
+  \phi_k(x) &= x^k = x\phi_{k-1}(x)
+\end{align}</math>
+and are produced by the recurrence coefficients <math>\alpha(x) = x</math> and <math>\beta = 0</math>.
+In this case, the recurrence formula to compute the sum is
+:<math>b_k(x) = a_k + x b_{k+1}(x)</math>
+and, in this case, the sum is simply
+:<math>S(x) = a_0 + x b_1(x) = b_0(x)</math>,
+which is exactly the usual [[Horner's method]].
+===Special case for Chebyshev series===
+Consider a truncated [[Chebyshev series]]
+:<math>p_n(x) = a_0 + a_1T_1(x) + a_2T_2(x) + \cdots + a_nT_n(x).</math>
+The coefficients in the recursion relation for the [[Chebyshev polynomials]] are
+:<math>\alpha(x) = 2x, \quad \beta = -1,</math>
+with the initial conditions
+:<math>T_0(x) = 1, \quad T_1(x) = x.</math>
+Thus, the recurrence is
+:<math>b_k(x) = a_k + 2xb_{k+1}(x) - b_{k+2}(x)</math>
+and the final sum is
+:<math>p_n(x) = a_0 + xb_1(x) - b_2(x).</math>
+One way to evaluate this is to continue the recurrence one more step, and compute
+:<math>b_0(x) = 2a_0 + 2xb_1(x) - b_2(x),</math>
+(note the doubled ''a''<sub>0</sub> coefficient) followed by
+:<math>p_n(x) = b_0(x)-x b_1(x)-a_0=\frac{1}{2}\left[b_0(x) - b_2(x)\right].</math>
+===Geodetic applications===
+Clenshaw's algorithm is extensively used in geodetic applications
+where it is usually referred to as '''Clenshaw summation'''.<ref>
+{{Citation
+| last1=Tscherning
+| first1=C. C.
+| last2=Poder
+| first2=K.
+| year=1982
+| title=Some Geodetic applications of Clenshaw Summation
+| journal=Bolletino di Geodesia e Scienze Affini
+| volume=41
+| number=4
+| pages=349–375
+| url=http://cct.gfy.ku.dk/publ_cct/cct80.pdf
+}}</ref>  A simple application is summing the trigonometric series to compute
+the [[meridian arc]].  These have the form
+:<math>m(\theta) = C_0\,\theta + C_1\sin \theta + C_2\sin 2\theta + \cdots + C_n\sin n\theta.</math>
+Leaving off the initial <math>C_0\,\theta</math> term, the remainder is a summation of the appropriate form. There is no leading term because <math>\phi_0(\theta) = \sin 0\theta = \sin 0 = 0</math>.
+The [[List of trigonometric identities#Chebyshev method|recurrence relation for <math>\sin k\theta</math>]] is
+:<math>\sin k\theta = 2 \cos\theta \sin (k-1)\theta - \sin (k-2)\theta</math>,
+making the coefficients in the recursion relation
+:<math>\alpha_k(\theta) = 2\cos\theta, \quad \beta_k = -1.</math>
+and the evaluation of the series is given by
+:<math>\begin{align}
+   b_{n+1}(\theta) &= b_{n+2}(\theta) = 0,\\[.3em]
+  b_k(\theta) &= C_k + 2 b_{k+1}(\theta)\cos \theta - b_{k+2}(\theta)\quad(n\ge k \ge 1).
+\end{align}</math>
+The final step is made particularly simple because <math>\phi_0(\theta) = \sin 0 = 0</math>, so the end of the recurrence is simply <math>b_1(\theta)\sin(\theta)</math>; the <math>C_0\,\theta</math> term is added separately:
+:<math>m(\theta) = C_0\,\theta + b_1(\theta)\sin \theta.</math>
+Note that the algorithm requires only the evaluation of two trigonometric quantities <math>\cos \theta</math> and <math>\sin \theta</math>.
+==See also==
+*[[Horner scheme]] to evaluate polynomials in [[monomial form]]
+*[[De Casteljau's algorithm]] to evaluate polynomials in [[Bézier form]]
+==References==
+<references/>
+{{DEFAULTSORT:Clenshaw Algorithm}}
+[[Category:Numerical analysis]]

Continuous phase modulation: Difference between revisions

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Contents

Clenshaw algorithm

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Special case for Chebyshev series

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Continuous phase modulation: Difference between revisions

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Clenshaw algorithm

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