Induced metric

In numerical analysis, Brent's method is a complicated but popular root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less reliable methods. The algorithm tries to use the potentially fast-converging secant method or inverse quadratic interpolation if possible, but it falls back to the more robust bisection method if necessary. Brent's method is due to Richard Brent^[1] and builds on an earlier algorithm by Theodorus Dekker.^[2] Consequently, the method is also known as Brent-Dekker.

Dekker's method

The idea to combine the bisection method with the secant method goes back to Template:Harvtxt.

Suppose that we want to solve the equation f(x) = 0. As with the bisection method, we need to initialize Dekker's method with two points, say a₀ and b₀, such that f(a₀) and f(b₀) have opposite signs. If f is continuous on [a₀, b₀], the intermediate value theorem guarantees the existence of a solution between a₀ and b₀.

Three points are involved in every iteration:

b_k is the current iterate, i.e., the current guess for the root of f.
a_k is the "contrapoint," i.e., a point such that f(a_k) and f(b_k) have opposite signs, so the interval [a_k, b_k] contains the solution. Furthermore, |f(b_k)| should be less than or equal to |f(a_k)|, so that b_k is a better guess for the unknown solution than a_k.
b_k−1 is the previous iterate (for the first iteration, we set b_k−1 = a₀).

Two provisional values for the next iterate are computed. The first one is given by linear interpolation, also known as the secant method:

s={\begin{cases}b_{k}-{\frac {b_{k}-b_{k-1}}{f(b_{k})-f(b_{k-1})}}f(b_{k}),&{\mbox{if }}f(b_{k})\neq f(b_{k-1})\\m&{\mbox{otherwise }}\end{cases}}

and the second one is given by the bisection method

$m={\frac {a_{k}+b_{k}}{2}}.$ If the result of the secant method, s, lies strictly between b_k and m, then it becomes the next iterate (b_k+1 = s), otherwise the midpoint is used (b_k+1 = m).

Then, the value of the new contrapoint is chosen such that f(a_k+1) and f(b_k+1) have opposite signs. If f(a_k) and f(b_k+1) have opposite signs, then the contrapoint remains the same: a_k+1 = a_k. Otherwise, f(b_k+1) and f(b_k) have opposite signs, so the new contrapoint becomes a_k+1 = b_k.

Finally, if |f(a_k+1)| < |f(b_k+1)|, then a_k+1 is probably a better guess for the solution than b_k+1, and hence the values of a_k+1 and b_k+1 are exchanged.

This ends the description of a single iteration of Dekker's method.

Dekker's method performs well if the function f is reasonably well-behaved. However, there are circumstances in which every iteration employs the secant method, but the iterates b_k converge very slowly (in particular, |b_k − b_k−1| may be arbitrarily small). Dekker's method requires far more iterations than the bisection method in this case.

Template:Harvtxt proposed a small modification to avoid this problem. He inserted an additional test which must be satisfied before the result of the secant method is accepted as the next iterate. Two inequalities must be simultaneously satisfied:

given a specific numerical tolerance $\delta$ ,

if the previous step used the bisection method, the inequality

|\delta |<|b_{k}-b_{k-1}|

must hold to perform interpolation, otherwise the bisection method is performed and its result used for the next iteration.

If the previous step performed interpolation, then the inequality

|\delta |<|b_{k-1}-b_{k-2}|

is used instead to perform the next action (to choose) interpolation (when inequality is true) or bisection method

(when inequality is not true).

Also, if the previous step used the bisection method, the inequality

|s-b_{k}|<{\begin{matrix}{\frac {1}{2}}\end{matrix}}|b_{k}-b_{k-1}|

must hold, otherwise the bisection method is performed and its result used for the next iteration.

If the previous step performed interpolation, then the inequality

|s-b_{k}|<{\begin{matrix}{\frac {1}{2}}\end{matrix}}|b_{k-1}-b_{k-2}|

is used instead.

This modification ensures that at the k^th iteration, a bisection step will be performed in at most $2\log _{2}(|b_{k-1}-b_{k-2}|/\delta )$ additional iterations, because the above conditions force consecutive interpolation step sizes to halve every two iterations, and after at most $2\log _{2}(|b_{k-1}-b_{k-2}|/\delta )$ iterations, the step size will be smaller than $\delta$ , which invokes a bisection step. Brent proved that his method requires at most N² iterations, where N denotes the number of iterations for the bisection method. If the function f is well-behaved, then Brent's method will usually proceed by either inverse quadratic or linear interpolation, in which case it will converge superlinearly.

Furthermore, Brent's method uses inverse quadratic interpolation instead of linear interpolation (as used by the secant method) .If f(b_k), f(a_k) and f(b_k−1) are distinct, it slightly increases the efficiency. As a consequence, the condition for accepting s (the value proposed by either linear interpolation or inverse quadratic interpolation) has to be changed: s has to lie between (3a_k + b_k) / 4 and b_k.

Algorithm

input a, b, and (a pointer to) a function for f
calculate f(a)
calculate f(b)
if f(a) f(b) >= 0 then exit function because the root is not bracketed.
if |f(a)| < |f(b)| then swap (a,b) end if
c := a
set mflag
repeat until f(b or s) = 0 or |b − a| is small enough (convergence)
  if f(a) ≠ f(c) and f(b) ≠ f(c) then
     $s:={\frac {af(b)f(c)}{(f(a)-f(b))(f(a)-f(c))}}+{\frac {bf(a)f(c)}{(f(b)-f(a))(f(b)-f(c))}}+{\frac {cf(a)f(b)}{(f(c)-f(a))(f(c)-f(b))}}$  (inverse quadratic interpolation)
  else
     $s:=b-f(b){\frac {b-a}{f(b)-f(a)}}$  (secant rule)
  end if
  if (condition 1) s is not between  ${\frac {3a+b}{4}}$  and  $b$  or
     (condition 2) (mflag is set and |Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.|/2) or
     (condition 3) (mflag is cleared and |Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.| ≥ |Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.|/2) or
     (condition 4) (mflag is set and |Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.| < |Template:Mvar|) or
     (condition 5) (mflag is cleared and |Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.| < |Template:Mvar|)
  then
     $s:={\frac {a+b}{2}}$  (bisection method)
    set mflag
  else
    clear mflag
  end if
  calculate f(s)
  d := c  (d is assigned for the first time here; it won't be used above on the first iteration because mflag is set)
  c := b
  if f(a) f(s) < 0 then b := s else a := s end if
  if |f(a)| < |f(b)| then swap (a,b) end if
end repeat
output b or s (return the root)

Example

Suppose that we are seeking a zero of the function defined by f(x) = (x + 3)(x − 1)². We take [a₀, b₀] = [−4, 4/3] as our initial interval. We have f(a₀) = −25 and f(b₀) = 0.48148 (all numbers in this section are rounded), so the conditions f(a₀) f(b₀) < 0 and |f(b₀)| ≤ |f(a₀)| are satisfied.

In the first iteration, we use linear interpolation between (b₋₁, f(b₋₁)) = (a₀, f(a₀)) = (−4, −25) and (b₀, f(b₀)) = (1.33333, 0.48148), which yields s = 1.23256. This lies between (3a₀ + b₀) / 4 and b₀, so this value is accepted. Furthermore, f(1.23256) = 0.22891, so we set a₁ = a₀ and b₁ = s = 1.23256.
In the second iteration, we use inverse quadratic interpolation between (a₁, f(a₁)) = (−4, −25) and (b₀, f(b₀)) = (1.33333, 0.48148) and (b₁, f(b₁)) = (1.23256, 0.22891). This yields 1.14205, which lies between (3a₁ + b₁) / 4 and b₁. Furthermore, the inequality |1.14205 − b₁| ≤ |b₀ − b₋₁| / 2 is satisfied, so this value is accepted. Furthermore, f(1.14205) = 0.083582, so we set a₂ = a₁ and b₂ = 1.14205.
In the third iteration, we use inverse quadratic interpolation between (a₂, f(a₂)) = (−4, −25) and (b₁, f(b₁)) = (1.23256, 0.22891) and (b₂, f(b₂)) = (1.14205, 0.083582). This yields 1.09032, which lies between (3a₂ + b₂) / 4 and b₂. But here Brent's additional condition kicks in: the inequality |1.09032 − b₂| ≤ |b₁ − b₀| / 2 is not satisfied, so this value is rejected. Instead, the midpoint m = −1.42897 of the interval [a₂, b₂] is computed. We have f(m) = 9.26891, so we set a₃ = a₂ and b₃ = −1.42897.
In the fourth iteration, we use inverse quadratic interpolation between (a₃, f(a₃)) = (−4, −25) and (b₂, f(b₂)) = (1.14205, 0.083582) and (b₃, f(b₃)) = (−1.42897, 9.26891). This yields 1.15448, which is not in the interval between (3a₃ + b₃) / 4 and b₃). Hence, it is replaced by the midpoint m = −2.71449. We have f(m) = 3.93934, so we set a₄ = a₃ and b₄ = −2.71449.
In the fifth iteration, inverse quadratic interpolation yields −3.45500, which lies in the required interval. However, the previous iteration was a bisection step, so the inequality |−3.45500 − b₄| ≤ |b₄ − b₃| / 2 need to be satisfied. This inequality is false, so we use the midpoint m = −3.35724. We have f(m) = −6.78239, so m becomes the new contrapoint (a₅ = −3.35724) and the iterate remains the same (b₅ = b₄).
In the sixth iteration, we cannot use inverse quadratic interpolation because b₅ = b₄. Hence, we use linear interpolation between (a₅, f(a₅)) = (−3.35724, −6.78239) and (b₅, f(b₅)) = (−2.71449, 3.93934). The result is s = −2.95064, which satisfies all the conditions. But since the iterate did not change in the previous step, we reject this result and fall back to bisection. We update s = -3.03587, and f(s) = -0.58418.
In the seventh iteration, we can again use inverse quadratic interpolation. The result is s = −3.00219, which satisfies all the conditions. Now, f(s) = −0.03515, so we set a₇ = b₆ and b₇ = −3.00219 (a₇ and b₇ are exchanged so that the condition |f(b₇)| ≤ |f(a₇)| is satisfied). (Correct : linear interpolation s = -2.99436, f(s) = 0.089961)
In the eighth iteration, we cannot use inverse quadratic interpolation because a₇ = b₆. Linear interpolation yields s = −2.99994, which is accepted. (Correct : s = -2.9999, f(s) = 0.0016)
In the following iterations, the root x = −3 is approached rapidly: b₉ = −3 + 6·10^{−8 and b₁₀ = −3 − 3·10^{−15. (Correct : Iter 9 : f(s) = -1.4E-07, Iter 10 : f(s) = 6.96E-12)}}

Example code

The above algorithm can be translated to C#-like code as follows:

        public static double BrentsMethodSolve(Func<double, double> function, double lowerLimit, double upperLimit, double errorTol)
        {
            double a = lowerLimit;
            double b = upperLimit;
            double c = 0;
            double d = double.MaxValue;

            double fa = function(a);
            double fb = function(b);

            double fc = 0;
            double s = 0;
            double fs = 0;

            // if f(a) f(b) >= 0 then error-exit
            if (fa * fb >= 0)
            {
                if (fa < fb)
                    return a;
                else
                    return b;
            }

            // if |f(a)| < |f(b)| then swap (a,b) end if
            if (Math.Abs(fa) < Math.Abs(fb))
            { double tmp = a; a = b; b = tmp; tmp = fa; fa = fb; fb = tmp; }

            c = a;
            fc = fa;
            bool mflag = true;
            int i = 0;

            while (!(fb==0) && (Math.Abs(a-b) > errorTol))
            {
                if ((fa != fc) && (fb != fc))
                    // Inverse quadratic interpolation
                    s = a * fb * fc / (fa - fb) / (fa - fc) + b * fa * fc / (fb - fa) / (fb - fc) + c * fa * fb / (fc - fa) / (fc - fb);
                else
                    // Secant Rule
                    s = b - fb * (b - a) / (fb - fa);

                double tmp2 = (3 * a + b) / 4;
                if ((!(((s > tmp2) && (s < b)) || ((s < tmp2) && (s > b)))) || (mflag && (Math.Abs(s - b) >= (Math.Abs(b - c) / 2))) || (!mflag && (Math.Abs(s - b) >= (Math.Abs(c - d) / 2))))
                {
                    s = (a + b) / 2;
                    mflag = true;
                }
                else
                {
                    if ((mflag && (Math.Abs(b - c) < errorTol)) || (!mflag && (Math.Abs(c - d) < errorTol)))
                    {
                        s = (a + b) / 2;
                        mflag = true;
                    }
                    else
                        mflag = false;
                }
                fs = function(s);
                d = c;
                c = b;
                fc = fb;
                if (fa * fs < 0) { b = s; fb = fs; }
                else { a = s; fa = fs; }

                // if |f(a)| < |f(b)| then swap (a,b) end if
                if (Math.Abs(fa) < Math.Abs(fb))
                { double tmp = a; a = b; b = tmp; tmp = fa; fa = fb; fb = tmp; }
                i++;
                if (i > 1000)
                    throw new Exception(String.Format("Error is {0}", fb));
            }
            return b;
        }

Implementations

Template:Harvtxt published an Algol 60 implementation.
Netlib contains a Fortran translation of this implementation with slight modifications.
The PARI/GP method solve implements the method.
Other implementations of the algorithm (in C++, C, and Fortran) can be found in the Numerical Recipes books.
The Apache Commons Math library implements the algorithm in Java.
The Scipy optimize module implements the algorithm in Python (programming language)
The Modelica Standard Library implements the algorithm in Modelica.
The optimize function implements the algorithm in R (software).

References

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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

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

External links

zeroin.f at Netlib.
Module for Brent's Method by John H. Mathews
module brent in C++ (also C, Fortran, Matlab) by John Burkardt
GSL implementation.

[1] Template:Harvnb

[2] Template:Harvnb

[1]

[2]

Induced metric

Contents

Dekker's method

Algorithm

Example

Example code

Implementations

References

Further reading

External links

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Induced metric

Dekker's method

Algorithm

Example

Example code

Implementations

References

Further reading

External links

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