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In [[numerical analysis]], the '''Crank–Nicolson method''' is a [[finite difference method]] used for numerically solving the [[heat equation]] and similar [[partial differential equations]].<ref>{{cite book | title = Convective Heat Transfer | author = Tuncer Cebeci | publisher = Springer | year = 2002 | isbn = 0-9668461-4-1 | url = http://books.google.com/?id=xfkgT9Fd4t4C&pg=PA257&dq=%22Crank-Nicolson+method%22 }}</ref> It is a [[Big O notation|second-order]] method in time, it is [[Explicit and implicit methods|implicit]] in time and can be written as an [[implicit Runge–Kutta method]], and it is [[Numerical stability|numerically stable]]. The method was developed by [[John Crank]] and [[Phyllis Nicolson]] in the mid 20th century.<ref>{{Cite journal
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| title = A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type
| journal = Proc. Camb. Phil. Soc.
| volume = 43
| issue = 1
| year = 1947
| pages = 50&ndash;67
| doi = 10.1007/BF02127704
| last1 = Crank
| first1 = J.
| last2 = Nicolson
| first2 = P.
| postscript = .
}}.</ref>
 
For diffusion equations (and many other equations), it can be shown the Crank–Nicolson method is unconditionally [[Numerical stability|stable]].<ref>{{Cite book | last1=Thomas | first1=J. W. | title=Numerical Partial Differential Equations: Finite Difference Methods | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Texts in Applied Mathematics | isbn=978-0-387-97999-1 | year=1995 |volume=22 | postscript=. }}. Example 3.3.2 shows that Crank–Nicolson is unconditionally stable when applied to <math>u_t=au_{xx}</math>.</ref> However, the approximate solutions can still contain (decaying) spurious oscillations if the ratio of time step &Delta;{{var|t}} times the [[thermal diffusivity]] to the square of space step, &Delta;{{var|x}}{{sup|2}}, is large (typically larger than 1/2 per [[Von Neumann stability analysis]]). For this reason, whenever large time steps or high spatial resolution is necessary, the less accurate [[backward Euler method]] is often used, which is both stable and immune to oscillations.
 
==The method==
[[File:Crank-Nicolson-stencil.svg|thumb|200px|right|The Crank–Nicolson stencil for a 1D problem.]]
 
The Crank–Nicolson method is based on [[central difference]] in space, and the [[trapezoidal rule (differential equations)|trapezoidal rule]] in time, giving second-order convergence in time. For example, in one dimension, if the [[partial differential equation]] is
 
:<math>\frac{\partial u}{\partial t} = F\left(u,\, x,\, t,\, \frac{\partial u}{\partial x},\, \frac{\partial^2 u}{\partial x^2}\right)</math>
 
then, letting <math>u(i \Delta x,\, n \Delta t) = u_{i}^{n}\,</math>, the equation for Crank–Nicolson method is a combination of the [[forward Euler method]] at <math>n</math> and the [[backward Euler method]] at ''n''&nbsp;+&nbsp;1 (note, however, that the method itself is ''not'' simply the average of those two methods, as the equation has an implicit dependence on the solution):
 
:<math>\frac{u_{i}^{n + 1} - u_{i}^{n}}{\Delta t} =
F_{i}^{n}\left(u,\, x,\, t,\, \frac{\partial u}{\partial x},\, \frac{\partial^2 u}{\partial x^2}\right) \qquad \mbox{(forward Euler)}</math>
 
:<math>\frac{u_{i}^{n + 1} - u_{i}^{n}}{\Delta t} =
F_{i}^{n + 1}\left(u,\, x,\, t,\, \frac{\partial u}{\partial x},\, \frac{\partial^2 u}{\partial x^2}\right) \qquad \mbox{(backward Euler)}</math>
 
:<math>\frac{u_{i}^{n + 1} - u_{i}^{n}}{\Delta t} =
\frac{1}{2}\left[
F_{i}^{n + 1}\left(u,\, x,\, t,\, \frac{\partial u}{\partial x},\, \frac{\partial^2 u}{\partial x^2}\right) +
F_{i}^{n}\left(u,\, x,\, t,\, \frac{\partial u}{\partial x},\, \frac{\partial^2 u}{\partial x^2}\right)
\right] \qquad \mbox{(Crank-Nicolson)}.</math>
 
The function ''F'' must be discretized spatially with a [[central difference]].
 
Note that this is an ''implicit method'': to get the "next" value of ''u'' in time and that a system of algebraic equations must be solved. If the partial differential equation is nonlinear, the [[temporal discretization|discretization]] will also be nonlinear so that advancing in time will involve the solution of a system of nonlinear algebraic equations, though linearizations are possible. In many problems, especially linear diffusion, the algebraic problem is [[tridiagonal]] and may be efficiently solved with the [[tridiagonal matrix algorithm]], which gives a fast <math>\mathcal{O}(n)</math> direct solution as opposed to the usual <math>\mathcal{O}(n^3)</math> for a full matrix.
 
==Example: 1D diffusion==
 
The Crank–Nicolson method is often applied to [[Diffusion equation|diffusion problems]]. As an example, for linear diffusion,
 
:<math>{\partial u \over \partial t} = a \frac{\partial^2 u}{\partial x^2}</math>
 
whose Crank–Nicolson discretization is then:
 
:<math>\frac{u_{i}^{n + 1} - u_{i}^{n}}{\Delta t} = \frac{a}{2 (\Delta x)^2}\left(
(u_{i + 1}^{n + 1} - 2 u_{i}^{n + 1} + u_{i - 1}^{n + 1}) +
(u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n})
\right)</math>
 
or, letting <math>r = \frac{a \Delta t}{2 (\Delta x)^2}</math>:
 
:<math>-r u_{i + 1}^{n + 1} + (1 + 2 r)u_{i}^{n + 1} - r u_{i - 1}^{n + 1} = r u_{i + 1}^{n} + (1 - 2 r)u_{i}^{n} + r u_{i - 1}^{n}\,</math>
 
which is a [[tridiagonal]] problem, so that <math>u_{i}^{n + 1}\,</math> may be efficiently solved by using the [[tridiagonal matrix algorithm]] in favor of a much more costly [[matrix inversion]].
 
A quasilinear equation, such as (this is a minimalistic example and not general)
 
:<math>\frac{\partial u}{\partial t} = a(u) \frac{\partial^2 u}{\partial x^2}</math>
 
would lead to a nonlinear system of algebraic equations which could not be easily solved as above; however, it is possible in some cases to linearize the problem by using the old value for <math>a</math>, that is <math>a_{i}^{n}(u)\,</math> instead of <math>a_{i}^{n + 1}(u)\,</math>. Other times, it may be possible to estimate <math>a_{i}^{n + 1}(u)\,</math> using an explicit method and maintain stability.
 
==Example: 1D diffusion with advection for steady flow, with multiple channel connections==
This is a solution usually employed for many purposes when there's a contamination problem in streams or rivers under steady flow conditions but information is given in one dimension only. Often the problem can be simplified into a 1-dimensional problem and still yield useful information.
 
Here we model the concentration of a solute contaminant in water. This problem is composed of three parts: the known diffusion equation (<math>D_x</math> chosen as constant), an advective component (which means the system is evolving in space due to a velocity field), which we choose to be a constant ''Ux'', and a lateral interaction between longitudinal channels (k).
 
{{NumBlk|:|<math>\frac{\partial C}{\partial t} = D_x \frac{\partial^2 C}{\partial x^2} - U_x \frac{\partial C}{\partial x}- k (C-C_N)-k(C-C_M)</math>|{{EquationRef|1}}}}
 
where ''C'' is the concentration of the contaminant and subscripts ''N'' and ''M'' correspond to ''previous'' and ''next'' channel.
 
The Crank–Nicolson method (where ''i'' represents position and ''j'' time) transforms each component of the PDE into the following:
 
{{NumBlk|:|<math>\frac{\partial C}{\partial t} = \frac{C_{i}^{j + 1} - C_{i}^{j}}{\Delta t}</math>|{{EquationRef|2}}}}
 
{{NumBlk|:|<math>\frac{\partial^2 C}{\partial x^2}= \frac{1}{2 (\Delta x)^2}\left(
(C_{i + 1}^{j + 1} - 2 C_{i}^{j + 1} + C_{i - 1}^{j + 1}) +
(C_{i + 1}^{j} - 2 C_{i}^{j} + C_{i - 1}^{j})
\right)</math>|{{EquationRef|3}}}}
 
{{NumBlk|:|<math>\frac{\partial C}{\partial x} = \frac{1}{2}\left(
\frac{(C_{i + 1}^{j + 1} - C_{i - 1}^{j + 1})}{2 (\Delta x)} +
  \frac{(C_{i + 1}^{j} - C_{i - 1}^{j})}{2 (\Delta x)}
\right)</math>|{{EquationRef|4}}}}
 
{{numBlk|:|<math>C= \frac{1}{2} (C_{i}^{j+1} + C_{i}^{j})</math>|{{EquationRef|5}}}}
 
{{NumBlk|:|<math>C_N= \frac{1}{2} (C_{Ni}^{j+1} + C_{Ni}^{j})</math>|{{EquationRef|6}}}}
 
{{numBlk|:|<math>C_M= \frac{1}{2} (C_{Mi}^{j+1} + C_{Mi}^{j}).</math>|{{EquationRef|7}}}}
 
Now we create the following constants to simplify the algebra:
 
:<math> \lambda= \frac{D_x\Delta t}{2 \Delta x^2}</math>
 
:<math> \alpha= \frac{U_x\Delta t}{4 \Delta x}</math>
 
:<math> \beta= \frac{k\Delta t}{2}</math>
 
and substitute ({{EquationNote|2}}), ({{EquationNote|3}}), ({{EquationNote|4}}), ({{EquationNote|5}}), ({{EquationNote|6}}), ({{EquationNote|7}}), ''α'', ''β'' and ''λ'' into ({{EquationNote|1}}). We then put the ''new time'' terms on the left (''j''&nbsp;+&nbsp;1) and the ''present time'' terms on the right (''j'') to get:
 
:<math> -\beta C_{Ni}^{j+1}-(\lambda+\alpha)C_{i-1}^{j+1} +(1+2\lambda+2\beta)C_{i}^{j+1}-(\lambda-\alpha)C_{i+1}^{j+1}-\beta C_{Mi}^{j+1} = \beta C_{Ni}^{j}+(\lambda+\alpha)C_{i-1}^{j} +(1-2\lambda-2\beta)C_{i}^{j}+(\lambda-\alpha)C_{i+1}^{j}+\beta C_{Mi}^{j}.</math>
 
To model the ''first'' channel, we realize that it can only be in contact with the following channel (''M''), so the expression is simplified to:
 
:<math> -(\lambda+\alpha)C_{i-1}^{j+1} +(1+2\lambda+\beta)C_{i}^{j+1}-(\lambda-\alpha)C_{i+1}^{j+1}-\beta C_{Mi}^{j+1} = +(\lambda+\alpha)C_{i-1}^{j} +(1-2\lambda-\beta)C_{i}^{j}+(\lambda-\alpha)C_{i+1}^{j}+\beta C_{Mi}^{j}.</math>
 
In the same way, to model the ''last'' channel, we realize that it can only be in contact with the previous channel (''N''), so the expression is simplified to:
 
:<math> -\beta C_{Ni}^{j+1}-(\lambda+\alpha)C_{i-1}^{j+1} +(1+2\lambda+\beta)C_{i}^{j+1}-(\lambda-\alpha)C_{i+1}^{j+1}= \beta C_{Ni}^{j}+(\lambda+\alpha)C_{i-1}^{j} +(1-2\lambda-\beta)C_{i}^{j}+(\lambda-\alpha)C_{i+1}^{j}.</math>
 
To solve this linear system of equations we must now see that boundary conditions must be given first to the beginning of the channels:
 
<math> C_0^{j}</math>: initial condition for the channel at present time step <br />
<math> C_{0}^{j+1}</math>: initial condition for the channel at next time step <br />
<math> C_{N0}^{j}</math>: initial condition for the previous channel to the one analyzed at present time step <br />
<math> C_{M0}^{j}</math>: initial condition for the next channel to the one analyzed at present time step.  
 
For the last cell of the channels (''z'') the most convenient condition becomes an adiabatic one, so
 
:<math>\frac{\partial C}{\partial x}_{x=z} =
\frac{(C_{i + 1} - C_{i - 1})}{2 \Delta x}  = 0.</math>
 
This condition is satisfied if and only if (regardless of a null value)
 
:<math> C_{i + 1}^{j+1} = C_{i - 1}^{j+1}. \, </math>
 
Let us solve this problem (in a matrix form) for the case of 3 channels and 5 nodes (including the initial boundary condition). We express this as a linear system problem:
 
:<math> \begin{bmatrix}AA\end{bmatrix}\begin{bmatrix}C^{j+1}\end{bmatrix}=[BB][C^{j}]+[d]</math>
 
where
 
:<math> \mathbf{C^{j+1}} = \begin{bmatrix}
C_{11}^{j+1}\\ C_{12}^{j+1} \\ C_{13}^{j+1} \\ C_{14}^{j+1}
\\ C_{21}^{j+1}\\ C_{22}^{j+1} \\ C_{23}^{j+1} \\ C_{24}^{j+1}
\\ C_{31}^{j+1}\\ C_{32}^{j+1} \\ C_{33}^{j+1} \\ C_{34}^{j+1}
\end{bmatrix}</math> &nbsp; and &nbsp;  <math>\mathbf{C^{j}} = \begin{bmatrix}
C_{11}^{j}\\ C_{12}^{j} \\ C_{13}^{j} \\ C_{14}^{j}
\\ C_{21}^{j}\\ C_{22}^{j} \\ C_{23}^{j} \\ C_{24}^{j}
\\ C_{31}^{j}\\ C_{32}^{j} \\ C_{33}^{j} \\ C_{34}^{j}
\end{bmatrix}.</math>
 
Now we must realize that ''AA'' and ''BB'' should be arrays made of four different subarrays (remember that only three channels are considered for this example but it covers the main part discussed above).
 
: <math>\mathbf{AA} = \begin{bmatrix}
AA1 & AA3 & 0\\
AA3 & AA2 & AA3\\
0 & AA3 & AA1\end{bmatrix}</math> &nbsp; and &nbsp;
 
: <math>\mathbf{BB} = \begin{bmatrix}
BB1 & -AA3 & 0\\
-AA3 & BB2 & -AA3\\
0 & -AA3 & BB1\end{bmatrix}</math> &nbsp;
 
where the elements mentioned above correspond to the next arrays and an additional 4x4 full of zeros. Please note that the sizes of AA and BB are 12x12:
 
: <math>\mathbf{AA1} = \begin{bmatrix}
(1+2\lambda+\beta) & -(\lambda-\alpha) & 0 & 0 \\
-(\lambda+\alpha) & (1+2\lambda+\beta) & -(\lambda-\alpha) & 0 \\
0 & -(\lambda+\alpha) & (1+2\lambda+\beta) & -(\lambda-\alpha)\\
0 & 0 & -2\lambda & (1+2\lambda+\beta)\end{bmatrix}</math> &nbsp; ,&nbsp;
 
: <math>\mathbf{AA2} = \begin{bmatrix}
(1+2\lambda+2\beta) & -(\lambda-\alpha) & 0 & 0 \\
-(\lambda+\alpha) & (1+2\lambda+2\beta) & -(\lambda-\alpha) & 0 \\
0 & -(\lambda+\alpha) & (1+2\lambda+2\beta) & -(\lambda-\alpha)\\
0 & 0 & -2\lambda & (1+2\lambda+2\beta) \end{bmatrix}</math> &nbsp; ,&nbsp;
 
: <math>\mathbf{AA3} = \begin{bmatrix}
-\beta & 0 & 0 & 0 \\
0 & -\beta & 0 & 0 \\
0 & 0 & -\beta & 0 \\
0 & 0 & 0 & -\beta \end{bmatrix}</math> &nbsp; ,&nbsp;
 
: <math>\mathbf{BB1} = \begin{bmatrix}
(1-2\lambda-\beta) & (\lambda-\alpha) & 0 & 0 \\
(\lambda+\alpha) & (1-2\lambda-\beta) & (\lambda-\alpha) & 0 \\
0 & (\lambda+\alpha) & (1-2\lambda-\beta) & (\lambda-\alpha)\\
0 & 0 & 2\lambda & (1-2\lambda-\beta)\end{bmatrix}</math> &nbsp; & &nbsp;
 
: <math>\mathbf{BB2} = \begin{bmatrix}
(1-2\lambda-2\beta) & (\lambda-\alpha) & 0 & 0 \\
(\lambda+\alpha) & (1-2\lambda-2\beta) & (\lambda-\alpha) & 0 \\
0 & (\lambda+\alpha) & (1-2\lambda-2\beta) & (\lambda-\alpha)\\
0 & 0 & 2\lambda & (1-2\lambda-2\beta) \end{bmatrix}.</math>
 
The ''d'' vector here is used to hold the boundary conditions. In this example it is a 12x1 vector:
 
: <math>\mathbf{d} = \begin{bmatrix}
(\lambda+\alpha)(C_{10}^{j+1}+C_{10}^{j}) \\ 0 \\ 0 \\ 0 \\ (\lambda+\alpha)(C_{20}^{j+1}+C_{20}^{j}) \\ 0 \\ 0 \\ 0 \\ (\lambda+\alpha)(C_{30}^{j+1}+C_{30}^{j}) \\
0\\
0\\
0\end{bmatrix}.</math>
 
To find the concentration at any time, one must iterate the following equation:
:<math> \begin{bmatrix}C^{j+1}\end{bmatrix}=\begin{bmatrix}AA^{-1}\end{bmatrix}([BB][C^{j}]+[d]).</math>
 
==Example: 2D diffusion==
When extending into two dimensions on a uniform [[Cartesian grid]], the derivation is similar and the results may lead to a system of [[Banded matrix|band-diagonal]] equations rather than [[Tridiagonal matrix|tridiagonal]] ones. The two-dimensional heat equation
 
:<math>\frac{\partial u}{\partial t} = a \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right)</math>
 
can be solved with the Crank–Nicolson discretization of
 
:<math>\begin{align}u_{i,j}^{n+1} &= u_{i,j}^n + \frac{1}{2} \frac{a \Delta t}{(\Delta x)^2} \big[(u_{i+1,j}^{n+1} + u_{i-1,j}^{n+1} + u_{i,j+1}^{n+1} + u_{i,j-1}^{n+1} - 4u_{i,j}^{n+1}) \\ & \qquad {} + (u_{i+1,j}^{n} + u_{i-1,j}^{n} + u_{i,j+1}^{n} + u_{i,j-1}^{n} - 4u_{i,j}^{n})\big]\end{align}</math>
 
assuming that a square grid is used so that <math>\Delta x = \Delta y</math>. This equation can be simplified somewhat by rearranging terms and using the [[Courant number|CFL number]]
 
:<math>\mu = \frac{a \Delta t}{(\Delta x)^2}.</math>
 
For the Crank–Nicolson numerical scheme, a low [[Courant number|CFL number]] is not required for stability, however it is required for numerical accuracy. We can now write the scheme as:
 
:<math>\begin{align}&(1 + 2\mu)u_{i,j}^{n+1} - \frac{\mu}{2}\left(u_{i+1,j}^{n+1} + u_{i-1,j}^{n+1} + u_{i,j+1}^{n+1} + u_{i,j-1}^{n+1}\right) \\ & \quad = (1 - 2\mu)u_{i,j}^{n} + \frac{\mu}{2}\left(u_{i+1,j}^{n} + u_{i-1,j}^{n} + u_{i,j+1}^{n} + u_{i,j-1}^{n}\right).\end{align}</math>
 
==Application in financial mathematics==
{{Further|Finite difference methods for option pricing}}
Because a number of other phenomena can be [[Mathematical model|modeled]] with the [[heat equation]] (often called the diffusion equation in [[financial mathematics]]), the Crank–Nicolson method has been applied to those areas as well.<ref>{{cite book | title = The Mathematics of Financial Derivatives: A Student Introduction | author1 = Wilmott, P. | author2 = Howison, S. | author3 = Dewynne, J. | publisher = Cambridge Univ. Press | year = 1995 | isbn = 0-521-49789-2 | url = http://books.google.co.in/books?hl=en&q=The%20Mathematics%20of%20Financial%20Derivatives%20Wilmott&um=1&ie=UTF-8&sa=N&tab=wp }}</ref> Particularly, the [[Black–Scholes]] option pricing model's [[differential equation]] can be transformed into the heat equation, and thus  [[numerical methods|numerical solutions]] for [[Valuation of options|option pricing]] can be obtained with the Crank–Nicolson method.
 
The importance of this for finance, is that option pricing problems, when extended beyond the standard assumptions (e.g. incorporating changing dividends), cannot be solved in closed form, but can be solved using this method.  Note however, that for non-smooth final conditions (which happen for most financial instruments), the Crank–Nicolson method is not satisfactory as numerical oscillations are not damped.  For [[vanilla option]]s, this results in oscillation in the [[Greeks_(finance)#Gamma|gamma value]] around the [[strike price]].  Therefore, special damping initialization steps are necessary (e.g., fully implicit finite difference method).
 
==See also==
*[[Financial mathematics]]
*[[Trapezoidal rule (differential equations)]]
 
==References==
 
<references/>
<!-- *Fitzpatrick R. (2003) ''[http://farside.ph.utexas.edu/~rfitzp/teaching/329/lectures/node80.html The Crank–Nicolson scheme]''. Retrieved May 4, 2005.
*Pitman E. Bruce. (1999) ''[http://www.math.buffalo.edu/~pitman/courses/mth438/na/node16.html Parabolic equations]''. Retrieved May 4, 2005. -->
 
==External links==
*[http://math.fullerton.edu/mathews/n2003/CrankNicolsonMod.html Module for Parabolic P.D.E.'s]
 
*[http://www3.nd.edu/~dbalsara/Numerical-PDE-Course/ Numerical PDE Techniques for Scientists and Engineers], open access Lectures and Codes for Numerical PDEs
 
*[http://scicomp.stackexchange.com/questions/7399/how-to-discretize-the-advection-equation-using-the-crank-nicolson-methods An example of how to apply and implement the Crank-Nicolson method for the Advection equation]
 
 
{{Numerical integrators}}
 
{{Numerical PDE}}
 
{{DEFAULTSORT:Crank-Nicolson Method}}
[[Category:Mathematical finance]]
[[Category:Numerical differential equations]]

Latest revision as of 19:05, 31 October 2014

But even in case you are not a PR you are entitled to buy non-landed property as Singapore is a really liberal market in comparison with other countries within the area. Non-landed refers to properties corresponding to flats and condominiums

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On account of restricted number of such properties, this category won't ever be in over-provide. Subsequently, costs needs to be reasonably properly-supported even with uncertain economic situations. The corporate arrange a manufacturing division located at Citimac Industrial Complicated. Aspial Company expanded its portfolio of businesses to offer pawnbroking providers through its wholly-owned subsidiary, Maxi-Money Group Pte. Ltd. Maxi-Money pledges its ongoing company social duty efforts by launching fourth consecutive 12 months of charity drive with the proceeds going to the Society for the Physically Disabled (SPD). Capitaland has US$11.96bn net value or accounts for a fifth of Singapore's market. Real Property Appraisers/ Valuers Actual Property Trade-Related Professionals

We use cookies to make sure that we give you the finest expertise on our website. Should you proceed with out changing your browser settings, we'll assume that you are completely satisfied to obtain all cookies on the Knight Frank web site. Find out more about how Knight Frank makes use of cookies Completed service apartments in singapore (pop over to this website) April 2009, The Coast boasts 249 units of three- and four-bed room residences as well as luxury penthouses, forty one personal berthing services and views of the South China Sea. It is one in all Ho Bee's 5 developments on the exclusive enclave of Sentosa Cove.

The Singapore Property Awards recognise excellence in actual property development projects or individual properties by way of design, aesthetics, performance, contribution to the built setting and neighborhood at large. It represents an excellent achievement which developers, professionals and property owners aspire to realize. It bestows upon the winner the correct to make use of the coveted award logo recognised extensively throughout the FIABCI network.

MY MANHATTAN PERSONAL CONDOMINIUM APARTMENT SIMEI AVENUE 3, SINGAPORE (DISTRICT 18) NATURA @ HILLVIEW NON-PUBLIC CONDOMINIUM HOUSE HILLVIEW TERRACE, SINGAPORE (DISTRICT 23) NATURALIS NON-PUBLIC CONDOMINIUM CONDO TELOK KURAU LORONG M, SINGAPORE (DISTRICT 15) NEWTON IMPERIAL (PREPARED HOMES) NON-PUBLIC CONDOMINIUM HOUSE NEWTON HIGHWAY, SINGAPORE (DISTRICT eleven) NOTTINGHILL SUITES PERSONAL CONDOMINIUM CONDO TOH TUCK STREET, SINGAPORE (DISTRICT 21) ONE ROSYTH PERSONAL CONDOMINIUM RESIDENCE ROSYTH STREET, SINGAPORE (DISTRICT 19) ONE SHENTON (PREPARED HOUSES) PRIVATE CONDOMINIUM RESIDENCE SHENTON METHOD, SINGAPORE (DISTRICT 01) OXLEY BIZHUB INDUSTRIAL ENTERPRISE SPACE (B1, FACTORY, WAREHOUSE) UBI HIGHWAY 1, PAYA LEBAR ROAD, SINGAPORE (DISTRICT 14) Killiney Street, Singapore 239519 Map

Personal residential properties investment will likely be thought-about for utility for Everlasting Resident application. A foreigner will be considered for PR status if he invests at the very least S$2 million in enterprise set-ups, different investment vehicles corresponding to venture capital funds, foundations or trusts, and/or private residential properties. Up to 50% of the funding can be in non-public residential properties, topic to international ownership restrictions below the Residential Property Act (RPA). This is to attract and anchor international talent in Singapore.

GPS Commercial department have an extensive expertise in gross sales and leasing of commercial properties similar to coffeeshop, foodcourt, family restaurant, clubs, retail outlets and industrial building like warehouse, light industrial buildings. The unique traits of the division is their road wise advertising model which has helped business purchasers to revenue of their business property investment. We provide a full range of services which include market entry analysis, tendering, feasibility examine and turn key initiatives and liaising with all government bodies for licensing and approval. Resale Department