Method of conditional probabilities

In numerical linear algebra, the conjugate gradient method is an iterative method for numerically solving the linear system

$\mathit{A}\mathit{x}=\mathit{b}$

where $\mathit{A}$ is symmetric positive-definite. The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems.

The intent of this article is to document the important steps in these derivations.

Derivation from the conjugate direction method

Template:Expand section The conjugate gradient method can be seen as a special case of the conjugate direction method applied to minimization of the quadratic function

$f(\mathit{x})={\mathit{x}}^{\mathrm{T}}\mathit{A}\mathit{x}-2{\mathit{b}}^{\mathrm{T}}\mathit{x}\text{.}$

The conjugate direction method

In the conjugate direction method for minimizing

$f(\mathit{x})={\mathit{x}}^{\mathrm{T}}\mathit{A}\mathit{x}-2{\mathit{b}}^{\mathrm{T}}\mathit{x}\text{.}$

one starts with an initial guess ${\mathit{x}}_0$ and the corresponding residual ${\mathit{r}}_0=\mathit{b}-{\mathit{A}\mathit{x}}_0$, and computes the iterate and residual by the formulae

$\begin{array}{rl}{\alpha }_i& =\frac{{\mathit{p}}_i^{\mathrm{T}}{\mathit{r}}_i}{{\mathit{p}}_i^{\mathrm{T}}{\mathit{A}\mathit{p}}_i}\text{,}\\ {\mathit{x}}_{i+1}& ={\mathit{x}}_i+{\alpha }_i{\mathit{p}}_i\text{,}\\ {\mathit{r}}_{i+1}& ={\mathit{r}}_i-{\alpha }_i{\mathit{A}\mathit{p}}_i\end{array}$

where ${\mathit{p}}_0,{\mathit{p}}_1,{\mathit{p}}_2,\dots$ are a series of mutually conjugate directions, i.e.,

${\mathit{p}}_i^{\mathrm{T}}{\mathit{A}\mathit{p}}_j=0$

for any $i\ne j$.

The conjugate direction method is imprecise in the sense that no formulae are given for selection of the directions ${\mathit{p}}_0,{\mathit{p}}_1,{\mathit{p}}_2,\dots$. Specific choices lead to various methods including the conjugate gradient method and Gaussian elimination.

Derivation from the Arnoldi/Lanczos iteration

Template:See The conjugate gradient method can also be seen as a variant of the Arnoldi/Lanczos iteration applied to solving linear systems.

The general Arnoldi method

In the Arnoldi iteration, one starts with a vector ${\mathit{r}}_0$ and gradually builds an orthonormal basis $\{{\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,\dots \}$ of the Krylov subspace

$\mathcal{𝒦}(\mathit{A},{\mathit{r}}_0)=\{{\mathit{r}}_0,{\mathit{A}\mathit{r}}_0,{\mathit{A}}^2{\mathit{r}}_0,\dots \}$

by defining ${\mathit{v}}_i={\mathit{w}}_i/\Vert {\mathit{w}}_i{\Vert }_2$ where

${\mathit{w}}_i=\{\begin{array}{ll}{\mathit{r}}_0& \text{if }i=1\text{,}\\ {\mathit{A}\mathit{v}}_{i-1}-\sum _{j=1}^{i-1}({\mathit{v}}_j^{\mathrm{T}}{\mathit{A}\mathit{v}}_{i-1}){\mathit{v}}_j& \text{if }i>1\text{.}\end{array}$

In other words, for $i>1$, ${\mathit{v}}_i$ is found by Gram-Schmidt orthogonalizing ${\mathit{A}\mathit{v}}_{i-1}$ against $\{{\mathit{v}}_1,{\mathit{v}}_2,\dots ,{\mathit{v}}_{i-1}\}$ followed by normalization.

Put in matrix form, the iteration is captured by the equation

${\mathit{A}\mathit{V}}_i={\mathit{V}}_{i+1}{\tilde{\mathit{H}}}_i$

where

$\begin{array}{rl}{\mathit{V}}_i& =\left[\begin{array}{cccc}{\mathit{v}}_1& {\mathit{v}}_2& \cdots & {\mathit{v}}_i\end{array}\right]\text{,}\\ {\tilde{\mathit{H}}}_i& =\left[\begin{array}{ccccc}{h}_{11}& h_{12}& h_{13}& \cdots & h_{1,i}\\ h_{21}& h_{22}& h_{23}& \cdots & h_{2,i}\\ & h_{32}& h_{33}& \cdots & h_{3,i}\\ & & \ddots & \ddots & ⋮\\ & & & h_{i,i-1}& h_{i,i}\\ & & & & h_{i+1,i}\end{array}\right]=\left[\begin{array}{c}{\mathit{H}}_i\\ h_{i+1,i}{\mathit{e}}_i^{\mathrm{T}}\end{array}\right]\end{array}$

with

$h_{ji}=\{\begin{array}{ll}{\mathit{v}}_j^{\mathrm{T}}{\mathit{A}\mathit{v}}_i& \text{if }j\le i\text{,}\\ \Vert {\mathit{w}}_{i+1}{\Vert }_2& \text{if }j=i+1\text{,}\\ 0& \text{if }j>i+1\text{.}\end{array}$

When applying the Arnoldi iteration to solving linear systems, one starts with ${\mathit{r}}_0=\mathit{b}-{\mathit{A}\mathit{x}}_0$, the residual corresponding to an initial guess ${\mathit{x}}_0$. After each step of iteration, one computes ${\mathit{y}}_i={\mathit{H}}_i^{-1}(\Vert {\mathit{r}}_0{\Vert }_2{\mathit{e}}_1)$ and the new iterate ${\mathit{x}}_i={\mathit{x}}_0+{\mathit{V}}_i{\mathit{y}}_i$.

The direct Lanczos method

For the rest of discussion, we assume that $\mathit{A}$ is symmetric positive-definite. With symmetry of $\mathit{A}$, the upper Hessenberg matrix ${\mathit{H}}_i={\mathit{V}}_i^{\mathrm{T}}{\mathit{A}\mathit{V}}_i$ becomes symmetric and thus tridiagonal. It then can be more clearly denoted by

${\mathit{H}}_i=\left[\begin{array}{cc}{a}_1& b_2\\ b_2& a_2& b_3\\ & \ddots & \ddots & \ddots \\ & & b_{i-1}& a_{i-1}& b_i\\ & & & b_i& a_i\end{array}\right]\text{.}$

This enables a short three-term recurrence for ${\mathit{v}}_i$ in the iteration, and the Arnoldi iteration is reduced to the Lanczos iteration.

Since $\mathit{A}$ is symmetric positive-definite, so is ${\mathit{H}}_i$. Hence, ${\mathit{H}}_i$ can be LU factorized without partial pivoting into

${\mathit{H}}_i={\mathit{L}}_i{\mathit{U}}_i=\left[\begin{array}{c}1\\ c_2& 1\\ & \ddots & \ddots \\ & & c_{i-1}& 1\\ & & & c_i& 1\end{array}\right]\left[\begin{array}{cc}{d}_1& b_2\\ & d_2& b_3\\ & & \ddots & \ddots \\ & & & d_{i-1}& b_i\\ & & & & d_i\end{array}\right]$

with convenient recurrences for $c_i$ and $d_i$:

$\begin{array}{rl}{c}_i& =b_i/d_{i-1}\text{,}\\ d_i& =\{\begin{array}{ll}{a}_1& \text{if }i=1\text{,}\\ a_i-c_i{b}_i& \text{if }i>1\text{.}\end{array}\end{array}$

Rewrite ${\mathit{x}}_i={\mathit{x}}_0+{\mathit{V}}_i{\mathit{y}}_i$ as

$\begin{array}{rl}{\mathit{x}}_i& ={\mathit{x}}_0+{\mathit{V}}_i{\mathit{H}}_i^{-1}(\Vert {\mathit{r}}_0{\Vert }_2{\mathit{e}}_1)\\ & ={\mathit{x}}_0+{\mathit{V}}_i{\mathit{U}}_i^{-1}{\mathit{L}}_i^{-1}(\Vert {\mathit{r}}_0{\Vert }_2{\mathit{e}}_1)\\ & ={\mathit{x}}_0+{\mathit{P}}_i{\mathit{z}}_i\end{array}$

with

$\begin{array}{rl}{\mathit{P}}_i& ={\mathit{V}}_i{\mathit{U}}_i^{-1}\text{,}\\ {\mathit{z}}_i& ={\mathit{L}}_i^{-1}(\Vert {\mathit{r}}_0{\Vert }_2{\mathit{e}}_1)\text{.}\end{array}$

It is now important to observe that

$\begin{array}{rl}{\mathit{P}}_i& =\left[\begin{array}{cc}{\mathit{P}}_{i-1}& {\mathit{p}}_i\end{array}\right]\text{,}\\ {\mathit{z}}_i& =\left[\begin{array}{c}{\mathit{z}}_{i-1}\\ {\zeta }_i\end{array}\right]\text{.}\end{array}$

In fact, there are short recurrences for ${\mathit{p}}_i$ and ${\zeta }_i$ as well:

$\begin{array}{rl}{\mathit{p}}_i& =\frac{1}{d_i}({\mathit{v}}_i-b_i{\mathit{p}}_{i-1})\text{,}\\ {\zeta }_i& =-c_i{\zeta }_{i-1}\text{.}\end{array}$

With this formulation, we arrive at a simple recurrence for ${\mathit{x}}_i$:

$\begin{array}{rl}{\mathit{x}}_i& ={\mathit{x}}_0+{\mathit{P}}_i{\mathit{z}}_i\\ & ={\mathit{x}}_0+{\mathit{P}}_{i-1}{\mathit{z}}_{i-1}+{\zeta }_i{\mathit{p}}_i\\ & ={\mathit{x}}_{i-1}+{\zeta }_i{\mathit{p}}_i\text{.}\end{array}$

The relations above straightforwardly lead to the direct Lanczos method, which turns out to be slightly more complex.

The conjugate gradient method from imposing orthogonality and conjugacy

If we allow ${\mathit{p}}_i$ to scale and compensate for the scaling in the constant factor, we potentially can have simpler recurrences of the form:

$\begin{array}{rl}{\mathit{x}}_i& ={\mathit{x}}_{i-1}+{\alpha }_{i-1}{\mathit{p}}_{i-1}\text{,}\\ {\mathit{r}}_i& ={\mathit{r}}_{i-1}-{\alpha }_{i-1}{\mathit{A}\mathit{p}}_{i-1}\text{,}\\ {\mathit{p}}_i& ={\mathit{r}}_i+{\beta }_{i-1}{\mathit{p}}_{i-1}\text{.}\end{array}$

As premises for the simplification, we now derive the orthogonality of ${\mathit{r}}_i$ and conjugacy of ${\mathit{p}}_i$, i.e., for $i\ne j$,

$\begin{array}{rl}{\mathit{r}}_i^{\mathrm{T}}{\mathit{r}}_j& =0\text{,}\\ {\mathit{p}}_i^{\mathrm{T}}{\mathit{A}\mathit{p}}_j& =0\text{.}\end{array}$

The residuals are mutually orthogonal because ${\mathit{r}}_i$ is essentially a multiple of ${\mathit{v}}_{i+1}$ since for $i=0$, ${\mathit{r}}_0=\Vert {\mathit{r}}_0{\Vert }_2{\mathit{v}}_1$, for $i>0$,

$\begin{array}{rl}{\mathit{r}}_i& =\mathit{b}-{\mathit{A}\mathit{x}}_i\\ & =\mathit{b}-\mathit{A}({\mathit{x}}_0+{\mathit{V}}_i{\mathit{y}}_i)\\ & ={\mathit{r}}_0-{\mathit{A}\mathit{V}}_i{\mathit{y}}_i\\ & ={\mathit{r}}_0-{\mathit{V}}_{i+1}{\tilde{\mathit{H}}}_i{\mathit{y}}_i\\ & ={\mathit{r}}_0-{\mathit{V}}_i{\mathit{H}}_i{\mathit{y}}_i-h_{i+1,i}({\mathit{e}}_i^{\mathrm{T}}{\mathit{y}}_i){\mathit{v}}_{i+1}\\ & =\Vert {\mathit{r}}_0{\Vert }_2{\mathit{v}}_1-{\mathit{V}}_i(\Vert {\mathit{r}}_0{\Vert }_2{\mathit{e}}_1)-h_{i+1,i}({\mathit{e}}_i^{\mathrm{T}}{\mathit{y}}_i){\mathit{v}}_{i+1}\\ & =-h_{i+1,i}({\mathit{e}}_i^{\mathrm{T}}{\mathit{y}}_i){\mathit{v}}_{i+1}\text{.}\end{array}$

To see the conjugacy of ${\mathit{p}}_i$, it suffices to show that ${\mathit{P}}_i^{\mathrm{T}}{\mathit{A}\mathit{P}}_i$ is diagonal:

$\begin{array}{rl}{\mathit{P}}_i^{\mathrm{T}}{\mathit{A}\mathit{P}}_i& ={\mathit{U}}_i^{-\mathrm{T}}{\mathit{V}}_i^{\mathrm{T}}{\mathit{A}\mathit{V}}_i{\mathit{U}}_i^{-1}\\ & ={\mathit{U}}_i^{-\mathrm{T}}{\mathit{H}}_i{\mathit{U}}_i^{-1}\\ & ={\mathit{U}}_i^{-\mathrm{T}}{\mathit{L}}_i{\mathit{U}}_i{\mathit{U}}_i^{-1}\\ & ={\mathit{U}}_i^{-\mathrm{T}}{\mathit{L}}_i\end{array}$

is symmetric and lower triangular simultaneously and thus must be diagonal.

Now we can derive the constant factors ${\alpha }_i$ and ${\beta }_i$ with respect to the scaled ${\mathit{p}}_i$ by solely imposing the orthogonality of ${\mathit{r}}_i$ and conjugacy of ${\mathit{p}}_i$.

Due to the orthogonality of ${\mathit{r}}_i$, it is necessary that ${\mathit{r}}_{i+1}^{\mathrm{T}}{\mathit{r}}_i=({\mathit{r}}_i-{\alpha }_i{\mathit{A}\mathit{p}}_i{)}^{\mathrm{T}}{\mathit{r}}_i=0$. As a result,

$\begin{array}{rl}{\alpha }_i& =\frac{{\mathit{r}}_i^{\mathrm{T}}{\mathit{r}}_i}{{\mathit{r}}_i^{\mathrm{T}}{\mathit{A}\mathit{p}}_i}\\ & =\frac{{\mathit{r}}_i^{\mathrm{T}}{\mathit{r}}_i}{({\mathit{p}}_i-{\beta }_{i-1}{\mathit{p}}_{i-1}{)}^{\mathrm{T}}{\mathit{A}\mathit{p}}_i}\\ & =\frac{{\mathit{r}}_i^{\mathrm{T}}{\mathit{r}}_i}{{\mathit{p}}_i^{\mathrm{T}}{\mathit{A}\mathit{p}}_i}\text{.}\end{array}$

Similarly, due to the conjugacy of ${\mathit{p}}_i$, it is necessary that ${\mathit{p}}_{i+1}^{\mathrm{T}}{\mathit{A}\mathit{p}}_i=({\mathit{r}}_{i+1}+{\beta }_i{\mathit{p}}_i{)}^{\mathrm{T}}{\mathit{A}\mathit{p}}_i=0$. As a result,

$\begin{array}{rl}{\beta }_i& =-\frac{{\mathit{r}}_{i+1}^{\mathrm{T}}{\mathit{A}\mathit{p}}_i}{{\mathit{p}}_i^{\mathrm{T}}{\mathit{A}\mathit{p}}_i}\\ & =-\frac{{\mathit{r}}_{i+1}^{\mathrm{T}}({\mathit{r}}_i-{\mathit{r}}_{i+1})}{{\alpha }_i{\mathit{p}}_i^{\mathrm{T}}{\mathit{A}\mathit{p}}_i}\\ & =\frac{{\mathit{r}}_{i+1}^{\mathrm{T}}{\mathit{r}}_{i+1}}{{\mathit{r}}_i^{\mathrm{T}}{\mathit{r}}_i}\text{.}\end{array}$

This completes the derivation.

References

1. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
   
   In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
   
   Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
   
   Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
   
   A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
   
   The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
   
   There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
2. 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
   
   My blog: http://www.primaboinca.com/view_profile.php?userid=5889534