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| In [[numerical linear algebra]], the [[conjugate gradient method]] is an [[iterative method]] for numerically solving the [[System of linear equations|linear system]]
| | <br><br>Knife sharpeners fall into a couple of totally different classes and we can divide them up a couple methods. There are manual knife sharpeners and electrical knife sharpeners. Each varieties of sharpeners use a particular material to remove metal from the knife blade, often stone, ceramic, or diamond. Whatever the type, the sharpening materials has varying grits, starting from very coarse to extremely advantageous. Coarse materials take away the metal in a short time, which is helpful for very dull blades or when reprofiling a knife edge to a different angle (More on reprofiling later!). The finer grits are used later within the course of, very like sanding down a bit of wooden furniture.<br><br>The knife could be very light weight however you'll be able to tell that it is very sturdy and candle a beating. It feels very snug in my hand with the finger grooves and it is very evident that the knife won't slip out of my hand because the grip is constructed of both exhausting and smooth polymers, therefore the double injection. It's a arduous durable deal with however has the grip that you simply need, I really feel like I have full management of the knife whereas holding it. I even let my spouse maintain the knife and requested her the way it felt in her hand.<br><br>Honing straightens the misaligned fringe of the knife blade. It's best to hone your knife each time you utilize it – simply before slicing and after you wash and dry the knife. A sharpening steel is used to hone a knife and is normally a cylindrical rod, roughly about 12 inches lengthy. Some sharpening steels have a flattened space. Most sharpening steels are made Ontario Rat 5 Knife Review - [http://www.thebestpocketknifereviews.com/ontario-knives-review/ Recommended Internet page] - of laborious steel, but some are fabricated from ceramic. To hone your knife, you run the business aspect of the knife blade across the sharpening steel to get the blade edge back in alignment. Honing can be accomplished as the final step of sharpening.<br><br>Benchmade is among the prime rated knife manufacturers with some of the most widely appreciated knives in the industry. The Mini Griptilian contains a drop-point stainless-steel blade that is lower than 3″ in size. The handle is crafted from stainless steel and options [http://Www.thebestpocketknifereviews.com/ontario-knives-review/ Ontario Rat Knife Reviews] an ergonomic design that allows for the proper grip. The defining characteristic of the Mini Griptilian is its unique Axis locking mechanism that is proprietary to Benchmade. The knife may be easily pulled into an open or closed place with using thumb studs. The blade has a easy satin finish that gives little resistance whereas opening or closing it.<br><br>Laguiole steak knives are the best steak knives yow will discover on the planet at the moment when you consider their attributes together with but not limited to their price and the quality that they provide on the stated worth. The reason for these steak knives being known as the perfect on this planet is twofold. One is the truth that these knives are manufactured to one of the best of their requirements through the use of the most effective metal. This ensures knives of the highest quality and of probably the most durable sort, thereby cementing their place as a number of the greatest steak knives out there available in the market.<br><br>On average, for the price of three Case knives, I should buy eleven Tough Riders. This was decided by comparing the common value of my Case knife Assortment divided by the average price of my Rough Riders. The prices used within the comparability have been the lowest worth quoted on both Amazon or SMKW web site for the knife in my collection. (As some of my Case knives are outdated collector knives that cost greater than at the moment’s market price, I looked for the closest present manufacturing Case knife that was comparable in [http://Ielaw.uibe.edu.cn/wiki/index.php/Ontario_Pilot_Knife_Review pattern] and deal with material for a extra equitable price). |
| | |
| :<math>\boldsymbol{Ax}=\boldsymbol{b}</math>
| |
| | |
| where <math>\boldsymbol{A}</math> is [[Symmetric matrix|symmetric]] [[Positive-definite matrix|positive-definite]]. The conjugate gradient method can be derived from several different perspectives, including specialization of the [[conjugate direction method]] for [[Optimization (mathematics)|optimization]], and variation of the [[Arnoldi iteration|Arnoldi]]/[[Lanczos iteration|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==
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| {{Expand section|date=April 2010}}
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| The conjugate gradient method can be seen as a special case of the conjugate direction method applied to minimization of the quadratic function
| |
| | |
| :<math>f(\boldsymbol{x})=\boldsymbol{x}^\mathrm{T}\boldsymbol{A}\boldsymbol{x}-2\boldsymbol{b}^\mathrm{T}\boldsymbol{x}\text{.}</math>
| |
| | |
| ===The conjugate direction method===
| |
| In the conjugate direction method for minimizing
| |
| | |
| :<math>f(\boldsymbol{x})=\boldsymbol{x}^\mathrm{T}\boldsymbol{A}\boldsymbol{x}-2\boldsymbol{b}^\mathrm{T}\boldsymbol{x}\text{.}</math>
| |
| | |
| one starts with an initial guess <math>\boldsymbol{x}_0</math> and the corresponding residual <math>\boldsymbol{r}_0=\boldsymbol{b}-\boldsymbol{Ax}_0</math>, and computes the iterate and residual by the formulae
| |
| | |
| :<math>\begin{align}
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| \alpha_i&=\frac{\boldsymbol{p}_i^\mathrm{T}\boldsymbol{r}_i}{\boldsymbol{p}_i^\mathrm{T}\boldsymbol{Ap}_i}\text{,}\\
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| \boldsymbol{x}_{i+1}&=\boldsymbol{x}_i+\alpha_i\boldsymbol{p}_i\text{,}\\
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| \boldsymbol{r}_{i+1}&=\boldsymbol{r}_i-\alpha_i\boldsymbol{Ap}_i
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| \end{align}</math>
| |
| | |
| where <math>\boldsymbol{p}_0,\boldsymbol{p}_1,\boldsymbol{p}_2,\ldots</math> are a series of mutually conjugate directions, i.e.,
| |
| | |
| :<math>\boldsymbol{p}_i^\mathrm{T}\boldsymbol{Ap}_j=0</math>
| |
| | |
| for any <math>i\neq j</math>.
| |
| | |
| The conjugate direction method is imprecise in the sense that no formulae are given for selection of the directions <math>\boldsymbol{p}_0,\boldsymbol{p}_1,\boldsymbol{p}_2,\ldots</math>. Specific choices lead to various methods including the conjugate gradient method and [[Gaussian elimination]]. | |
| | |
| ==Derivation from the Arnoldi/Lanczos iteration==
| |
| {{see|Arnoldi iteration|Lanczos iteration}}
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| 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 <math>\boldsymbol{r}_0</math> and gradually builds an [[orthonormal]] basis <math>\{\boldsymbol{v}_1,\boldsymbol{v}_2,\boldsymbol{v}_3,\ldots\}</math> of the [[Krylov subspace]]
| |
| | |
| :<math>\mathcal{K}(\boldsymbol{A},\boldsymbol{r}_0)=\{\boldsymbol{r}_0,\boldsymbol{Ar}_0,\boldsymbol{A}^2\boldsymbol{r}_0,\ldots\}</math> | |
| | |
| by defining <math>\boldsymbol{v}_i=\boldsymbol{w}_i/\lVert\boldsymbol{w}_i\rVert_2</math> where
| |
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| :<math>\boldsymbol{w}_i=\begin{cases}
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| \boldsymbol{r}_0 & \text{if }i=1\text{,}\\
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| \boldsymbol{Av}_{i-1}-\sum_{j=1}^{i-1}(\boldsymbol{v}_j^\mathrm{T}\boldsymbol{Av}_{i-1})\boldsymbol{v}_j & \text{if }i>1\text{.}
| |
| \end{cases}</math>
| |
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| In other words, for <math>i>1</math>, <math>\boldsymbol{v}_i</math> is found by [[Gram-Schmidt orthogonalization|Gram-Schmidt orthogonalizing]] <math>\boldsymbol{Av}_{i-1}</math> against <math>\{\boldsymbol{v}_1,\boldsymbol{v}_2,\ldots,\boldsymbol{v}_{i-1}\}</math> followed by normalization.
| |
| | |
| Put in matrix form, the iteration is captured by the equation
| |
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| :<math>\boldsymbol{AV}_i=\boldsymbol{V}_{i+1}\boldsymbol{\tilde{H}}_i</math>
| |
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| where
| |
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| :<math>\begin{align}
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| \boldsymbol{V}_i&=\begin{bmatrix}
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| \boldsymbol{v}_1 & \boldsymbol{v}_2 & \cdots & \boldsymbol{v}_i
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| \end{bmatrix}\text{,}\\
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| \boldsymbol{\tilde{H}}_i&=\begin{bmatrix}
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| h_{11} & h_{12} & h_{13} & \cdots & h_{1,i}\\
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| h_{21} & h_{22} & h_{23} & \cdots & h_{2,i}\\
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| & h_{32} & h_{33} & \cdots & h_{3,i}\\
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| & & \ddots & \ddots & \vdots\\
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| & & & h_{i,i-1} & h_{i,i}\\
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| & & & & h_{i+1,i}
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| \end{bmatrix}=\begin{bmatrix}
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| \boldsymbol{H}_i\\
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| h_{i+1,i}\boldsymbol{e}_i^\mathrm{T}
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| \end{bmatrix}
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| \end{align}</math>
| |
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| with
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| :<math>h_{ji}=\begin{cases}
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| \boldsymbol{v}_j^\mathrm{T}\boldsymbol{Av}_i & \text{if }j\leq i\text{,}\\
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| \lVert\boldsymbol{w}_{i+1}\rVert_2 & \text{if }j=i+1\text{,}\\
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| 0 & \text{if }j>i+1\text{.}
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| \end{cases}</math>
| |
| | |
| When applying the Arnoldi iteration to solving linear systems, one starts with <math>\boldsymbol{r}_0=\boldsymbol{b}-\boldsymbol{Ax}_0</math>, the residual corresponding to an initial guess <math>\boldsymbol{x}_0</math>. After each step of iteration, one computes <math>\boldsymbol{y}_i=\boldsymbol{H}_i^{-1}(\lVert\boldsymbol{r}_0\rVert_2\boldsymbol{e}_1)</math> and the new iterate <math>\boldsymbol{x}_i=\boldsymbol{x}_0+\boldsymbol{V}_i\boldsymbol{y}_i</math>.
| |
| | |
| ===The direct Lanczos method===
| |
| For the rest of discussion, we assume that <math>\boldsymbol{A}</math> is symmetric positive-definite. With symmetry of <math>\boldsymbol{A}</math>, the [[upper Hessenberg matrix]] <math>\boldsymbol{H}_i=\boldsymbol{V}_i^\mathrm{T}\boldsymbol{AV}_i</math> becomes symmetric and thus tridiagonal. It then can be more clearly denoted by
| |
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| :<math>\boldsymbol{H}_i=\begin{bmatrix}
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| a_1 & b_2\\
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| b_2 & a_2 & b_3\\
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| & \ddots & \ddots & \ddots\\
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| & & b_{i-1} & a_{i-1} & b_i\\
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| & & & b_i & a_i
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| \end{bmatrix}\text{.}</math>
| |
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| This enables a short three-term recurrence for <math>\boldsymbol{v}_i</math> in the iteration, and the Arnoldi iteration is reduced to the Lanczos iteration.
| |
| | |
| Since <math>\boldsymbol{A}</math> is symmetric positive-definite, so is <math>\boldsymbol{H}_i</math>. Hence, <math>\boldsymbol{H}_i</math> can be [[LU factorization|LU factorized]] without [[partial pivoting]] into
| |
| | |
| :<math>\boldsymbol{H}_i=\boldsymbol{L}_i\boldsymbol{U}_i=\begin{bmatrix}
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| 1\\
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| c_2 & 1\\
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| & \ddots & \ddots\\
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| & & c_{i-1} & 1\\
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| & & & c_i & 1
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| \end{bmatrix}\begin{bmatrix}
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| d_1 & b_2\\
| |
| & d_2 & b_3\\
| |
| & & \ddots & \ddots\\
| |
| & & & d_{i-1} & b_i\\
| |
| & & & & d_i
| |
| \end{bmatrix}</math>
| |
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| with convenient recurrences for <math>c_i</math> and <math>d_i</math>: | |
| | |
| :<math>\begin{align}
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| c_i&=b_i/d_{i-1}\text{,}\\
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| d_i&=\begin{cases}
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| a_1 & \text{if }i=1\text{,}\\
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| a_i-c_ib_i & \text{if }i>1\text{.}
| |
| \end{cases}
| |
| \end{align}</math>
| |
| | |
| Rewrite <math>\boldsymbol{x}_i=\boldsymbol{x}_0+\boldsymbol{V}_i\boldsymbol{y}_i</math> as
| |
| | |
| :<math>\begin{align}
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| \boldsymbol{x}_i&=\boldsymbol{x}_0+\boldsymbol{V}_i\boldsymbol{H}_i^{-1}(\lVert\boldsymbol{r}_0\rVert_2\boldsymbol{e}_1)\\
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| &=\boldsymbol{x}_0+\boldsymbol{V}_i\boldsymbol{U}_i^{-1}\boldsymbol{L}_i^{-1}(\lVert\boldsymbol{r}_0\rVert_2\boldsymbol{e}_1)\\
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| &=\boldsymbol{x}_0+\boldsymbol{P}_i\boldsymbol{z}_i
| |
| \end{align}</math>
| |
| | |
| with | |
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| :<math>\begin{align}
| |
| \boldsymbol{P}_i&=\boldsymbol{V}_{i}\boldsymbol{U}_i^{-1}\text{,}\\
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| \boldsymbol{z}_i&=\boldsymbol{L}_i^{-1}(\lVert\boldsymbol{r}_0\rVert_2\boldsymbol{e}_1)\text{.}
| |
| \end{align}</math>
| |
| | |
| It is now important to observe that
| |
| | |
| :<math>\begin{align}
| |
| \boldsymbol{P}_i&=\begin{bmatrix}
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| \boldsymbol{P}_{i-1} & \boldsymbol{p}_i
| |
| \end{bmatrix}\text{,}\\
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| \boldsymbol{z}_i&=\begin{bmatrix}
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| \boldsymbol{z}_{i-1}\\
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| \zeta_i
| |
| \end{bmatrix}\text{.}
| |
| \end{align}</math>
| |
| | |
| In fact, there are short recurrences for <math>\boldsymbol{p}_i</math> and <math>\zeta_i</math> as well:
| |
| | |
| :<math>\begin{align}
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| \boldsymbol{p}_i&=\frac{1}{d_i}(\boldsymbol{v}_i-b_i\boldsymbol{p}_{i-1})\text{,}\\
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| \zeta_i&=-c_i\zeta_{i-1}\text{.}
| |
| \end{align}</math>
| |
| | |
| With this formulation, we arrive at a simple recurrence for <math>\boldsymbol{x}_i</math>:
| |
| | |
| :<math>\begin{align}
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| \boldsymbol{x}_i&=\boldsymbol{x}_0+\boldsymbol{P}_i\boldsymbol{z}_i\\
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| &=\boldsymbol{x}_0+\boldsymbol{P}_{i-1}\boldsymbol{z}_{i-1}+\zeta_i\boldsymbol{p}_i\\
| |
| &=\boldsymbol{x}_{i-1}+\zeta_i\boldsymbol{p}_i\text{.}
| |
| \end{align}</math>
| |
| | |
| 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 <math>\boldsymbol{p}_i</math> to scale and compensate for the scaling in the constant factor, we potentially can have simpler recurrences of the form:
| |
| | |
| :<math>\begin{align}
| |
| \boldsymbol{x}_i&=\boldsymbol{x}_{i-1}+\alpha_{i-1}\boldsymbol{p}_{i-1}\text{,}\\
| |
| \boldsymbol{r}_i&=\boldsymbol{r}_{i-1}-\alpha_{i-1}\boldsymbol{Ap}_{i-1}\text{,}\\
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| \boldsymbol{p}_i&=\boldsymbol{r}_i+\beta_{i-1}\boldsymbol{p}_{i-1}\text{.}
| |
| \end{align}</math>
| |
| | |
| As premises for the simplification, we now derive the orthogonality of <math>\boldsymbol{r}_i</math> and conjugacy of <math>\boldsymbol{p}_i</math>, i.e., for <math>i\neq j</math>,
| |
| | |
| :<math>\begin{align}
| |
| \boldsymbol{r}_i^\mathrm{T}\boldsymbol{r}_j&=0\text{,}\\
| |
| \boldsymbol{p}_i^\mathrm{T}\boldsymbol{Ap}_j&=0\text{.}
| |
| \end{align}</math>
| |
| | |
| The residuals are mutually orthogonal because <math>\boldsymbol{r}_i</math> is essentially a multiple of <math>\boldsymbol{v}_{i+1}</math> since for <math>i=0</math>, <math>\boldsymbol{r}_0=\lVert\boldsymbol{r}_0\rVert_2\boldsymbol{v}_1</math>, for <math>i>0</math>,
| |
| | |
| :<math>\begin{align}
| |
| \boldsymbol{r}_i&=\boldsymbol{b}-\boldsymbol{Ax}_i\\
| |
| &=\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_0+\boldsymbol{V}_i\boldsymbol{y}_i)\\
| |
| &=\boldsymbol{r}_0-\boldsymbol{AV}_i\boldsymbol{y}_i\\
| |
| &=\boldsymbol{r}_0-\boldsymbol{V}_{i+1}\boldsymbol{\tilde{H}}_i\boldsymbol{y}_i\\
| |
| &=\boldsymbol{r}_0-\boldsymbol{V}_i\boldsymbol{H}_i\boldsymbol{y}_i-h_{i+1,i}(\boldsymbol{e}_i^\mathrm{T}\boldsymbol{y}_i)\boldsymbol{v}_{i+1}\\
| |
| &=\lVert\boldsymbol{r}_0\rVert_2\boldsymbol{v}_1-\boldsymbol{V}_i(\lVert\boldsymbol{r}_0\rVert_2\boldsymbol{e}_1)-h_{i+1,i}(\boldsymbol{e}_i^\mathrm{T}\boldsymbol{y}_i)\boldsymbol{v}_{i+1}\\
| |
| &=-h_{i+1,i}(\boldsymbol{e}_i^\mathrm{T}\boldsymbol{y}_i)\boldsymbol{v}_{i+1}\text{.}\end{align}</math>
| |
| | |
| To see the conjugacy of <math>\boldsymbol{p}_i</math>, it suffices to show that <math>\boldsymbol{P}_i^\mathrm{T}\boldsymbol{AP}_i</math> is diagonal:
| |
| | |
| :<math>\begin{align}
| |
| \boldsymbol{P}_i^\mathrm{T}\boldsymbol{AP}_i&=\boldsymbol{U}_i^{-\mathrm{T}}\boldsymbol{V}_i^\mathrm{T}\boldsymbol{AV}_i\boldsymbol{U}_i^{-1}\\
| |
| &=\boldsymbol{U}_i^{-\mathrm{T}}\boldsymbol{H}_i\boldsymbol{U}_i^{-1}\\
| |
| &=\boldsymbol{U}_i^{-\mathrm{T}}\boldsymbol{L}_i\boldsymbol{U}_i\boldsymbol{U}_i^{-1}\\
| |
| &=\boldsymbol{U}_i^{-\mathrm{T}}\boldsymbol{L}_i
| |
| \end{align}</math>
| |
| | |
| is symmetric and lower triangular simultaneously and thus must be diagonal.
| |
| | |
| Now we can derive the constant factors <math>\alpha_i</math> and <math>\beta_i</math> with respect to the scaled <math>\boldsymbol{p}_i</math> by solely imposing the orthogonality of <math>\boldsymbol{r}_i</math> and conjugacy of <math>\boldsymbol{p}_i</math>.
| |
| | |
| Due to the orthogonality of <math>\boldsymbol{r}_i</math>, it is necessary that <math>\boldsymbol{r}_{i+1}^\mathrm{T}\boldsymbol{r}_i=(\boldsymbol{r}_i-\alpha_i\boldsymbol{Ap}_i)^\mathrm{T}\boldsymbol{r}_i=0</math>. As a result,
| |
| | |
| :<math>\begin{align}
| |
| \alpha_i&=\frac{\boldsymbol{r}_i^\mathrm{T}\boldsymbol{r}_i}{\boldsymbol{r}_i^\mathrm{T}\boldsymbol{Ap}_i}\\
| |
| &=\frac{\boldsymbol{r}_i^\mathrm{T}\boldsymbol{r}_i}{(\boldsymbol{p}_i-\beta_{i-1}\boldsymbol{p}_{i-1})^\mathrm{T}\boldsymbol{Ap}_i}\\
| |
| &=\frac{\boldsymbol{r}_i^\mathrm{T}\boldsymbol{r}_i}{\boldsymbol{p}_i^\mathrm{T}\boldsymbol{Ap}_i}\text{.}
| |
| \end{align}</math>
| |
| | |
| Similarly, due to the conjugacy of <math>\boldsymbol{p}_i</math>, it is necessary that <math>\boldsymbol{p}_{i+1}^\mathrm{T}\boldsymbol{Ap}_i=(\boldsymbol{r}_{i+1}+\beta_i\boldsymbol{p}_i)^\mathrm{T}\boldsymbol{Ap}_i=0</math>. As a result,
| |
| | |
| :<math>\begin{align}
| |
| \beta_i&=-\frac{\boldsymbol{r}_{i+1}^\mathrm{T}\boldsymbol{Ap}_i}{\boldsymbol{p}_i^\mathrm{T}\boldsymbol{Ap}_i}\\
| |
| &=-\frac{\boldsymbol{r}_{i+1}^\mathrm{T}(\boldsymbol{r}_i-\boldsymbol{r}_{i+1})}{\alpha_i\boldsymbol{p}_i^\mathrm{T}\boldsymbol{Ap}_i}\\
| |
| &=\frac{\boldsymbol{r}_{i+1}^\mathrm{T}\boldsymbol{r}_{i+1}}{\boldsymbol{r}_i^\mathrm{T}\boldsymbol{r}_i}\text{.}
| |
| \end{align}</math>
| |
| | |
| This completes the derivation.
| |
| | |
| ==References==
| |
| #{{cite journal|last1 = Hestenes|first1 = M. R.|authorlink1 = David Hestenes|last2 = Stiefel|first2 = E.|authorlink2 = Eduard Stiefel|title = Methods of conjugate gradients for solving linear systems|journal = Journal of Research of the National Bureau of Standards|volume = 49|issue = 6|date=December 1952|url = http://nvl.nist.gov/pub/nistpubs/jres/049/6/V49.N06.A08.pdf|format=PDF}}
| |
| #{{cite book|last = Saad|first = Y.|title = Iterative methods for sparse linear systems|edition = 2nd|chapter = Chapter 6: Krylov Subspace Methods, Part I|publisher = SIAM|year = 2003|isbn = 978-0-89871-534-7}}
| |
| | |
| {{Numerical linear algebra}}
| |
| | |
| [[Category:Numerical linear algebra]]
| |
| [[Category:Optimization algorithms and methods]]
| |
| [[Category:Gradient methods]]
| |
| [[Category:Articles containing proofs]]
| |
Knife sharpeners fall into a couple of totally different classes and we can divide them up a couple methods. There are manual knife sharpeners and electrical knife sharpeners. Each varieties of sharpeners use a particular material to remove metal from the knife blade, often stone, ceramic, or diamond. Whatever the type, the sharpening materials has varying grits, starting from very coarse to extremely advantageous. Coarse materials take away the metal in a short time, which is helpful for very dull blades or when reprofiling a knife edge to a different angle (More on reprofiling later!). The finer grits are used later within the course of, very like sanding down a bit of wooden furniture.
The knife could be very light weight however you'll be able to tell that it is very sturdy and candle a beating. It feels very snug in my hand with the finger grooves and it is very evident that the knife won't slip out of my hand because the grip is constructed of both exhausting and smooth polymers, therefore the double injection. It's a arduous durable deal with however has the grip that you simply need, I really feel like I have full management of the knife whereas holding it. I even let my spouse maintain the knife and requested her the way it felt in her hand.
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