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# least squares method excel

An Algorithm for Least-Squares Estimation of Nonlinear Parameters. The least-squares method is generally used in linear regression that calculates the best fit line for observed data by minimizing the sum of squares of deviation of data points from the line. The results of the calculations in each cell in column, Now we need to calculate the sum of the differences of the squares of the values of, In the window insert function, choose the category. Enter your data into the spreadsheet. Donald Marquardt. As in Method of Least Squares, we express this line in the form Thus, Given a set of n points ( x 11 , …, x 1 k , y 1 ), … , ( x n 1 , …, x nk , y n ), our objective is to find a line of the above form which best fits the points. The spreadsheet programmer probably felt that it was too difficult (or impossible) to express this as a "linear least squares" regression, so he/she opted for a non-linear least squares using Solver. The Least Squares Method is an iterative method that can be applied, for example, with the Excel Solver. So, we just dismantled a practical example how to apply the method of least squares in Excel. Let’s look at the application of the method of least squares, solving a concrete example. The MSE now is about 14 million – 14,138,888 to be exacti. One of the most often used functions of a For example, if This dependence can be described by the equation below: Also, we know that if X=0 and Y=0. That is, they find the coefficients of a straight line (or higher dimension shape) so that the sum of the squares of the distances of each data point from the line is a minimum. Instead, numerical analysts have Simple spreadsheet programs such as Microsoft Excel or Quattro Pro are powerful tools For a polynomial equation, we do that by using array constants.An advantage to using LINEST to get the coefficients that define the polynomial equation is that we can return the coefficients directly to cells. Before proceeding to the solution of the main task, you will need to activate the add-in “solver” in the program. In the previous two Excel tutorials, we have discovered two ways to obtain least-squares estimates of the slope and intercept of a best-fit line: use the slope() and intercept() functions; add a trendline to a scatterplot Excel provides us with a couple of tools to perform Least Squares calculations, but they are all centered around the simpler functions: simple Linear functions of the shape y=a.x+b, y-a.exp(b.x), y=a.x^b and etcetera. The package, named nlls11.xla , might be loaded automatically when you launch Excel. It minimizes the sum of the residuals of points from the plotted curve. It gives the trend line of best fit to a time series data. Covariance is a process to help us find the Correlation, which is the number … Formatting a Least-Squares Graph Using Microsoft Excel. Methods for Using Linear Regression in Excel. Text Font Double-click on the graph title, axis title, legend or the equation of the line. Excel is a powerful spreadsheet editor that enables you to perform many different operations and tasks. Both columns Constructing a Least-Squares Graph Using Microsoft Excel. Dragging is used not only to move objects but also to highlight text, data, etc. Under Border, select the style, color and weight (or none) of the box around the graph.Under Area, select white for a white background.Click OK.. To answer that question, first we have to agree on what we mean by the “best The given example explains how to find the equation of a straight line or a least square line by using the method of least square, which is very useful in statistics as well as in mathematics. It is often easier to put similar types of data Installation An add-in package for Excel, which performs certain specific non-linear least squares analyses, is available for use in Chem 452. (See Accessing Excel data from the computer lab) Insert a row at the top and add titles to the columns if necessary or desired. Nonetheless, formulas for total fixed costs (a) and variable cost per unit (b)can be derived from the above equations. Least squares linear regression in Excel is easy. Taught By. The LINEST function calculates the statistics for a line by using the "least squares" method to calculate a straight line that best fits your data, and then returns an array that describes the line. Ordinary Least Squares regression (OLS) is more commonly named linear regression (simple or multiple depending on the number of explanatory variables).In the case of a model with p explanatory variables, the OLS regression model writes:Y = β0 + Σj=1..p βjXj + εwhere Y is the dependent variable, β0, is the intercept of the model, X j corresponds to the jth explanatory variable of the model (j= 1 to p), and e is the random error with expec… Kenneth Levenberg. Excel will output a table containing 4 cells. The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals made in the results of every single equation.. (e.g., cells 1-4 of columns A and C), place the mouse cursor on cell A1 and drag of data should be highlighted. The value of r is the number in the first row, second column. Note that the method described above is not precisely how we solve least-squares problems numerically in practice, since cond(A0A) ∼ cond(A2) so that this new linear system as written may be ill-conditioned. We now look at the line in the x y plane that best fits the data ( x 1 , y 1 ), …, ( x n , y n ). Method of Least Squares In Correlation we study the linear correlation between two random variables x and y. Fataneh Taghaboni-Dutta. Now you need to fill in function arguments: After the function, a window will appear with the results of the search solution and will replace the values in column. (2):164-201, 1944. least-squares fitting (LLSF) refers to regression equations that are linear in their parameters (this, of course, includes but is not limi- ted to equations that ar e linear themsel ves). Let’s say we have two rows of numeric values X and Y. Clinical Professor of Business Administration. Example: Fit a least square line for the following data. Least Squares Regression Equation Using Excel. SIAM Journal on Applied Mathematics 11 (2):431-441, 1963. into columns rather than rows (although this is not a requirement). means to place the mouse cursor on a position, press and hold the left mouse button, move the Select (highlight) the data that you want to include in the graph. Note that "click" means to put the mouse cursor on a position and press the left mouse In this article we will learn how we can apply the method of least squares (OLS), which is used for solving different problems with minimization of sum of squared deviations of certain functions from the decision variables. In this article we will learn how we can apply the method of least squares (OLS), which is used for solving different problems with minimization of sum of squared deviations of … Least squares method, also called least squares approximation, in statistics, a method for estimating the true value of some quantity based on a consideration of errors in observations or measurements. ∑y = na + b∑x ∑xy = ∑xa + b∑x² Note that through the process of elimination, these equations can be used to determine the values of a and b. Let’s say we have some data of pressure drop vs. flow rate through a water valve, and after plotting the data on a chart we see that the data is quadratic.Even though this data is nonlinear, the LINEST function can also be used here to find the best fit curve for this data. He collected the following data on the number of direct labor hours and associated factory overhead cost for the months of January through August. Let us discuss the Method of Least Squares in detail. This example teaches you the methods to perform Linear Regression Analysis in Excel. Non-Linear Least Squares Analysis with Excel 1. The mean of the Sales is £7333.33 so this is our prediction. Least Square Method in Excel 3:35. Exploring Vintage Actuarial Methods: The Least Squares IBNR Method of F.E. That’s what the Linest and Trend functions do. The general polynomial regression model can be developed using the method of least squares. (You can get this on the model sheet by putting £7333.33 in for the constant, Insert your data into an Excel spreadsheet. A section on the general formulation for nonlinear least-squares tting is now available. mouse cursor on cell A1 and drag the mouse cursor to cell B4. With some tricks you can also perform LS on polynomes using Excel. Quarterly of Applied Mathematics. Method of Least Squares Using Computer Spreadsheet Program The controller for Beckham Company believes that the number of direct labor hours is associated with overhead cost. This is an example that I have shown you in the PowerPoint and I'm going to now show you how I calculated the least square method. using Microsoft Excel is outlined below. keyboard, place the mouse cursor on cell C1 and drag to cell C4. the least squares problem, in our case A0A will always have a solution, even if it is singular.) In practice may meet more challenges, however, in General the logic of action is similar to what we have described. Try the Course for Free. Set the checkbox next to. the mouse cursor to cell A4. Direct the mouse pointer over the cell with the result. button once; to "double-click" means to press the left mouse button twice rapidly; to "drag" It appears they are using a variation of Euler's method, assuming V0 is exact, then computing V(i) from V(i-1) as you've described above. de Vylder By ERIC BLANCKE, ACAS, AND JEREMY SMITH, FCAS Posted on May 1, 2015 April 24, 2018 Business and Technology Research As soon as a black plus sign (the fill handle), hold down the left mouse button and pull it down to the last row of the table. One of the most often used functions of a spreadsheet program is to construct graphs. If the data you want to include happen to be in two columns that are not adjacent Then, while holding down the CTRL key on the This method is most widely used in time series analysis. Transcript. spreadsheet program is to construct graphs. Coefficients for the Least Squares Regression Line . So, this equation can be written as: Get down to our task, which is to find the sum of the squared differences. This document describes least-squares minimization algorithms for tting point sets by linear structures or quadratic structures. You can calculate the values of the residuals manually in excel or you can use Assayfit Pro or another curve fitting program to do it. When we plot a scattered graph to see the relationship between two interval variables, we can see how straight a line is formed (whether the relationship is strong) and whether the relationship is positive or negative (upward or download). you want to plot the data contained in cells 1-4 of columns A and B, place the mouse cursor to a different position, then release the mouse button. A Method for the Solution of Certain Non-Linear Problems in Least Squares. However, when we want to look deeper to see the strength of the relationship, we need to calculate covariance and correlation. The method of least squares – using the Excel Solver Michael Wood 5 advertising. The method of least squares aims to minimise the variance between the values estimated from the polynomial and the expected values from the dataset.The coefficients of the polynomial regression model (ak,ak−1,⋯,a1) may be determined by solving the following system of linear equations.This system of equations is derived from the polynomial residual function (derivation may be seen in this Wolfram M… Assayfit Pro will calculate the best values for the parameters without the need of providing start values as the program will calculate these. Background Color Right-click anywhere on the background. Excel is a powerful spreadsheet editor that enables you to perform many different operations and tasks. Preparatory phase: activate the add-in “solver”, Step 2: solve the problem with the use of MNK, How to solve the transportation problem in Excel, A window will appear to select the desired add-ins. The organization is somewhat di erent from that of the previous version of the document. Least squares method excel. Simple spreadsheet programs such as Microsoft Excel or Quattro Pro are powerful tools for many types of calculations in chemistry. The least squares method is a statistical procedure to find the best fit for a set of data points by minimizing the sum of the offsets or residuals of points from the plotted curve. Linear Least-squares Regression in Excel. The least-squares method provides the closest relationship between the dependent and independent variables by minimizing the distance between the residuals, and the line of best fit, i.e., the sum of squares of residuals is minimal under this approach. Modi cations include the following. The Excel Solver can be easily configured to determine the coefficients and Y-intercept of the linear regression line that minimizes the sum of the squares of all residuals of each input equation. 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## *DEFORMATION PLASTICITY

Keyword type: model definition, material

This option defines the elasto-plastic behavior of a material by means of the generalized Ramberg-Osgood law. The one-dimensional model takes the form:

 E\epsilon=\sigma + \alpha \left( \frac {\left| \sigma \right| }{\sigma_0} \right)^{n-1}\sigma (787)

where \epsilon is the logarithmic strain and \sigma the Cauchy stress. In the present implementation, the Eulerian strain is used, which is very similar to the logarithmic strain (about 1.3 % difference dat 20 % engineering strain). All coefficients may be temperature dependent.

First line:

• *DEFORMATION PLASTICITY

Following line:

• Young's modulus (E).
• Poisson's ratio (\nu).
• Yield stress (\sigma_0)
• Exponent (n).
• Yield offset (\alpha).
• Temperature.
Repeat this line if needed to define complete temperature dependence.

Example:

*DEFORMATION PLASTICITY
210000.,.3,800.,12.,0.4

defines a Ramberg-Osgood law. No temperature dependence is introduced.

Example files: beampl.

format: plain


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# Kinematics in two dimensions

1. Nov 6, 2004

### FancyNut

This problem looks so simple but every answer I get is wrong...

A skateboarder starts up a 1.0-m-high, 30 degree ramp at a speed of 7.0 m/s. The skateboard wheels roll without friction.

How far from the end of the ramp does the skateboarder touch down?

EDIT : I realized several mistakes.. this is how I did it the last time:

I used the y-component of motion (free fall) to calculate time..

$$0 = 1 + v_0 sin (30) t- \frac {g}{2} t^2$$

then I plug into this:

$$x_f = v_0 cos (30) t$$

I have no idea what I'm doing wrong here... Ancy help would be much appreciated.

Last edited: Nov 6, 2004
2. Nov 6, 2004

### boaz

i think you chose the wrong y and x axises. i suggest you to choose the x axis directed along the ramp surface line. then you will have to separate the mg force to y and x components. it will help me a lot, if you also type the answer which is written in your workbook.

3. Nov 6, 2004

### FancyNut

I'll try that new coordinate system...

I don't have the final answer. I'm using www.masteringphysics.com and the only way to get it is to request it which will end the problem for me-- 10 points off. :(

4. Nov 6, 2004

### FancyNut

Ok I guess I should've calculated the final velocity the skater takes off..

acceleration is - 9.8 sin 30 and after doing everything again it's still.... wrong.

5. Nov 6, 2004

### boaz

ok, then we go to plan b. can you give me more details, such as the weight of the skater, how long is the ramp and etc?

6. Nov 6, 2004

### FancyNut

There is no weight given.

I just tried it again... this time I calculated how long the ramp is which is 2 (since hight is 1 and sin 30 = 1/2). Now using this distance I calculated the final velocity the dude leaves the ramp with... then using that velocity I took the y-component of it and calcuated the time it takes to hit ground... used that time to calculate final position... and it was wrong.

7. Nov 6, 2004

### FancyNut

I think I have it right this time except one thing... this second degree equation. I used the quadratic formual... and after applying it and using the new t value I got the answer is STILL wrong. Why does this have to happen.

$$0 = 1 + 14.7 t - \frac {g}{2} t^2$$

$$4.9 t^2 - 14.7 t - 1 = 0$$

Last edited: Nov 6, 2004
8. Nov 6, 2004

anybody... ?

9. Nov 7, 2004

### Skomatth

Where did u get 14.7 from?

To get the speed at the top of the ramp.

10. Nov 7, 2004

### Duarh

I recommend you use conservation of energy.

We have MVinitial^2/2=MVfinal^2/2+Mgh
V^2=Vfinal^2+2gh

where Vfinal - speed at top of ramp, g - acceleration of gravity - h height of ramp. once you have vfinal, you can calculate its y component by trig and see how long it'll take for the skater to go up and down to the earth again (under acceleration -g), and then use that time to calculate the horizontal distance from the horizontal component of Vfinal

11. Nov 7, 2004

### FancyNut

whew. I finally got the right answer. Thanks guys!

Is it me or is masteringphysics.com the scariest educational tool in existence? Typing in my number and pressing the 'submit' tab while dreading the 'try again' message is a horrible, horrible experience. :(


text:
• Create Account

### #Actualboops boops

Posted 19 September 2012 - 07:36 AM

Hi everyone,
I'm making a custom control in VB.Net which provides image panning, zooming and rotation.It's working the way I like in GDI+ except that it's slow and can't deal with anything over about 4096x4096 pixels. So I decided to try rendering in DirectX instead. It's definitely zippier but there is still an upper limit on image size: now, after column/row 4095 or so, the last column/row is repeated in the render to produce stripes of colour. I gather from this thread that the upper size limit is a long known problem. I'm a bit of a dinosaur (WinXPsp3 with DX3d10, 2GB system memory, 256MB video) so I guess I am running into the same thing.

This is my first time using DirectX and I arrived at the following code by taking available examples and stripping out whatever seemed superflous.
Private Sub InitiateDevice()
Dim pParams As New PresentParameters With {.Windowed = True, .SwapEffect = SwapEffect.Discard, .BackBufferCount = 0}
_Device = New Device(0, DeviceType.Hardware, Me.Handle, CreateFlags.MixedVertexProcessing, pParams)
_Device.SetRenderState(RenderStates.AlphaBlendEnable, True)
_Device.RenderState.SourceBlend = Blend.SourceAlpha
End Sub

Private Function GetTextureFromFile(filename As String) As Texture
Return TextureLoader.FromFile(_Device, filename, _ImageSize.Width, _ImageSize.Height, 0, 0, 0, Pool.Managed, Filter.Linear, Filter.Linear, 0)
End Function

Private Sub Render()
If _Device IsNot Nothing Then
Dim srcRectangle As New Rectangle(Point.Empty, _ImageSize)
Dim destSize As New Size(CInt(_ImageSize.Width * _ZoomFactor), CInt(_ImageSize.Height * _ZoomFactor))
_Device.Clear(ClearFlags.Target, Me.BackColor, 0, 0)
_Device.BeginScene()
Using sprite As New Sprite(_Device)
sprite.Begin(SpriteFlags.AlphaBlend)
sprite.Draw2D(_Texture, srcRectangle, destSize, _
_RotationCentre, _Rotation, _ImageCentre, Color.White)
sprite.End()
End Using
_Device.EndScene()
_Device.Present()
End If
End Sub

The identifiers beginning with underscores are defined at class level. I have tried variations on arguments such as Width, Height, Pool and Filter without solving the problem. Assuming there isn't something simple I need to change, can anyone recommend a better technique for handling large images (e.g. > 10K * 10K pixels) with DirectX?

cheers, BB

EDIT: I just found a tool called DirectX Caps Viewer which reveals that my graphics card is indeed limited to a maximum texture size of 4096 * 4096. Can I overcome this limit by dividing large images into multiple textures? If so, can anyone recommend an example?

### #2boops boops

Posted 19 September 2012 - 07:20 AM

Hi everyone,
I'm making a custom control in VB.Net which provides image panning, zooming and rotation.It's working the way I like in GDI+ except that it's slow and can't deal with anything over about 4096x4096 pixels. So I decided to try rendering in DirectX instead. It's definitely zippier but there is still an upper limit on image size: now, after column/row 4095 or so, the last column/row is repeated in the render to produce stripes of colour. I gather from this thread that the upper size limit is a long known problem. I'm a bit of a dinosaur (WinXPsp3 with DX3d10, 2GB system memory, 256MB video) so I guess I am running into the same thing.

This is my first time using DirectX and I arrived at the following code by taking available examples and stripping out whatever seemed superflous.
Private Sub InitiateDevice()
Dim pParams As New PresentParameters With {.Windowed = True, .SwapEffect = SwapEffect.Discard, .BackBufferCount = 0}
_Device = New Device(0, DeviceType.Hardware, Me.Handle, CreateFlags.MixedVertexProcessing, pParams)
_Device.SetRenderState(RenderStates.AlphaBlendEnable, True)
_Device.RenderState.SourceBlend = Blend.SourceAlpha
End Sub

Private Function GetTextureFromFile(filename As String) As Texture
Return TextureLoader.FromFile(_Device, filename, _ImageSize.Width, _ImageSize.Height, 0, 0, 0, Pool.Managed, Filter.Linear, Filter.Linear, 0)
End Function

Private Sub Render()
If _Device IsNot Nothing Then
Dim srcRectangle As New Rectangle(Point.Empty, _ImageSize)
Dim destSize As New Size(CInt(_ImageSize.Width * _ZoomFactor), CInt(_ImageSize.Height * _ZoomFactor))
_Device.Clear(ClearFlags.Target, Me.BackColor, 0, 0)
_Device.BeginScene()
Using sprite As New Sprite(_Device)
sprite.Begin(SpriteFlags.AlphaBlend)
sprite.Draw2D(_Texture, srcRectangle, destSize, _
_RotationCentre, _Rotation, _ImageCentre, Color.White)
sprite.End()
End Using
_Device.EndScene()
_Device.Present()
End If
End Sub

The identifiers beginning with underscores are defined at class level. I have tried variations on arguments such as Width, Height, Pool and Filter without solving the problem. Assuming there isn't something simple I need to change, can anyone recommend a better technique for handling large images (e.g. > 10K * 10K pixels) with DirectX?

cheers, BB

### #1boops boops

Posted 19 September 2012 - 07:18 AM

Hi everyone,
I'm making a custom control in VB.Net which provides image panning, zooming and rotation.It's working the way I like in GDI+ except that it's slow and can't deal with anything over about 4096x4096 pixels. So I decided to try rendering in DirectX instead. It's definitely zippier but there is still an upper limit on image size: now, after column/row 4095 or so, the last column/row is repeated in the render to produce stripes of colour. I gather from this thread that the upper size limit is a long known problem. I'm a bit of a dinosaur (WinXPsp3 with DX3d10, 2GB system memory, 256KB video) so I guess I am running into the same thing.

This is my first time using DirectX and I arrived at the following code by taking available examples and stripping out whatever seemed superflous.
Private Sub InitiateDevice()
Dim pParams As New PresentParameters With {.Windowed = True, .SwapEffect = SwapEffect.Discard, .BackBufferCount = 0}
_Device = New Device(0, DeviceType.Hardware, Me.Handle, CreateFlags.MixedVertexProcessing, pParams)
_Device.SetRenderState(RenderStates.AlphaBlendEnable, True)
_Device.RenderState.SourceBlend = Blend.SourceAlpha
End Sub

Private Function GetTextureFromFile(filename As String) As Texture
Return TextureLoader.FromFile(_Device, filename, _ImageSize.Width, _ImageSize.Height, 0, 0, 0, Pool.Managed, Filter.Linear, Filter.Linear, 0)
End Function

Private Sub Render()
If _Device IsNot Nothing Then
Dim srcRectangle As New Rectangle(Point.Empty, _ImageSize)
Dim destSize As New Size(CInt(_ImageSize.Width * _ZoomFactor), CInt(_ImageSize.Height * _ZoomFactor))
_Device.Clear(ClearFlags.Target, Me.BackColor, 0, 0)
_Device.BeginScene()
Using sprite As New Sprite(_Device)
sprite.Begin(SpriteFlags.AlphaBlend)
sprite.Draw2D(_Texture, srcRectangle, destSize, _
_RotationCentre, _Rotation, _ImageCentre, Color.White)
sprite.End()
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# Property of action of a group ring on cyclotomic Galois extension

I want to understand one of the properties of the action of a group ring on cyclotomic Galois extension.

If $\zeta_n$ is primitive root of unity, $G = Gal(\mathbb{Q}(\zeta_n) / \mathbb{Q})$ and $f : G \rightarrow \mathbb{Z}$ is an element of the group ring $\mathbb{Z}[G]$, the action of $f$ on an element $x \in \mathbb{Q}(\zeta_n)$ is defined as: $$x^f = \prod_{\sigma \in G} \sigma(x)^{f(\sigma)}$$

The property I'm trying to prove is:

$$(x_1 + x_2)^f = x_1^f + x_2^f$$

My calculations go like this:

$$(x_1 + x_2)^f = \prod_{\sigma \in G}\sigma(x_1 + x_2)^{f(\sigma)} = \prod_{\sigma \in G}(\sigma(x_1) + \sigma(x_2))^{f(\sigma)}$$

$$x_1^f + x_2^f = \prod_{\sigma \in G} \sigma(x_1)^{f(\sigma)} + \prod_{\sigma \in G} \sigma(x_2)^{f(\sigma)}$$

Don't know how to finish the proof. Any help with this? Thanks in advance!

• As you see it is not true. $x^f = \prod_{\sigma \in G} \sigma(x)^{f(\sigma)}$ is an action of $\mathbb{Z}[G]$ on the multiplicative group $\mathbb{Q}(\zeta_n)^*$ – reuns Jun 8 '17 at 2:24
• What user1952009 says! $\Bbb{Q}(\zeta_n)^*$ becomes a $G$-module, but the "addition" operation of this module is the multiplication of $\Bbb{Q}(\zeta_n)^*$. Your identity cannot possible hold. For an obvious example consider the case of $f=2*1_G$, when $x^f=x^2$. – Jyrki Lahtonen Jun 8 '17 at 6:03
• @user1952009 thank you for the answer. I saw the statement in this article (on the 3rd page) and in the Henri Cohen's book A course in computational Algebraic number theory (447 page). Is the statement really completely wrong? – brick Jun 8 '17 at 12:53
• Did you read what Jyrki wrote ? It is true if you replace $+$ by $\times$ the multiplication of $\mathbb{Q}(\zeta_n)^*$. – reuns Jun 8 '17 at 13:17
• Yeah, @user1952009, I read it. But did you saw the article I linked in my previous comment? There are given both of the properties $(x_1 + x_2)^f = x_1^f + x_2^f$ and $(x_1x_2)^f = x_1^fx_2^f$. I agree that the second one is OK, but the first one... – brick Jun 8 '17 at 13:27


text:
# What is the force between two magnetic dipoles?

If I have two current loops parallel to each other with currents I1 and I2 and radii R1 and R2 a distance z from each other, what is the force between them?

What would change if they were two solenoids instead of current loops?

Would the same hold if it was two magnets? Or a magnet and a solenoid?

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Hi sirconnorstack, welcome to Physics Stack Exchange! This is actually a pretty basic question, at least the first part... have you tried searching the web? Did you find anything, and if so, what problems were you having with understanding it? I ask so that we can give you an answer that will be useful to you. –  David Z Nov 22 '11 at 5:39
–  Qmechanic Jan 10 '13 at 21:47
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## 2 Answers

The field of a magnetic dipole is, at long distances (in some naturalized unit convention), equal to

$$B_i = \partial_i ({\mu \cdot r \over 4\pi r^2})$$

where $\mu$ is the magnetic moment vector (the current times area of the loop, in the direction perpendicular to the area of the loop, times N for a solenoid with N windings, or just a given value for a fixed magnet). This form is universal for all magnetic dipoles at long distances, so it's the same for small loops, for small magnets and for small solenoids.

The form is easy to understand, because it's the form of the field for an electric dipole

The response of a second rigid magnetic dipole to this field is a torque which tends to align the magnetic moment along the direction of the field, plus a force proportional to the field gradient. These are also determined by the magnetic dipole of the magnet.

$$T = \mu \times B$$

$$F_j = - \mu^i \partial_i B_j$$

The first tends to align the magnetic dipole with the field, and this tends to happen quickly, the second moves the aligned dipole to the region of stronger field, and this happens more slowly.

These equations give a complete determination of the forces and torques acting between two dipoles, but if you substitute in the dipole field, the expressions become complicated and unilluminating. The picture is that there is a $1/r^3$ magnetic field which will align magnets so that their magnetic moment points along it (in the presence of dissipation), and then will lead to an attractive force as the aligned magnet drifts to a region of stronger field.

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What is ∂i in these equations? –  cstack Nov 28 '11 at 18:07
$\partial_i$ means derivative with respect to the i-th coordinate, it's just a tensor-notation version of $\nabla$. This is needed when you want to use the summation convention, or when taking the gradient of a vector (as above). –  Ron Maimon Nov 28 '11 at 20:13
add comment

The equation for the force depends on how far you are from the current source. If you are far away, then the dipole-dipole interaction formulas can be used. In that case, all of the configurations will give the same force if they have the same dipole moments. If you are closer, then more complicated specific equations must be used for each configuration.

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# Netzschleuder network catalogue, repository and centrifuge

Problems with this dataset? Open an issue.
You may also take a look at the source code.
The network in this dataset can be loaded directly from graph-tool with:
import graph_tool.all as gt
g = gt.collection.ns["florentine_families"]

# florentine_families — Padgett Florentine families

Description

Multiplex network with 2 edge types representing marriage alliances and business relationships between Florentine families during the Italian Renaissance. Data hosted by Manlio De Domenico.1

1. Description obtained from the ICON project.

Tags
Social Relationships Multilayer Unweighted
Citation
Upstream URL OK
https://manliodedomenico.com/data.php
Networks
Tip: hover your mouse over a table header to obtain a legend.
Name Nodes Edges $\left<k\right>$ $\sigma_k$ $\lambda_h$ $\tau$ $r$ $c$ $\oslash$ $S$ Kind Mode NPs EPs gt GraphML GML csv
florentine_families 16 35 4.38 2.57 2.96 4.20 -0.46 0.30 4 0.94 Undirected Unipartite name nodeLabel weight layer 1 KiB 1 KiB 1 KiB 1 KiB
Ridiculograms


text:
The variational principle and the quantum mechanical particle-in-a-box

Question P8.4.4

Consider a one-dimensional quantum mechanical particle in a box ($-1\le x \le 1$) described by the Schrödinger equation: $$-\frac{\mathrm{d}^2\psi}{\mathrm{d}x^2} = E\psi,$$ in energy units for which $\hbar^2/(2m) = 1$ with $m$ the mass of the particle. The exact solution for the ground state of this system is given by $$\psi = \cos\left(\frac{\pi x}{2}\right), \quad E = \frac{\pi^2}{4}.$$ An approximate solution may be arrived at using the variational principle by minimizing the expectation value of the energy of a trial wavefunction, $$\psi_\mathrm{trial} = \sum_{n=0}^N a_n\phi_n(x)$$ with respect to the coefficients $a_n$. Taking the basis functions to have the following symmetrized polynomial form, $$\phi_n = (1-x)^{N-n+1}(x+1)^{n+1},$$ use scipy.optimize.minimize and scipy.integrate.quad to find the optimum value of the expectation value (Rayleigh-Ritz ratio): $$\mathcal{E} = \frac{\langle \psi_\mathrm{trial} | \hat{H} | \psi_\mathrm{trial} \rangle}{\langle \psi_\mathrm{trial} | \psi_\mathrm{trial} \rangle} = -\frac{\int_{-1}^1\psi_\mathrm{trial}\frac{\mathrm{d}^2}{\mathrm{d}x^2}\psi_\mathrm{trial}\;\mathrm{d}x}{\int_{-1}^1\psi_\mathrm{trial}\psi_\mathrm{trial}\;\mathrm{d}x}.$$ Compare the estimated energy, $\mathcal{E}$, with the exact answer for $N=1,2,3,4$.

[Hint: Use np.polynomial.Polynomial objects to represent the basis and trial wavefunctions].

Solution

To access solutions, please obtain an access code from Cambridge University Press at the Lecturer Resources page for my book (registration required) and then sign up to scipython.com providing this code.


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“…It is clear that the OARs and OAAs of diesel fuels decrease gradually with the increasing of oxygen absorption time at given temperatures and pressures. The change of HCDF quality is clearly greater than that of SRDF during storage, that is, the stability on oxygen absorption of SRDF is remarkably better than that of HCDF (Qi, 1999;Qi, 2000). Especially at 350 K and 0.25 MPa they change most evidently.…”
Section: Oar and Oaa Of Diesel Fuel Versus Timementioning
“…The experimental apparatus for diesel fuel oxygen absorption (DFOA), the properties of test samples, SRDF and HCDF, and the test program have already been reported in the literature (Qi, 1999;Qi, 2000). The oxygen absorption quantity (OAQ, mL/g) is defined as the standard oxygen quantity absorbed by diesel fuel sample.…”
Section: Methodsmentioning
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# How to speed up scientific Python code

written by on 2016-06-07 | tags: python programming optimization

### Introduction

For scientists who are looking to take their Python code to the next level, options are abound. I’ve decided to put down in writing a short summary of what I’ve learned so far.

Disclaimers:

• the "best use cases" part comes from personal experience, not from referenced literature, and is by necessity going to be of limited scope.
• your mileage… ahem, speed increases may vary. (yada yada yada)
• your algorithm complexity matters more than these tools below.
• learn how to profile your code, and then profile, profile, and profile before you optimize.

Now that the disclaimers are out of the way, here’s the shortlist, in no particular order:

#### PyPy

Great if you have only pure Python code, have to deal with things in Python objects, and import other modules that are only pure Python.

My use cases: Agent-based simulations, re-implementing random draws from a variety of distributions.

Upsides: Speedups can be massive; in my hands anywhere between 8-100X. PyPy is a really simple drop-in replacement for the regular CPython that we’re all used to.

Downsides: Have to download an alternate Python interpreter. I have yet to successfully use PyPy as my jupyter backend. Need to tweak environment variables to get pip to install packages/modules such that they’re accessible by pypy.

#### numpy and scipy

Great if you’re writing things that are, by nature, arrays of stuff, with operations that can be vectorized across all rows (or columns).

My use cases: Matrix math.

Upsides: numpy and scipy are well-supported, community-favoured, thoroughly tested and all-round great standard scientific computing libraries to use. Easily installable via conda!

Downsides: Very few, except that Python overhead is incurred every time a function is called, so one has to get used to avoid calling same function over and over in a for loop.

#### numba

Exciting stuff from Continuum Analytics developer team! Basically a JIT compiler (not unlike PyPy) but for numeric code instead. Can speed up numpy code, according to their claims.

My use cases: Matrix math and numeric code.

Upsides: Uses the CPython interpreter that we’re all used to. Speedups gained by simply adding a function decorator. Also easily installable by conda!

Downsides: Doesn’t work with Python objects very well right now.

#### Cython

Only for the brave of heart! A simple way to continue writing Python-ish code that then gets compiled to C, which is blazing fast.

My use cases: Not sure yet. It is good, don’t get me wrong.

Upsides: I’ve done a head-to-head numpy vs. Cython version of what I’m trying to do. 25X speed-up. And again, easily installable by conda. (It’s probably clear by now that I’m a Continuum Analytics fanboi.)

Downside: Have to learn a new-ish language with some intricacies in order to really get the massive speed-ups. I initially could only write Cython code that didn’t break, but would end up with little speed-up (1-1.5X max). Going faster needed a lot more tweaking, probably lots more practice.

### Bonus!

How did I pick up each of these? It was some mixture of following online tutorials, trying to modify examples myself, reading Stack Overflow, and coding with another advanced person who was willing to teach me stuff. I found the last one to be the most effective of them all, as the back-and-forth is great for learning.

Hope this post helps you decide what to try next for speeding up your code!


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# ONE KING DIMENSION

 0 How many bombs are required to destroy [7,8] and [7,8] i.e two same kingdoms asked 08 Jan '15, 08:53 2★ambika93 81●2●8 accept rate: 0%

 3 As you can see in the sample input, putting a bomb on x=2 destroys two kingdoms. In the case you mentioned, only one bomb is required. answered 08 Jan '15, 09:32 5★mogers 649●5●16 accept rate: 25% by x=2 you meant x=7 or x=8 right? (08 Jan '15, 13:11) I was talking about the sample input, but now I see that the my last sentence was unclear. I edited it. (08 Jan '15, 17:50) mogers5★ @mogers, i'm getting wrong answer from last two days , any possible test case,i'm unable to trace it. (08 Jan '15, 22:46) ambika932★
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