Using Alglib least-squares solvers

Following the last post on using the Python version of Alglib from Excel, via xlwings, this post looks in more detail at alternatives for fitting a non-linear function to a set of data, using the Levenberg-Marquardt method.  The spreadsheet with examples and full open-source code may be downloaded from:

xlAlglib.zip

The spreadsheet requires xlwings, Python (including Numpy and Scipy) and Alglib to be installed.  All are available for free download, see Excel to Alglib via xlwings for more details.

In the examples below a non-linear function has been fitted to data generated using the same function plus some random scatter using two different Alglib modules:

  • The lsfit module (least-squares fitting)
  • The minlm module (Levenberg-Marquardt optimizer)

For each of these modules there is one user defined function (UDF) that uses a function entered as text on the spreadsheet, and another that calls a python function.

The lsfit versions (  xl_NLFitwFunc and  xl_NLFitwText) are described with examples on the Fit NonLin sheet:

The minlm versions (xl_minlm_vb and xl_minlm_vbtext) are found on the LM Solvers sheet:

The example shown below uses all four functions to fit a function of the form:
a * exp(b*x) + c * x + d
to the data shown in the screenshot below:

The function xl_minlm_vb (shown below) has the following arguments:

  • x (B42:B45), Initial guess of function parameters, single column range
  • xydat (B22:C34), Data to be fitted, two column range
  • Func (C37), Fitting function; a function in NLFuncs.py
  • bndl (B48:E48), Lower bound values, single row range
  • bndu (B49:E49), Upper bound values, single row range
  • epsx (B51), Maximum error
  • maxits (B52), Maximum iterations
  • out (0), Out = 0 returns results, anything else returns termination type code

Results of the four functions are shown below:

Code for the xl_minlm_vb function is described below.  Open source code for all the others is available in the download zip file.

The function xl_minlm_vb (automatically generated by xl wings) calls the Python function of the same name:

@xw.func
@xw.arg('x', ndim=1)
@xw.arg('bndl', ndim=1)
@xw.arg('bndu', ndim=1)
@xw.ret(transpose= True)
def xl_minlm_vb(x, xydat, funcname, bndl, bndu, epsx, maxits, out = 0):
    """
        Minimise a named function with Alglib minlmcreatev  Levenberg–Marquardt algorithm
        Initial guess of function parameters, single column range
        Data to be fitted, two column range
        Fitting function; a function in NLFuncs.py
        Lower bound values, single row range
        Upper bound values, single row range
        Maximum error, 0 = automatic
        Maximum iterations, 0 = unlimited
        out = 0 returns results, anything else returns termination type code
    """
    if type(x) == tuple: x = list(x)
    if type(bndl) == tuple: bndl = list(bndl)
    if type(bndu) == tuple: bndu = list(bndu)
    nx = len(x)
    nf = len(xydat)
    func = getattr(NLFuncs, funcname)
    state = xal.minlmcreatev(nx, nf, x, 0.00001)
    xal.minlmsetbc(state, bndl, bndu)
    xal.minlmsetcond(state, epsx, maxits)
    xal.minlmoptimize_v(state, func, None, xydat)
    x, rep = xal.minlmresults(state)
    if out == 0:
        return x 
    else:
        return rep.terminationtype
  • if type(x) == tuple:  … Arrays are passed from Excel to Python as tuples, but Alglib requires a list.
  • func = getattr(NLFuncs, funcname): looks in NLFuncs.py for a Python function with the name given by funcname, and assigns this function to “func”.
  • state = xal.minlmcreatev( … x, rep = xal.minlmresults(state):  calls Alglib minlm functions to adjust the parameters passed to func in the “state”  object to minimize the square of the results.

“func” in this case is xl_expfunc:

def xl_expfunc(x, fi, func_dat):
    nf = len(func_dat)
    func_dat  = np.array(func_dat) 
    for i in range(0,nf):
        X = func_dat[i,0]
        fi[i] = x[0]*np.exp(x[1]*X)+x[2]*X+x[3] - func_dat[i,1]
    return

Note that Alglib minimises the square of the array returned by xl_expfunc by adjusting the values in the x array.  To use this for curve fitting purposes we have to subtract the supplied Y value from the exponential function:
fi[i] = x[0]*np.exp(x[1]*X)+x[2]*X+x[3] – func_dat[i,1]

Alglib adjusts the function parameters until the error or maximum iterations criteria are satisfied, then extracts the required results from the “state” object with:
x, rep = xal.minlmresults(state)

The x array is then returned to the spreadsheet, via the VBA function.

Advertisements
Posted in AlgLib, Curve fitting, Excel, Link to Python, Maths, Newton, NumPy and SciPy, UDFs, VBA, xlwings | Tagged , , , , , , , , , , | Leave a comment

Working with implied units

The Units4Excel spreadsheet provides a number of functions to convert between different units and evaluate functions entered as text on the spreadsheet, taking account of the units of both the inputs and the results.  The latest version can be downloaded from:

Units4Excel.zip

I have now added some examples comparing reinforced shear capacity to the ACI 318 design code, using the “customary US units” and SI units versions of the code:

The examples shown below use the user defined functions (UDFs) Eval and EvalU.  Eval returns the value of any function entered as text on the spreadsheet, substituting given values for any parameters.  EvalU is a “unit aware” version, returning the result in base SI units, or in any specified units.  EvalU returns the calculated units if they are not specified, or an error message if the specified output units are not consistent with those calculated from the input units.

Evaluating formula 11-4 as given in the code produces strange results, even if consistent input units are used.  If the input units are ignored (using the Eval function) the formula returns the correct results, provided that the input has consistent units, but using EvalU with no specified return unit finds a unit with non-integer exponents, and including m, rather than just N.  If the return units are specified, as a force unit, this results in an error message, since the specified return units are different to those calculated from the input units:

The problem is that the code formula contains implied units, since Nu/(2000*Ag) has units of stress (rather than no units), and fc^0.5 has units of stress^0.5, rather than stress.  The units can be made consistent by converting the 2000 (or 14 for the SI version) to an input parameter, Kc1, with stress units, and adding a new parameter, Kc2, also with stress units, as shown below:

EvalU now returns calculated units of N, as expected, and both versions of the formula return very similar results in N.  Also note that when the output units are consistent with the input units (either lbf or N), EvalU and Eval return exactly the same value, as would be expected.

It is also possible to incorporate the initial constants (2 or 0.17) in the Kc1 and KC2 factors, and to use inconsistent input units.  In the first example below Kc1 and Kc2 have been changed to ksi units, and Kc2 is factored by 2^2, so the initial 2 factor can be removed.  In the SI example the units of Kc1 and Kc2 are unchanged, but Kc2 is factored by 0.17^2:

With the revised formulas the EvalA results are correct, but the Eval results, ignoring the input units, are now incorrect, because the input units are inconsistent (stresses are in ksi or MPa, but forces are in lbf or kN).

Formula 11-6 has no factors with implied units, so can be evaluated with no additional parameters, but as before, the input and output units must be consistent if Eval is used.  EvalU will return correct results with any recognised moment output units:

The results of 11-6 can now be used to evaluate 11-5 for sections with combined axial load and shear:

Finally the upper limit to the shear force for combined axial load and shear is found with formula 11-7.  Note that in this case the final expressions in the formula have a different form in the two versions of the code (1 + Nu/(500*Ag) for the US units version and 1 + 0.29*Nu/Ag for the SI version).  Both have been converted to the form 1 + Nu/(Kc2*Ag), so Kc2 can have units of stress, so for the SI version the value of Kc2 becomes 1/0.29:

Two additional points to note when using EvalU:

  • The units are case sensitive, so if Excel helpfully insists on correcting MPa to Mpa, the units will not be recognised.  To avoid this problem go to File-Options-Proofing, click AutoCorrect Options, and either unselect “Correct TWo Initial CApitals, or click Exceptions and add MPa to the exceptions list on the Initial CAps tab.  For a quick one-off you can also enter a space after MPa, then press Ctrl-Z.
  • Both input units and output units (if used) must be recognised units with the correct abbreviation.  The ListNonSI UDF will list all recognised units for any specified unit type, for example “area” as shown in the screen-shot below.  To display all the results enter the function with the unit type, then press Ctrl-Shift-S to resize the output array.

 

Posted in Concrete, Excel, Maths, Newton, UDFs, VBA | Tagged , , , , , , , , , , , , | 1 Comment

Stepping through 3DFrame

Following a recent comment this post looks at the code for setting up the stiffness matrix for a 3D frame, and how to step through any selected part of the code.

The general procedure for setting up a 3D stiffness matrix, and rotating it to the global coordinate system is given at the link above.  The 3DFrame spreadsheet (including full open-source code) can be downloaded from:

3DFrame.zip

The main program for the 3D analysis is the sub-routine p44, found in the mFrameP44 module in the Visual Basic Editor (VBE), which can be accessed by:

  • From the spreadsheet, press Alt-F11, or click the Visual Basic icon on the Developer tab*
  • Double-click the mFrameP44 module in the Project Explorer window.
  • Click the right-hand dropdown box above the editor window, and select P44

*  If the Developer tab is not visible, go to File-Options-Customize Ribbon, and select it as one of the main tabs.

The code for setting up the global stiffness matrix is:

    ' !---------------------global stiffness matrix assembly------------------------
elements_2:
' Range("kga").ClearContents
    For iel = 1 To Nels
        For i = 1 To nod
            Num(i) = g_num(i, iel)
            For j = 1 To NDim
                Coord(i, j) = g_coord(j, Num(i))
            Next j
        Next i

        ReDim km(ndof, ndof)
        i = BeamConER(iel, 6)
        If i = 0 Then
            ReDim R0(1 To 3, 1 To 3)
            Call rigid_jointed3(km, Prop, Gamma, Etype, iel, Coord, R0)
        Else
            For j = 1 To 12
                SpringA(1, j) = BeamHinge(i, j + 1)
            Next j
            Call spring_jointed3(km, Prop, Gamma, Etype, iel, Coord, R0, SpringA)
        End If

        For i = 1 To ndof
            g(i) = g_g(i, iel)
        Next i

        Call formkv(kv, km, g, neq, iel, g_coord, Num)
    If iel = Range("kmbeam").Value2 Then Range("km") = km
    Next iel    ' elements_2

This code loops through each beam element in the frame and calls either the rigid_jointed3 or spring-jointed3 routines for beams with fixed or spring end conditions respectively. The 12×12 matrix returned for each beam is then added to the global stiffness matrix with the formkv routine.

To step through the code in the p44 routine simply make sure that the cursor is somewhere within the body of the routine and press F8, or Debug-Step Into.  To run the routine to a selected point either:

  • Set a breakpoint at the required location by clicking in the left hand margin, then press F5 or Run-Run or
  • Select the line to run to, then press Ctrl-F8 or Debug-Run to cursor.

There is also a separate function, rigid_jointed3TT, that can be called from the spreadsheet for a specified beam, and returns the beam 12×12 stiffness matrix to the spreadsheet as an array function.  The instructions below show how to step through this function line by line.

To step through a VBA function it must either be called from a sub-routine, or initiated from the spreadsheet.  To do the latter:

  • Create a breakpoint somewhere within the VBA code by clicking in the left-hand margin.
  • Select a cell in the spreadsheet that calls the chosen function.  The screenshots below use cell C35 in the Matrix examples spreadsheet.
  • Press F2 (edit), then either enter for a function that returns a single value, or in this case Ctrl-Shift Enter to return an array.  The function will then run to the selected breakpoint:

You can now step through the code one line at a time by pressing the F8 key (or Debug-Step Into).  Note that the Locals Window, under the main Editor window displays the status and value of all variables, and you can also see the value of a variable by simply placing the cursor over the variable name anywhere in the code:

Procedures for stepping through the code are then the same as for a sub-routine.  To run to a selected line press Ctrl-F8 or right click and Run to cursor:

At the end of the function it will either return the end result (the km2 array), or one of the four intermediate arrays, depending on the value of the Out argument:

The Debug menu provides further options, for example allowing control to jump directly to a specified point, or for called routines to be run in a single step.

For more details of VBA debugging options see Chip Pearson’s Debugging VBA.

Posted in Beam Bending, Excel, Finite Element Analysis, Frame Analysis, Newton, UDFs, VBA | Tagged , , , , , | Leave a comment

Setting up UDF Applications

Following a recent comment, this post looks at setting up a user defined function (UDF) on a blank worksheet, using the ConBeamU function as an example.

To use a UDF, the VBA code for the function must either be included in the active workbook, or another open file.  In most cases the VBA code can be copied from the master file to a new workbook, but in the case of the ConbeamU function the units related code refers to a database of units data in the master file.  The easiest way to set up a new file for this case is to save the master file with a new name, and delete all the unwanted sheets.  I have added a ConbeamU-template file to the ConbeamU zip file, that has all the required code and units data, two sheets with notes and instructions, and a single blank sheet.  The new zip file can be downloaded from:

ConbeamU.zip

The screen shots below illustrate how to set up a new application of the ConbeamU function, starting from a blank worksheet.

First copy details of the function arguments from the Functions sheet, and enter or copy and paste the input data.  Each data range may be placed in any convenient position.  For the ConbeamU function the required data ranges are:

  • Segments: details of each beam segment with different section properties, with distance of the segment end from the start (left hand end) of the beam, flexural stiffness (EI), and optionally shear stiffness (GA).
  • Outpoints: a single column with either the number of output sections for each span, or a list of output point positions (see below for more details).
  • Supports: position of each support, with optionally translational stiffness, rotational stiffness, and displacement.
  • Dloads: extent and magnitude of distributed transverse loads.
  • PLoads: location and magnitude of point transverse loads and moments.
  • OutUnits: single row with units for output results: position, shear force, bending moment, slope (leave blank), and deflection.
  • Out: output index, optional, default = 1 (output as listed above).
  • ListOutPoints, optional, default = False, output at specified number of sections/span.  See below for details of True option.

Note that for the ConBeamU function (and other unit aware functions) all data ranges with values that have a unit must be headed by a valid unit symbol:

After entering all required input data, enter the function in the top-left cell of the output range.  The easiest way to ensure that all arguments are provided, in the right order, is to enter the function name, including the opening bracket, then click the “insert function” button, just to the left of the edit bar, as shown below

As each function argument is selected the contents of the cell or range are shown to the right of the data box.  When all the required arguments are selected the function results appear under the data boxes.  In this example the default values are what we want for the two optional arguments, so we can leave them blank.  Click OK or press enter to enter the function:

At this stage only the first value of the results array is visible:

The standard Excel way to display all the contents of an array function is:

  • Select the required output range, with the array function in the top-left corner.
  • Press F2 (edit).
  • Press Ctrl-Shift-Enter

However the ConbeamU spreadsheet has a macro that will do this automatically, so with the cell containing the entered function still selected, just press Ctrl-Shift-S.

The default output option is to specify the number of output sections for each span.  An alternative is to list the position of each section where output is required.  In the example below two points are entered at each internal support, and each point with a point load or moment.  To display the output set the final “ListOutPoints” argument to True, and select the list of locations range as the “OutPoints” argument.  Having edited the function, press Ctrl-Shift-Enter to enter:

As before, to re-set the output range to display only the cells with valid output values, press Ctrl-Alt-S:

Note that the list of output point locations does not need to be adjacent to the output range, it can be anywhere in the workbook, or even in another workbook.

The procedures described above are applicable to any Excel function or UDF that returns an array function, except that resetting the output range to display the whole array by pressing Ctrl-Shift-S only works in those workbooks where the required macro has been added.

Posted in Arrays, Beam Bending, Excel, Finite Element Analysis, Newton, UDFs, VBA | Tagged , , , , | 1 Comment

UDF to replace cell references with values

I recently posted an update to the Eval2 spreadsheet with a subroutine to display a cell formula as a text string, with the cell references replaced by the associated values.

I have now added three user defined functions (UDFs) to do the same job:

  • Addr2Val1 takes a text string as input and returns the formula with all cell references or range names converted to the value in the referenced cell.  Cell references may be anywhere on any spreadsheet.
  • The Eval function has been modified to work with cell references and range names, as well as optional lists of parameter symbols, and their associated values.
  • Addr2Val2 works the same as Addr2Val1, except the input is an active cell formula, rather than a text string.

The new file may be downloaded from:

Eval2.zip

Examples of each function are shown in the screenshot below, followed by source code for the Addr2Val2 function.  Full open-source code for the other functions is included in the download file.

Updated 3 Sep 2017: There seem to be issues with the WordPress system removing line breaks in some places, and inserting them in others. I have corrected the code below, but if you want to copy the code I recommend doing so from the download spreadsheet, rather than from the listing below, which probably still has some errors.

Function Addr2Val2(FuncRng As Range, Optional CommaDec As Boolean = False) As Variant
 Dim NumChar As Long, ParamDict As Scripting.Dictionary, i As Long, CheckC As String, AscCheckC As Long, CheckP As String, NewFunc As String, iErr As Long
 Dim ParamRng As Range, NumParam As Long, PVal As Variant, IsRng As Boolean, CheckRng As String, Func As String

    ' Evaluate a cell formula (Func), replacing cell addresses or range names with the values in the referenced cells.

    'Func is a single cell containing the formula to be evaluated
    
    ' CommaDec = True to convert commas to decimal point and semi-colons to commas
    ' CommaDec = False (default) for no convertion.
    
    Func = Trim(FuncRng.Formula)
    Func = Replace(Func, "$", "")
    
    If CommaDec = True Then
    ' Replace all , with . and ; with ,
        Func = Replace(Func, ",", ".")
        Func = Replace(Func, ";", ",")
    End If
    
    NumChar = Len(Func)
    i = 1
    Do While i <= NumChar
        CheckP = "" 
            Do 
                CheckC = Mid(Func, i, 1) 
                AscCheckC = Asc(CheckC) 
                If (AscCheckC > 64 And AscCheckC < 91) Or (AscCheckC > 96 _               And AscCheckC < 123) Or AscCheckC = 95 Or AscCheckC = 33 Then
                CheckP = CheckP & CheckC
                i = i + 1

       ElseIf CheckP <> "" And Asc(CheckC) > 47 And Asc(CheckC) < 58 Then 
                CheckP = CheckP & CheckC
                i = i + 1
            Else
                ' Check if CheckP is a cell address or range name
                On Error Resume Next
                CheckRng = ""
                IsRng = False
                CheckRng = TypeName(Range(CheckP))
                If CheckRng = "Range" Then IsRng = True
                If IsRng Then
                    NewFunc = NewFunc & Range(CheckP).Value2 & CheckC
                                
                ' else leave it unchanged
                Else
                    NewFunc = NewFunc & CheckP & CheckC
                End If
                i = i + 1
                Exit Do
            End If
        Loop

    Loop

    Addr2Val2 = NewFunc
End Function

Posted in Excel, UDFs, VBA | Tagged , , , , | Leave a comment

Jansch and Renbourn live

Three recently uploaded You Tube recordings of live performances by guitarists Bert Jansch and John Renbourn.  As further evidence that the quality of a You Tube video is inversely proportional to the number of views, none of the three have over 1000 views, and the first has just 6.

Bett’s Dance; Bert Jansch live at the 12 bar:

Bert Jansch and John Renbourn; un-named instrumental, live from Massachusetts, 1990

From the same concert, John Renbourn plays So Early in the Spring:

 

Posted in Bach | Tagged , , | Leave a comment

Scottish Fiddlers

Yesterday I saw “Scotland the Brave” at the Sydney Opera House, featuring the Willoughby Symphony Orchestra and Choir, and fiddler extraordinaire Marcus Holden:

whose playing of Scottish fiddle music reminded me of Robin Williamson and the Incredible String Band:

and lead to the discovery that Marcus Holden also plays in a string band, The Jugalug String Band:

Posted in Bach | Tagged , , , , , | Leave a comment