# The mcx matrix calculator

### Table of Contents

- Table of Contents
- Preliminaries
- 1. Introduction
- 2. Examples
- 3.
**mcx**manual - Repeated Iteration of Command Line Arguments
- Other resources

### Preliminaries

You should be aware of the standard Questaal format (which, among other formats, **mcx** can read) which has programming language capabilities.

**mcx** is required and is assumed to be in your path.

This manual is written for version 1.058. To see what version you are using, do:

```
$ mcx --version
```

### 1. Introduction

**mcx** is a matrix extension of an ordinary calculator. It is mainly designed to work with 2D arrays; it operates on numerical arrays. Usually the arrays are stored in ASCII format, though **mcx** has the ability to read binary files.

**mcx** is command-line driven and can efficiently manipulate arrays for convenient analysis. Some Questaal testing scripts make use of this calculator to analyze whether a test passes a certain tolerance.

**mcx** is stack-based: each time you read an array from a file it is pushed onto the stack. You can assign the top-level array to a name, in which case an independent copy is made.

The ASCII format of the file is quite flexible; you can specify the number of rows *nr* and number of columns *nc* in a number of ways:

You can specify the number of columns on the command line, with

**−nc=#**; similarly with the number of rows (**−nr=#**).**−nc=#**; this switch is used in Example 2.2.You can include a directive at the start of the file, before any data is read.

**mcx**passes the input through a file preprocessor first; in addition a directive specifying some combination*nr*and*nc*, e.g.`% rows #1 cols #2`

supplies the needed information to

**mcx**.In the absence of an explicit specification,

**mcx**will infer*nc*by counting how many numbers (more generally expressions) are on the first line.

If *nc* is obtained somehow but not *nr*, **mcx** determines it by counting the number of expressions in the entire file and dividing by *nc*.

*Note:* : if the **−nc=#** or **−nr=#** switches are not used, **mcx** determines them from the standard Questaal algorithm for reading 2D arrays, described in more detail here.

### 2. Examples

This section develops a few examples to provide an intuitive feel for how **mcx** works and to illustrate some features.

A systematic description of **mcx**’s features and arguments is given in Section 3.

Cut and paste the following data into array into file *mat1*

```
1.1 2.2
3.3 4.4
```

and this data into file ** mat2**.

```
5 6
7 8
```

#### Example 2.1. Add *mat1* to *mat2*

*mat1*

*mat2*

If you do the following:

```
$ mcx mat1 mat2 -+
```

you should see

```
% rows 2 cols 2 real
6.100000 8.200000
10.300000 12.400000
```

Any argument that begins with **“−“** is a switch or an operator (as distinct from data). The string following **“−“** names the operator.

Thus:

**mat1**: reads filefrom disk and pushes it onto the stack; call it*mat1**s*_{0}.**mat2**: reads fileand pushes it onto the stack so*mat2**s*_{0}→*s*_{1}and the contents of→*mat2**s*_{0}.**−+**: is the binary operation “add.” If*s*_{0}and*s*_{1}exist, they are summed and popped off the stack. Their sum is pushed onto top-level array*s*_{0}.

*Note:* if operations occur before arrays are given, they are push on an operations stack and execute when arrays become available. The following commands accomplish the same thing

```
$ mcx -+ mat1 mat2
$ mcx mat1 -+ mat2
$ mcx mat1 mat2 -+
```

Use **−show** to see the stack. Try this

```
$ mcx -f2f8.1 mat1 mat2 -show -+
```

You should get

```
# 0 named arrays, 2 on stack; pending 0 unops 0 bops (vsn 1.057)
# stack nr nc cast
# 2 2 2 real
# 1 2 2 real
% rows 2 cols 2 real
6.1 8.2
10.3 12.4
```

**−show** prints out what arrays are in the calculator; the addition is subsequently performed and the result printed.

**“−f…“** is a formatting statement; **2f8.1** follows the fortran convention for formatting real numbers.

#### Example 2.2 Standard input and standard output

After all the command-line arguments are parsed, usually **mcx** prints the top-level array *s*_{0}, and exits silently.

However,

If

*s*_{0}is not present,**mcx**waits for you to enter an array from standard input.

For example try`$ mcx -f2f8.1 $ 1 2 $ 3 $ 4 <ctrl-D>`

**mcx**should print out`% rows 2 cols 2 real 1.0 2.0 3.0 4.0`

If the last command is

**−show**,**mcx**just prints out information about named arrays and the stack and exits (even if the stack has no arrays). Thus the following command`$ mcx -f2f8.1 mat1 mat2 -show -+ -show`

does the same as it did in Example 2.1 but prints out stack contents a second time instead of

*s*_{0}.

You can also tell **mcx** to read the next array from standard input by using a full stop (**“ . “**) in lieu of file name. Thus

```
$ echo 1 -2 3 -4 | mcx .
```

pushes a 1×4 array onto *s*_{0}, while

```
$ echo 1 -2 3 -4 | mcx -nc=2 . mat2 -+
```

pushes a 2×2 array onto *s*_{0}, then pushes ** mat2**, and adds the two so that a single array remains on the stack.

**−nc=2**tells

**mcx**to treat the line from

**as an array with 2 columns.**

*stdin*#### Example 2.3 Manipulations of eigenvalues and eigenvectors of an array

This example finds the eigenvalues *e* and eigenvectors *z* of ** mat2**, and shows that

*mat2 z = z e*.

First try

```
$ mcx mat2 -evl
```

You should see a 2×1 complex array

```
% rows 2 cols 1 complex
-0.152067
13.152067
0.000000
0.000000
```

The eigenvalues are complex because the matrix is not hermitian. The imaginary part follows the real part; this is how **mcx** displays and reads complex arrays in ASCII format.

You can force ** mat2** to be hermitian (symmetric since

**is real) with**

*mat2***−herm**. Now do

```
$ mcx mat2 -herm -a h h -evl h -evc -ap z -tog -v2dia -x h z -x --
```

These instructions do the following:

**mat2**: reads fileand pushes it onto*mat2**s*_{0}.**−herm**: symmetrizes*s*_{0}.**−a h**: copies*s*_{0}to a named array*h*and pops it from the stack. The stack is now empty.**h −evl**: pushes*h*onto*s*_{0}and replaces*s*_{0}with its eigenvalues. Because*h*is hermitian,*s*_{0}is real.**h −evc**: pushes*h*onto the stack and replaces*s*_{0}with its eigenvectors. Now the stack has two arrays,*s*_{0}=*z*and*s*_{1}=*e*.**−ap z**: copies*s*_{0}to a named array*z*.**−tog**: toggles*s*_{0}and*s*_{1}, making*s*_{1}=*z*and*s*_{0}=*e*.**−v2dia**: Turns*s*_{0}(a vector or 2×1 array of eigenvalues) into a diagonal 2×2 array.**−x**: multiplies*s*_{1}×*s*_{0}. One array remains on the stack,*s*_{0}=*z × e*.**h z −x**: pushes*h*and*z*onto the stack and multiplies them. Now*s*_{0}=*h × z*, while*s*_{1}=*z × e***−−**: Adds*s*_{1}to −*s*_{0}.*s*_{1}and*s*_{1}are mathematically identical so the difference should be zero.

*Note:* this formula should still work even if *h* is not hermitian.

#### Example 2.4 Numerical integration, differentiation, and interpolation of a function

Integrate and differentiate the function , by tabulating it on a mesh and evaluating integrals and derivatives numerically. This example also uses the tabulated data to interpolate it to another mesh.

Copy the contents of the box below into file *dat*.

```
% const n=100 p=1 lam=2
% save
% macro iy(z) exp(-lam*z)*(-lam*z-1)/lam^2
% repeat i= 0:n
% var x=10*i/n
{x} {x^p*exp(-lam*x)}
% end
```

This creates 101 rows of *xy* pairs with *x* ranging between 0 and 10.

*Note:* : The number of points *n*+1, and also *p* and λ are declared in the file with the **const** preprocessor directive. You can override the values assigned there with command-line arguments, e.g. **−vp=#**. The **save** directive retains the variables declared in this file after it is read. The **macro** directive will be used for the indefinite integral, below.

##### Derivative

The derivative is readily found to be

Try some of the following commands

```
$ mcx -vp=2 dat -diff
$ mcx -vp=2 dat -diff -e3 x1 x2 'x1^(p-1)*(p-lam*x1)*exp(-lam*x1)'
$ mcx -vp=2 -f3f15.10 dat -diff:nord=5 -e3 x1 x2 'x1^(p-1)*(p-lam*x1)*exp(-lam*x1)' -e3 x1 x2 x2-x3
```

All of them differentiate the second column with respect to the first, using *p*=2.

- The first returns two columns with
*x*and*y*′. - The second returns three columns with
*x*,*y*′, and the analytic derivative of*y*′. - The third returns three columns (more decimals) with
*x*,*y*′, and the error in the numerical estimate for*y*′.

Unpacking the third command:

Argument | Function |

−vp=2 | Declares variable p and assigns the value 2. This overrides the assignment in dat. |

−f3f15.10 | Formats output (fortran format 3f15.10) |

dat | read s_{0} from dat. |

−diff:nord=5 | Replace column 2 with a numerical estimate for y′Use a 5-point polynomial to interpolate the data; estimate is derivative of polynomial interpolation |

−e3 x1 x2 ‘x1^(p−1)*(p−lam*x1)*exp(−lam*x1)’ | replace s_{0} with a three column array consisting of x, y′, and the analytic derivative of y′ |

−e3 x1 x2 x2−x3 | replace s_{0} with a three column array consisting of x, y′, and difference between the numerical and analytical y′ |

Some observations:

- The largest error appears for
*x*→0. The interpolation is less accurate when all the data lies on one side of the interpolating point. - The error improves when higher order polynomials (
**nord=#**) are used, or when the mesh is made finer (**−vn**=#). - If you use
*p*<1,*y*′ diverges at the origin. The numerical derivative cannot reproduce this.

##### Integral

*y* is small at *x*=10 if λ=−2, provided *p* is 3 or less; so we will use 10 in place of ∞.

Try some of the following commands

```
$ mcx -vp=1 dat -int 0 10
$ mcx -vp=2 dat -int 0 10
$ mcx -vp=3 dat -int 0 10
```

These commands calculate *I* between 0 and 10 for three values of *p*.

You should find that *I* is close to , i.e. 1/4, 1/4, and 3/8 for *p*=1,2,3. Some numerical errors appear in the 6^{th} digit. The integral is carried out by fitting the data to a polynomial of order **nord−1**, and integrating the polynomial. You can reduce the error by using a higher order than the default value of 4 for **nord**, viz

```
$ mcx -vp=3 dat -int:nord=6 0 10
```

Also for *p*>3 you should increase the upper bound beyond 10 since the integral from 10 to ∞ is on the order of 10^{−6}.

##### Indefinite Integral

As noted *I* must be evaluated between definite limits. However, you can make **mcx** simulate an indefinite integral by integrating over a range of upper bounds.

To compare with exact results, note that when *p*=1 the indefinite integral is

The **macro** in *dat* evaluates this integral.

Try the following:

```
$ mcx dat -int 0 0:2:.2 -e3 x1 x2 'iy(x1)-iy(0)'
```

This evaluates *I* for a lower bound of 0 and a uniform mesh of points for the upper bound between 0 and 2. In the third column the analytic integral is evaluated from the macro **iy** at the upper and lower bounds.

You should that the numerical and analytic integrals agree to about 6 decimal places:

```
% rows 11 cols 3 real
0.000000 0.000000 0.000000
0.200000 0.015387 0.015388
0.400000 0.047801 0.047802
0.600000 0.084343 0.084343
0.800000 0.118767 0.118767
1.000000 0.148498 0.148499
1.200000 0.172890 0.172890
1.400000 0.192230 0.192230
1.600000 0.207200 0.207200
1.800000 0.218578 0.218578
2.000000 0.227105 0.227105
```

##### Interpolation

Interpolation proceeds much in the same was as integration; only interpolation has lower bound. Try

```
$ mcx -vp=2 dat -intrp .5:1:.05 -e3 x1 x2 'x1^p*exp(-lam*x1)' -e3 x1 x2 x2-x3
```

This returns the abscissa on a mesh twice finer than the original mesh. Every odd point is perfectly interpolated (they lie on the original mesh); the even points reflect the error of the interpolation.

### 3. **mcx** manual

**mcx** is a stack-based, command-line driven calculator for matrices. Matrices reside on the stack, ordered as *s*_{0}, *s*_{1}, … . There are unary operators that operate on the top-level element *s*_{0}, replacing it with some transformation, and binary operators that operate on *s*_{1} and *s*_{0} replacing both of them with the result of some operation, e.g. *s*_{1} × *s*_{0}.

Usage:

mcx [−switches]data-file-ops …

Arguments that do not begin with **“−“** or **”[“** must be files, stored in the form of a 2D array. For ASCII files, data is read using the standard Questaal format which features programming language capabilities.

Any argument that begins with **“−“** is a switch, a unary operator, or a binary operator.

Any argument that begins with **”[“** is a declaration of a command-line looping construct where command arguments between **”[“** and **”]”** are iterated over, as described below.

When a file is read, its contents (together with the number of rows *nr* and columns *nc*) is pushed onto the stack and becomes the top-level stack element *s*_{0}. Elements already existing on the stack get pushed down one level. If there are *n* such elements, *s*_{i-1} → *s _{i}* for

*i*=

*n*,

*n*−1, …, and the new element becomes

*s*

_{0}.

Data is normally read from a file; however if *data-file* is a full stop (“ **.** ”), data is read from standard input in lieu of a file. (It can occur only once). See Example 2.2.

#### Switches

**−nc=# (nr=#)**

Stipulate that next matrix read has # columns (rows). See Example 2.2 for an illustration.**−vvar=#**

define variable**−vvar**assign value to**#**. This is the standard way Questaal programs assign variables from the command line. Since data files are parsed by the preprocessor, such variables may enter into preprocessor directives or as part of expressions in the data itself.**−show**

show data stack and any operations pending. See Example 2.1.**−w[l=string]***fname*| −bw[l=string]*fname*

write*s*_{0}to file*fname*.**−bw**writes to a binary file.**−wap**

(complex arrays only) write*s*_{0}as (amplitude,phase) rather than (Re, Im).**−a[***nr*=#|:*nc*=#] nam | −ap nam

assign*s*_{0}to**nam**, and pop*s*_{0}off the stack.**−ap nam**performs the assignment but does not pop*s*_{0}off the stack.**−av[***ir*,*ic*] var

assigns scalar variable**var**to element from*s*_{0}(*ir*,*ic*). If*ir*and*ic*are specified, the (1,1) element is used.**-r~switches**

switches are separated by**’~’**; you can use a different character instead. Switches are:~qr read with fortran read (fast, no algebra) ~s=# skips # records before reading ~open leaves file open after reading; thus if you read the file again it will read the next array ~br read from binary file, using first record to read *nr*and*nc*~spc load in sparse format ~br, *nr*,*nc*binary file has no first record; user supplies *nr*,*nc***−px[:nprec] | −pxc**

write in row (column) compressed storage format, displaying only elements larger than a tolerance By default the tolerance is 10^{−8}. If**nprec**is supplied, the tolerance is By default the tolerance is 10^{nprec}.

#### Unary operators

These operators act on the top level matrix, *s*_{0}, and replace it with the result of the unop.

Where *expr* appears in the switches below, algebraic variables declared from directives may be used; also you can use **x n** for the contents of the

*n*

^{th}column.

**−p:**

Pushes top array*s*_{0}onto stack, duplicating it.**−p+n (-p-n):**

Pushes nth array (from bottom) onto stack**−pop:**

Pops*s*_{0}from stack, shifting the other elements up one level.**−csum[:list]**

Sums the columns of*s*_{0}, replacing it with a matrix of one column.**−rsum[:list]**

Sums the rows of*s*_{0}, replacing it with a matrix of one row.**−s#:**

Scale*s*_{0}by # (may use -s#real,#imag)**−shft=#:**

Adds a constant # to*s*_{0}**−sort***expr*

Sorts rows*s*_{0}, ordering them by the result of*expr***−i | − iq:**

Inverts*s*_{0}**−1:***n*

Pushes unit matrix, dimension*n*, onto stack**−tp [nc~]***list*

Generates matrix from*list*. Elements in*list*are real numbers; see here for syntax.**−evl | −evc**

Replaces*s*_{0}by its eigenvalues (eigenvectors)**−t:**

Transposes*s*_{0}**−cc:**

Take complex conjugate of*s*_{0}**−herm:**Replaces*s*_{0}with its hermitian part**−real:**

Replaces*s*_{0}with its real part**−v2dia:**

If*s*_{0}is a vector, it is expanded into diagonal matrix.

If*s*_{0}is a square matrix, its diagonal elements are used to form a vector.**−split nam x1,x2…,xn y1,y2,…,yn:**

Splits*s*_{0}into subblocks; assign them names**nam**.*ij***−rep:n1,n2**

Concatenates replicas of*s*_{0}to create (*nr*×*n1*,*nc*×*n2*) array.**−roll:#1[,#2]**

Cyclically shifts rows (and columns, respectively) by**#1**(and**#2**) elements.**−pwr=#:**

Raises (col-1) matrix to a power

##### Row and column Unary Operators

These unops treat *s*_{0} as an array of nr rows and nc columns.

**−rowl***list*| −coll*list*

Creates a new array from a list of rows (columns) of*s*_{0}**−rowl:mirr | −coll:mirr**

Rearranges rows (columns) in reverse order**−rowl:pf=***fnam*| −coll:pf=*fnam*| −rowl:ipf=*fnam*| −coll:ipf=*fnam*

Same as −rowl (−coll) but*list*is read from permutation file*fnam*.**ipf**reverses the sense of the permutation.

*Example:*file*perm*contains 2 3 1 4. Then**−rowl:pf=perm**returns*s*_{0}with rows 2,3,1,4 permuted into 1,2,3,4**−rowl:ipf=perm**returns*s*_{0}w/ rows 3,1,2,4 permuted into 1,2,3,4

**−inc***expr*

Retains rows from*s*_{0}for which*expr*is nonzero.**−sub t,b,l,r | -sub t,l**

Extracts a subblock of*s*_{0}. In second form, bottom right corner = (*nr*,*nc*).**−subs:# t,b,l,r**

Scales a subblock of*s*_{0}by**#**.**−subv:# t,b,l,r**

Copies # to subblock of*s*_{0}.**−e#***expr1 expr2*…*expr*#

Create new matrix of**#**columns with values*expr1 expr2*…*expr*# .

Variables**x1**,**x2**, … can be used in the expressions. These variables refer to the elements of*s*_{0}in columns 1, 2, …, and change with each row.

This switch is used in Example 2.4.

*Note:*: at present this switch does not work for complex elements.

##### Unary Operators that treat data as discretized continuous functions of the first column

In these unops, column 1 (**x1**) is treated as an the independent variable.

Three of these unops fit data in other columns with a polynomial in **x1**. The polynomial is used to differentiate (**−diff**), integrate (**−int**), or interpolate (**−intrp**) data in the remaining columns. There are some modifiers (called **[:opts]** in the documentation below):

**:nord=#**will change the number of points in fitting polynomial to**#**(polynomial order is**#−1**). The default value is**#=4**.**:rat**will use a rational polynomial, rather than an ordinary one. Good for data with singularities.**:mesh**(applies to**−int**only) replaces*s*_{0}with its integral on the same mesh as the original set of values**x1**. A trapezoidal rule is used at present.

You can string the options together. These three unops are demonstrated in Example 2.4.

**−diff[:opts]**

differentiates columns 2…*nc*wrt column 1, evaluated at each**x1**.

*s*_{0}is returned as a table of points within the first column and the result of the derivative in the remaining columns.*list***−int[:opts] xlo***list*

integrates columns 2…*nc*wrt column 1 from the lower bound**xlo**to a set of upper bounds, given by.*list*

*s*_{0}is returned as a table of points within the first column and the result of the integral in the remaining columns.*list***−intrp[:opts]***list*

interpolates column 2 to points in*list*.

*s*_{0}is returned as a table of points within the first column and the result of the interpolation in the remaining columns.*list***−smo width,xmin,xmax,dx:**

smooths vector of delta-functions with gaussians**−abs**

takes the absolute value of each element**−max[:i|g]**

puts the largest element into the first column.

Optional g returns max value of the entire array

Optional i returns index to max value.**−unx[:i1] #1,#2**

(uncross) exchanges points in columns #1 and #2 after their point of closest approach**−unx2[:i1] #1,#2**

(cross) exchanges points in columns #1 and #2 at their point of closest approach, if they do not cross.**−at[:i1] val***expr*

find adjacent rows that bracket*expr*=*val*. Array contains linearly interpolated**x1**and*expr*.**−nint[#1]**

Replaces each element with nearest integer.

Optional #1: scale array by #1 before operation, and by 1/#1 after. Thus -nint:1000 rounds to nearest thousandth.

#### Binary Operators

These operate on the top two matrices in the stack, *s*_{1} and *s*_{0}. *s*_{1} and *s*_{0} are popped from the stack, and the result of the binary operation is put into *s*_{0}.

Dimensions for *s*_{0} and *s*_{1} are not independent; for example if they are added they must have the same number of rows and columns.

**−tog**

Toggle*s*_{0}and*s*_{1}.**−+**

Add*s*_{1}+*s*_{0}**−−**

Add*s*_{1}−*s*_{0}**−x**

Multiply*s*_{1}×*s*_{0}**−xe**

Multiply*s*_{1}and*s*_{0}element by element**−de**

Divide*s*_{1}/*s*_{0}element by element**−x3**

Multiply*s*_{1}*s*_{0}are thou they are 3D arrays:*s*_{1}=*s*_{1}(n11,n21,n31)*s*_{0}=*s*_{0}(n10,n20,n30) where n10=nr(0)/n20,n20=nc(1),n30=nc(0); n11=n10,n21=nr(1)/n11,n31=n20 Result(i,j,k) = sum_m s1(i,j,m) s0(i,m,k) is condensed into 2D (nr(1),nc(0))**−gevl | −gevc**

Same as**−evl | −evc**, but for the generalized eigenvalue problem.*s*_{1}is the nonorthogonal matrix.**−orthos:**

Replace*s*_{0}with*s*_{1}^{-1/2}*s*_{0}*s*_{1}^{-1/2}**−ccat:**

Concatenate columns of*s*_{1}and*s*_{0}into a single array**−rcat:**

Concatenate rows of*s*_{1}and*s*_{0}into a single array**−cross:**

Cross product*s*_{1}(1,1..3) x*s*_{0}(:,1..3)**−suba[#] t,b,l,r | -suba[#] t,l**

Copy*s*_{1}to subblock of*s*_{0}. Conventions for subblock are the same as for**−suba t,b,l,r | -sub t,l**.

Optional # copies #×*s*_{1}into*s*_{0}.**−index:**

Use*s*_{0}as an row index to*s*_{1}.*s*_{0}(i) is overwritten by*s*_{1}(*s*_{0}(i)).*s*_{1}is preserved. New*s*_{0}has row dimensions of the original*s*_{0}and column dimensions of*s*_{1}.

### Repeated Iteration of Command Line Arguments

Command line arguments can be repeated by enclosing them in brackets, with the syntax

```
[ name=list arg1 arg2 arg3 ... argn ]
```

`list`

is a standard Questaal integer list.

**mcx** executes the sequence of command line arguments **arg1**, **arg2**, **arg3**, …, **argn** for each element in **list**. Within this special construct, **arg1**, **arg2**, … are treated as strings that are parsed for expression substitution. In particular, variables *name* and *i* are loaded and recalculated each pass. *name* is the value of the current integer, and *i* is the index in the list. **name=** is not required; **mcx** will use *irpt* as the variable if it is omitted.

The scheme is best explained by a concrete illustratation. Suppose files *a0*, *a1* and *a4* reside on disk, with *a0* a 4×4 symmetrix matrix, *a1* 4×4 hermitian, and *a4* 1×3 real. The command

```
mcx [ k=0,1,4 'a{k}' ] -show
```

should get expanded to a sequence of four arguments

i | k | argument | action | |

1 | 1 | 0 | a{k} | a0 loaded onto stack |

2 | 2 | 1 | a{k} | a1 loaded onto stack |

3 | 3 | 4 | a{k} | a4 loaded onto stack |

4 | - | - | -show | displays stack |

You should see the following:

```
# 0 named arrays, 3 on stack; pending 0 unops 0 bops (vsn 1.058)
# stack nr nc cast
# 3 1 3 real
# 2 4 4 herm
# 1 4 4 symm
```

##### Conditional evaluation of an argument

Within the **[ … ]** construct, an argument that begins with **?** is treated as an expression. The result of that expression determines whether the subsequent argument should be evaluated or skipped. Thus the construct

```
mcx ... [ ... '?expr' argi ... ]
```

parses *expr* as an expression. If it evaluates nonzero, **argi** is executed. Otherwise, **argi** is passed over.

*Example:*

```
mcx [ ix=1:4 'a{ix}' '?ix>1' -rcat ]
```

pushes *a1*, *a2*, *a3*, *a4*, onto the stack. **-rcat** is evaluated after each of the last three arrays are loaded, thus appending that array to the prior one on the stack. The final stack consists of a single array which concatenates *a1*, … , *a4*.

##### Special handling of the last iteration

You can prevent the last argument (or arguments) in the loop from executing, by appending a **/** to the **[**. A single slash suppresses the last argument, two slashes suppress the last two arguments, and so on.

*Example:* cut and paste the 2×2 matrix in the box below to file *a*.

```
1 x
x x*x
```

Then do:

```
mcx [/ 1:3 '-vx={i}' a -+ ]
```

This sums three instances of *a* with *x*=1, *x*=2, and *x*=3. Note that three arrays are loaded but there are only two additions, so **−+** should be suppressed the last iteration.

You should see the following output:

```
% rows 2 cols 2 real
3.000000 6.000000
6.000000 14.000000
```

The (1,1) element is the sum of *x*^{0} for *x*=1…3, while (1,2) and (2,1) are the sum of *x*^{1} and (2,2) is the sum of *x*^{2}.

To see the arguments being parsed in detail, use the `-debug`

switch:

```
mcx -debug [/ 1:3 '-vx={i}' a -+ ]
```

### Other resources

The source code to **mcx** can be found in the bitbucket repository.