Here is the general «three-sweep» technology of construction of a graph in Cartesian coordinates in Mathcad (see the upper part of the fig. 29):
• first step: the appearance of analyzable function is given;
• second step: the range variable is introduced (Range Variable – in our case it is x), which assigns the graph scattering by the axis of ordinates (x start and x end) and number of points on it (not number of points but the interval between them is assigned);
• third step: we give a command to construct the graph, which we can format if it is necessary; for it we call corresponding dialogue windows; the size of the graph is changed with the help of transport of its contour (general «window» technology in Windows).
Starting with 7-th version and upper in Mathcad it is allowable quick (Quick Graph – see the second graph in the fig. 29) technology of construction of «flat» graphs (in Cartesian and in polar coordinates):
• first step: see above; by the way it is possible not to give the analyzable function as the user’s function – it is enough to write it near axis y (it is done on the second graph in the fig. 29)
• second step: we give a command to construct the graph; here some more defaults appear: the graph is constructed in the following range from -10 till 10 by the variable number of points, which depends on the size of the graph; as rule, the second default (number of points) meets user’s wishes (here the compromise between quality of the graph and speed of its construction is kept up). User has a right to break the second default (graph scattering) and to give the other range of argument scattering if he changes the numbers at the ends of the axis x.
The second technology has one more advantage (the main point of the Tip): the variable arg are free from some numerical value, which later on can confuse, for example, analytical (symbolic) solution of the problem. (Continuation of the theme in Tips 65 and 73.)
Apart construction of a graph with prior marking of the argument (three-sweep technology) has the following advantage: the graph, which is construct with the help of two-sweep technology, is badly drawn in Mathcad Explorer (the range is divided on low number of points and the graph is broken line).
This and some next Tips (see Tip 35, Tip 36, Tip 37, Tip 69 and others) will be illustrated by simple program, which forms user’s function with the name V that returns volume of regular (right) cone and that has two arguments: diameter of its base (d) and length of its generatrix (l).
Programming language Mathcad does not have special tools (operators) for commentation of programs. It is considered that one or another operators of a program can be commented by placed right (or around) the program textual insertions (see point 1 in the fig. 30). Sometimes it is difficult to place the commentary in necessary place, i.e. exactly near operators because of possible difference in prints of formulas and textual insertions.
It is possible to insert the commentaries in programs as separate operators – textual constants (first line of the program in point 2 in the fig. 30). But it is possible to do the other way (see 2-3 lines): in the program to write row vector with two elements. The first element is commentary but the second one is the operator of the program.
The commentation by the matrix, which first element is a textual constant has one essential defect: the function stops working with dimensional arguments because of an array of Mathcad can not keep values of «different» dimensions. One more defect is the impossibility of spell checking of the commentaries. In this case we can advise to do the following: to write the commentaries in Word, for example, and then to carry them in Mathcad.
Remark. For it would be possible to write the commentaries (textual constants) using Russian print (Cyrillic) it is necessary to attach the print, which works with Russian letters to the constants (Constants – see the upper part of the fig. 30).
The commentation methods of Mathcad-programs that we have illustrated in Tip 30 do not describe in the documentation and in Íelp of Mathcad. That’s why they can not be considered as illegal ones – it is not known how the hollow textual constants will behave in future versions of Mathcad. As since developers far from always provide complete compatibility of versions with respect to even documentary methods. Reasoning from this, as well taking into account the fact that there are no textual constants in Mathcad PLUS 6.0 (but programming in Mathcad began starting exactly with this version) it is possible to recommend the legal method of program commentation (see fig. 31): we assign something the variable Rem (in our case it is zero) out of the program, then we assign «something» the variable inside the program, which name is commentary. If you wish it is possible to insert gaps in the name (see Tip 14) – in our case «down» dash is used instead of them.
Fig. 31 with variable-remarks prompts more natural, legal and documentary method for program commentation – use of long speaker names of the variables and of the user’s functions. It should be removed «primitive» commentaries in the fig. 31, but using the command Replace from menu Edit it is necessary to change V to Cone_volume, d to Cone_diameter and etc.
In Tip 31 we recommend to comment the operators of Mathcad-program by setting remarks in form, which is accepted in the programming language BASIC. Indeed in the programming language the key word Rem is placed at the beginning but not at the end of the commentary:
Rem is commentary
Followers of the other programming languages can vest Mathcad-commentaries with the other forms that are showed in the fig. 32. At that if you make use in addition Tip34, then you can «mask» Mathcad-program one of traditional programming language.
We have recommended placing operators in column when you write Mathcad-documents in Tip 5. Such order you’ll obtain if you press the key Enter after input of every operator. When you have input the finished notional fragment it is expedient to place some operators on one line for the Mathcad-document will be more compact. Such organization of the operators is recommended by the documentation of Mathcad and almost all figures of the book have been designed corresponding method.
If you really want you can give the «second change» Mathcad-program, which operators will be executed not only top-down but also from left to right. There is the key Add Line in the programming toolbar of Mathcad but there is not the key for addition empty space for new operator on the line. However it is possible to do using the other tools.
To add new operator on the right to the operator on program line in Mathcad 7 Pro (see the first function with the name Min_3_Ñ_mc7 in the fig. 33) it is necessary to cover the first operator by cursor (not part of the operator but the complete one!) and to press the key «,». After this comma and placeholder appear on the right of registered operator where new operator is inserted. Everything would be perfect if there was possibility to write enough extensional program, which occupied on one screen of a display, one standard paper À4 (see Tip 79). But Mathcad 8 Pro appeared and it turned out that «two-dimensional» program, which keeps two and more operators on one line divided by commas, does not work in new version of Mathcad. It stresses once more that two and more operators are the undocumented method, which can give the bug at any moment. Capacity for work rebuilds if we turn into one dimension: to leave one operator on every line (to insert new lines in the program and to carry there «odd» string operators).
But writing down a few operators on one program line is still very alluring method. Such programs can be tried to save and to make them working in Mathcad 8 Pro and 2000 Pro. If we delete commas between the string operators (to place cursor near comma and press the key Del or BackSpace) then comma will disappear but the adjacent operators will be enclosed in brackets and the program will work. It is possible to delete originated brackets during resuscitation of the program (see the second function with the name min_3_C_mc8 in the fig. 33) but it is possible to keep them. At that it is necessary to remember that a few operators are the undocumented method. Two or three operators on one line are delayed-action mine, which is ready «to burst» at any moment. (Indeed, in Mathcad there are « documentary» mines. Very often even quite simple Mathcad-document, which is written following all the rules, stops working and requires the update if you carry over it to new version. The author has a number of programs, which kernel is solution of the systems of nonlinear algebraic equations. The block Given-Find works excellently in the fifth version of Mathcad but gives error massage instead of the answer in the sixth version. Unfortunately such troubles exist quite often on changeover new version of Mathcad: «Better thing is enemy of good »: see Tip 62.)
It is possible to turn two lines with single operators to one line with two operators the following way: upper operator is covered by cursor
a ¬ 1½
b ¬ 2
c ¬ a + b
and the key Del is pressed. It leads the following result:
A ¬ 1 b ¬ 2
C ¬ a + b
Then we can cover by cursor the operator b ¬ 2 and press the key Del. This way we get the line with three operators. There is nothing special in these operations: two lines (two indentions) are turned to one line (to one indention) by the set of cursor at the end of a line and press the key Del in the environment of any textual editor.
But we repeat that if reader sees Mathcad-program, where there are a few operators on one line, in this or the other book or magazine, he has to take it as « shrink-wrapped» program. It should be «unarchived» before input it for the avoidance of possible mistakes – to place one operator on one line.
Existence of a few operators on one line is the cause of one more trouble. We look through the following «horizontal» program:
A := a ¬ 2 b ¬ 3 c ¬ 4
It is asked what value the variable A will be equaled after execution of the program. In the theory it is four: Mathcad-program has to return the last value, which it operates (ñ ¬ 4). But it is not so. The variable A will equal twenty-four – it is production of the numbers, which appear on the given program line. In Mathcad 7 Pro the value of the variable A, which is returned by our program equals four that corresponds to logic of our program. Here is the compatibility of the versions! The compiler of Mathcad 8 Pro and 2000 Pro suggests that the gaps between operators are missing mulTiplication signs (see Tip 9). The program returns right answer if it is «given back the vertical line» or even the last operator ñ ¬ 4 is written on separate line.
Hence we have the following summary: if reader takes the risk of writing a few operators on one program line in Mathcad then he has to remember, in the first place, that it is undocumented method and, in the second place, that the last string of the program (or operand of the operator return) has to have only one operator.
One more method for placing a few operators on one program line is described in Tip 30 – the embedding of a matrix with one line, which elements are the operators of the program. Both the first and the second methods have the limitations not to mention that they are undocumented methods of programming.
The programming language Mathcad is «young» enough. Many programming languages (BASIC, fortran and etc.) in their earlier versions allowed only on operator on a line. Then («at a mature age») this limitation was removed for a programmer can decide himself if to write a few operators on one line or only one. We’ll hope that the key Add Line will be supplement with the key Add Operator on the programming toolbar and above mentioned method, which makes more compact programs will become legal one in new versions of Mathcad.
One more remark (warning). As we have said above Mathcad 2000 Pro understands a few operators on one line as product. That’s why it is impossible to work with a few operators on one line if one of them has the other type (all operators are numbers but one of them is text, for example). Execution of such program will be interrupted by error message.
Many Mathcad users, with experience in conventional programming languages, find the «multi-stored» technique of formula recording quite natural outside the program body, and rather «wild» within the program. Such users have gotten used to the text («one-stored», single-level) formula recording technique of other languages. To placate old habits (which are, as is known — second nature), and to make programs more compact, one may remember the technique of redefining and renaming built-in function described in Tip 20. It is also possible to redefine Mathcad’s basic built-in operators. For greater program compactness and for further indulgence to the users habits the «multi-stored» Mathcad operators can be recorded like «single-stored» ones: integration, differentiation, summation, product etc. In the figure, the function V returns the value of the cone volume using «basic-like» operators and functions (sqr, «^» and so on). The Mathcad way to formulate the function V is presented on the figure for Tip 30. For this Tip, functions with non-standard names (+, -, *, /, ^, è **) are created by means of the technology of the non-standard symbol insertion, described in Tips 14 and 16. (Editing by Thomas Porritt firstname.lastname@example.org)
Before to write a program it is desirable «to reconnoiter» the operators included in it outside of the program. In the point 1 in the fig. 35 the operators are written with the help of which the volume of our cone is calculated using measured parameters – diameter of its base and length of its generatrix. In the point 2 the given operators are gathered in the program, where the variables r, S and h are already local ones and they are inaccessible for control (they will be inaccessible if you do not become familiar with Tip 36).
If a loop is put in a program (for the sake of loops we use the programming Mathcad very often) then we can try to realize it by step-by-step approach outside of the program. The features of realization of it are described in Tip 53.
By the time of debugging it is worth to add a program so that it will return values of all variables, which are used in calculation. In the fig. 36 the matrix is written on the last line of the program, which first column is commentaries, but the second one is local variables of the program devoid of their dimension (corresponding unit is written in commentaries). We have to do it as since the elements of the matrix can not have different dimensions (see also Tip 30 and Tip 45). It is possible to remove the matrix after debugging of the program and instead of it to write the local variable V for the sake of which we wrote the program. Also it is possible to come back to the program, which we have considered in Tip 35 (to the program without the local variable V). But it is possible to set one more function – Volume (see the end of fig. 36), which will «exclude» necessary line or necessary element of the matrix from the function V.
One of the effective methods of debugging is BreakPoint. It is prescheduled breaking of the execution of a program by break points. Such debugging tool equips many «modern» tools of programming.
The construction return…if… is inserted in the function V three times in the fig. 37. It returns (return) value of a local variable if (if) user set break point (the third, additional argument of the debugging function V; it is possible not to introduce the argument, but simply to write ÂÐ:=1, ÂÐ:=2 and etc. before call the function). When we change the values of the variable ÂÐ we can make the function V to return values of its local variables by turns not all at once as in the fig. 36. When we «move» break point along the program then it will be possible to localize an error if it has crept into the program. After debugging of the program you can remove debugging operators and debugging argument. But if you set the values of the argument unequal 1, 2 or 3 when you call the function BP then you will be able to keep everything as is.
It is possible to dispute about if the alternative is the main or the subsidiary structural directive construction (see Tip 39). But the loop with precheck is more «mainstream» than the loop with postcheck. It is the fact in – there is the loop while (at first it checks and only then it executes) but there is not the loop repeat (terminology of Pascal – at first it executes and only then it checks). The operators of the loop body while can execute not once whereas the operators of the loop body repeat have to execute at the list once. But loop exit is able to be situated not only in its «head» (the loop while) not only in its «feet» (the loop repeat) but also in the middle of it. Apart such exits can be a few ones, but Boolean expressions, which direct the loops, can be different. The user’s functions are written in the fig. 38. They return the root of the algebraic equation with the help of Newton’s method (tangent method). The loop with precheck requires that Boolean expression, which directs the loop, has been defined before entry in the loop. But this is missed in Newton’s method. The variants of solution of the problem are showed in the fig. 38. The loop with precheck and the method of artificial assign of necessary first value of Boolean expression, which direct the loop x1←x + 2.TOL, are used in the first function. The loop with exit from the midpoint is used in the second function. This does the program more natural: break (break) the loop of approximation to the root, if (if) new approximation (x1) almost equals previous one (x). For the title of the loop while 1 does not confuse users (basically any number nonzero can be written in this case), it is possible to write one in white and white (see Tip 19) and to hide it. It will be good idea if we write the symbol infinity «∞» in the title of the loop while: as if we comment this way the method of construction of the endless loop, which body has the operators break (return or ñontinue).
If we raise the construction break if¼ (return¼ if¼) to the word while, then we’ll get the loop with precheck (while), if we lower it to «feet» of the loop body, we’ll get the loop with postcheck (repeat). The constructions, which break the loop, can be a few ones. At that different Boolean expressions can be written after the key words if.
The function in the fig. 38 has one more distinguishing feature. The second function works with use of auxiliary user’s function «approximately equal», which for its turn relies on the built-in variable TOL (accuracy of calculations – by default the variable TOL equals 0.001).
 This is not always. If, for example, when x < 0 a function either isn’t defined or returns complex values or has some other problems, then the graph will be constructed only on right part of the axis of ordinates.
 Short-cut version of Mathcad 8 Pro, which is meant for reading of Mathcad-documents (mainly in Internet).
 Advisedly the author illustrates Tips of the book by simple problems as he considers that it is impossible to give reader unknown algorithm and new program method at the same time.
 The variable with the name l is not quite convenient as since it (depending on selected print) can be confused with one or with the Latin capital letter I. But the letter l has already fixed for marking some length in geometry: the word «length» begins with the letter «l» both in Latin (longetudo) and in main European languages. When you create Mathcad-document it is very important to give the variables that names, which were fixed after them during forming one or another scientific discipline long before computers. We’ll be able to work without commentaries at all in Mathcad-document if we mark right way the variables.
 Mathcad PLUS 6.0 is popular in out country because of russified version as well as because it works under Windows 3.x.
 In Mathcad 2000 the command became to work either in text or only in formulas or in both places.
 It is very comfortable when you work with the program – when you look into it, search errors, optimize.
 It is possible to consider the program with a few operators on one line as «shrink-wrapped» one. To use the program it is necessary to «unarchive» it– to bring every operator on separate line.
 These brackets will appear once more if the program, which was saved in Mathcad 8 Pro, we open in Mathcad 2000 Pro.
 It is possible to lead cursor to the gap, to click right key of a mouse and to change the gap to some mulTiplication sign using the first command of contextual menu.
 It became possible to use the symbol of «horizontal» division in addition to the symbol of «vertical»: à ¸ â. This operator has completely «legalized» in Mathcad 2001 – separate key has appeared for it in the toolbar of arithmetical operators.
 Or «across»; «along» is when one moves something across. But we move the break point «down».
 In our programs in the fig. 38 the condition of the loop exit is minimization of the distance between two contiguous (scheduled) approximations to the root x1 ≈ x2. In it possible to choose the other condition – y(x) < TOL, for example. These conditions can be checked by turns – here is the loop with two exits from the middle.
 In this case we have to assign the variable X1 such value so our loop will execute at least once without fail (imitation of the loop via t).