The part of the book «Tips for Mathcad users»
(without
pictures)
Tip 1. Do not use assignment statement
01 do not use
assignment statement
Mathcad
program package is frequently called supercalculator: it displays a numerical
answer for any expression, containing constants, variables, operators and
functions. To obtain this result you should simply type «=» (equality sign)
after the expression:
125 + a2 – sin (b) =
2.753
Of course, variables, included in this expression (a and b), should be defined with any numerical value
beforehand. An assignment (input)
statement in the Mathcad environment is symbolized like «:=». But it is typed
only with «:» (colon) button.
Mathcad kindly prints a second symbol «=» itself:
a: =
The main point of this Tip.
It is worth to push not «:», but «=» button inserting an assignment statement to a Mathcad
worksheet: if the entered variable has not been defined yet, operator «=»
(output) will automatically turn into «:=»
(input). And if this variable has been defined, then its numerical value is
displayed. This operation controls whether the given variable is free ¾ neither user nor system had assigned nothing to
it. By doing so we secure ourselves against a number of mistakes. Firstly, a system variable can be «spoiled»: å: = 5, m: = 1, A: = 2
(variable å –
stores the base of natural logarithm, m and A – store respectively measurement units of length and
current strength). Secondly, one may forget, that the variable has been already
«loaded» and to assign a new value to it. The event when a variable has one
value in the first part of a Mathcad-worksheet and another in the second part ¾ is not quite normal.
But one can assign a new value to the variable not only due to forgetfulness. The variable redefinition technique was in vogue in those times when the computer memory was limiting problem solving and a «vacated» variable should have been immediately «loaded» with another value. Nowadays one have to act this way storing in the computer memory bulky arrays (vectors and matrixes, elementary and aggregate — this is a not numbered Tip[1]).
You can shut off the automatic replacement mode, when operator «=» is replaced by «:=» (such hybrid of two operators is called Smart Operator), by choosing option Preference… from the View menu (see the dialog box, called by this command, at the end of the figure).
The variable numerical value input operator was a kind of stumbling block for those who had tried to master Mathcad about ten years ago, having heard much about its extraordinary ability to plot diagrams, solve equations and systems. People, from force of habit acquired while handling with fortran or BASIC programming languages, had been keying in «a=» instead of «a:» in the Mathcad environment and … had been abandoning further work with Mathcad because of a strange error — the Mathcad system had displayed such information: variable a is not defined yet. There were made attempts to define it with the help of the variable type assignment statement, but the Mathcad environment does not offer such facility so far. The MathSoft company has «given up» only in the seventh version — it has abandoned the attempt to teach users to key in «a:» instead of more customary and natural «a=». And now the pendulum is swinging to another direction: by means of the first Tip the authors of the book are trying to make (to teach anew) users keying in again not «a:», but «a=».
The note. The figure shows a Mathcad-worksheet in a mode
when separate operators are displayed on a white background, while free space
of the document is gray. This mode will be used to make some Mathcad-worksheets
more readable: white frame against a light-gray background portrays separate
operators more clearly.
Not numbered Tip. The Mathcad beginner is advisable to operate in the operator background marking mode (see the figure). It is particularly good for the situation when several operators are situated on one line close to each other and their bounds are not always clear.
Tip 2. Symbolical output instead of numerical
02 Symbolical output
instead of numerical
Sometimes
it is worth to use operator «®» (output of the symbolical
value of an expression or a variable) instead
of «=» operator (numerical value
output — see the 1-st Tip). The figure 11[2]
illustrates three reasons for such substitution:
Item 1: operator «=» displays up to 15 decimal points in mantissa (see the number formatting window in the figure of the Tip7, for example); operator «®» ¾ up to 250 points (our figure presents twenty points — they are the key of the command keyword float[3]).).
Item 2: operator «®» allows absolutely precise answers at the expense of output of all significant digits (item 2.2) or by representing a number as a simple fraction (item 2.4).
Item 3:
operator «®» in conjunction with operator k:=k (releasing a variable
from its numerical value concerning operators of symbolical mathematics)
displays a numerical answer, including also symbolical constants (this result
can be obtained, truth to tell, simply by printing a variable after the answer
like physical units of measurements ¾ see item 2 in the figure of the Tip 4).
Attention! In the figure of the item 3.2 there is
written 123/2
rather than 66.5:
these numbers are different for the symbolical mathematics of Mathcad. In the
case that there was written 66.5 in the item 3.2, the answer would be quite different — 259.5×k. A symbolical answer has to
be anticipated by a «symbolical» question.
Tip 3. Combination of symbolical and analytical output
03 Combination of
symbolical and analytical output
Sometimes it is worth to unite symbolical (see the 2-nd Tip) and numerical (see the 1-st Tip) output operators into a single expression. As in the previous Tip, the figure shows three examples of such combination:
Item 1: arithmetic expression enables both a simple and decimal fraction answers at the same time.
Item 2:
while solving an equation one can obtain both absolutely precise and
approximate, but «speaking» answers («speaking» in the sense that the root
position on the numeric axis becomes evident at once).
Item 3:
expression can be simplified before calculation. This operation is implemented
by Optimize mode, but in our example we once again observe both analytical (a
polynomial of degree four) and numerical answers.
Tip 4. Constants in the numerical answer
04 Constants in the
numerical answer
Many
users do not know, that the basic operator of Mathcad «=» (the numerical value
output operator — see the Tip 1) has two operands rather than a single one:
˜ = ˜[˜]
The Mathcad user fills in the first (left) small square
(placeholder) with an algebraic expression (or a variable), which numerical
value appears in the second (medium) placeholder (see the Tip 1). The third
(right) placeholder may contain
(square brackets accentuate that this operand is not necessary) any constant or
variable, which has been defined earlier. Usually this placeholder is filled in
by the built-in or user’s (see the Tip 26) physical units, that permits various
answers:
L = 200 ◦ cm or L = 2 ◦ m
The figure demonstrates another two examples:
Example 1:
In the cartoon «Thirty-eight parrots » the length of Boa is measured in
Parrots, Marmosets and Elephants. If to assign (to select) animal’s size, than
Boa will «fit» in 5 Marmosets, in 2 Elephants or in 38 parrots and one more
carrot’s wing. «You see, in parrots I am much longer!».
Example 2:
The trigonometric equation solution quite often contains number p. If to input this number as the optional operand
of the operator «=», then it will make possible absolutely precise numerical
answers (one fourth p, rather than 0.7855…), and secondly, all answers (one fourth p
with the period of 1). The answer can be
supplemented by constants and even expressions containing p: 2 × p, 3 × p
etc.
Tip 5. Typical set of three operators
05 Typical set of
three operators
The
above figure presents a typical fragment of a Mathcad-worksheet with a heading
«Initial version of the calculation»: the user types a text commenting further
actions (Time), puts into calculation a
variable (t), assigning a numerical value to it (360 seconds), and «prints» it out accompanying with another
(minutes) dimensionality just for a change. One can pack the whole
Mathcad-worksheet with such sets of three operators and solve a problem put by
(the velocity estimation as in our figure). But this technology is inconvenient
for carrying fragments over a new place, because it is possible to lose
operators included in this fragments «on the way». Furthermore, if the output
operator («=») is disposed a little bit higher than the input one, a
relationship of cause and effect will be broken and error message will be
displayed: «The so-and-so variable is not defined above».
Not numbered Tip: Creating a new Mathcad-worksheet
it is worth to place operators in columns with only one on a line. In the ready
document you may proceed «from a line to a surface», placing some operators on
one line of the worksheet, making it thereby more compact and satisfying the
European tradition to write initially from left to right and only afterwards
starting a new line.
Another technology of calculation is displayed at
the bottom of the figure: a comment (Time) is being written with six spaces on
the end. Then, the mathematical areas
(Math Region command in the menu Insert) are being inserted in the center and
in the end of this chain of spaces. And these mathematical areas were filled in
with input and output operators. Thus, an obtained conglomerate (text—three
spaces—input operator—three spaces—output operator) can be dragged from place to
place without any caution. Besides it is possible to transform the three bottom
operators of the figure, constituting a certain semantic fragment of
calculations, into a single operator: text of three lines including six
mathematical areas (three input operators and three output operators). One may
try to «push» the whole calculation
into the single commentary text region, but such operator becomes difficult to
edit: it is necessary to know where to stop.
Mathematical insertions in comments differ from
the text, firstly, by color and font (the Mathcad default). Secondly, the
background marking mode gives more broad white surroundings to the mathematical
insertions: in the figure this white background of mathematical formulas is
moreover thickened in the interactive mode (modification using the graphics
editor environment).
Tip 6. Check of the data input
06 Check of the data
input
As a rule a variable in computation has a limited range of reasonable values. In this case it is expedient to check the input value correctness. The figure «asks» the user to assign a room temperature value to a variable t. Let us assume that this value can not be less than minus 20 degrees Celsius (at this point addition statement is functioning as logic addition — see the Tip 12) and more than 40 degrees Celsius (units of measurement — are accepted by default in this example). And if the user makes an inadvertent mistake, then the variable t takes a string value (“This is impossible!”) and blocks further computations. (The 21-st Tip informs how to control even the input value dimensionality).
The second expression containing function if is enclosed in frameworks (the Mathcad-worksheet area) which collapse in order to protect the user from embarrassment — see the command Collapse in the context-sensitive menu. To «slip out» this menu one should place cursor inside the area and push the right button of the mouse. The area itself (two horizontal lines) is inserted by the command Area from the Insert menu.
Tip 7. Input of a bulky matrix with standard elements
07 Input of a bulky
matrix with standard elements
To put into calculation a vector or a matrix the Mathcad users usually resort to the command Matrix… in the menu Insert (or press button with the matrix image). After that there will be displayed a dialog box requesting for the number of rows and columns of the matrix to be created. This technique displays a number of disadvantages. First, in this case the size of a matrix is bounded by the value of 100 elements, while any matrix can keep up to 8 millions elements in Mathcad. Many users solve this problem by creating several little matrices and «gluing» them together into a large one using stack and/or augment operators (see the Tip 13). The second disadvantage is in that all elements of a matrix defined through the Matrix command are unfilled. In this case matrix elements have to be inserted manually. And maybe this is a reason for a limitation of 100 elements, which can be inserted manually yet. A large amount of elements requires a different input technology, and its example constitutes the main point of the Tip.
The figure illustrates the Mathcad-worksheet where
a bulky matrix containing a standard
element (we have set it as 1.0) is automatically
created and displayed. This matrix size is set also by the user[4].
Then this matrix is copied to the right operand of the assignment statement and
subsequently edited: 1.0 is substituted by something else. We have assumed
here, that the matrix elements — are numbers ranged from 1 to 2. We should not
type units and points manually — it is sufficient to edit zeroes. Through the
use of the function Matrix,
or the technique described in the 8-th Tip it becomes possible to record various
numbers in such semi-finished matrix. For further copying of the matrix M it is important to format it correctly at output. The
fragment of the formatting window is showed at the bottom of the figure, where
the user notes, that:
· There is only one number displayed after the decimal point (Displayed precision — there are
displayed three decimal points by default, while the maximum allowed number of
points is — 15);
· It is necessary to display
non-significant zeroes too (1.0, rather than simply 1 — Show trailing zeros);
they are not shown by default;
· There should be displayed
the whole matrix in a matrix style (Matrix), rather than in the form of a table
with scrolling bars (see the figure in the following Tip); by default the
Mathcad system selects the matrix display style itself.
Not numbered Tip. It is worth to create such
matrix by reading a file from the disk (see the Tip 57) or by introducing the
Excel table into a Mathcad-worksheet (see the second part of the book).
Tip 8. Output of a bulky matrix
08 Output of a bulky
matrix
There may appear some difficulties not only at input of a bulky matrix (see the Tip 7), but also at displaying and printing its elements. By default there is displayed only upper left corner of a bulky matrix in the form of a table. Its sidehead keeps row numbers and headline keeps column numbers (we have broken this default in the figure of the Tip 7). A matrix which is not entirely displayed has vertical and horizontal scrolling bars with a runner to the right and at the bottom of a table. To look through the whole matrix one should move the runner: the table scrolling technique. Another possibility — output of a bulky matrix not in the form of a table but in the form of a matrix (see the Tip 7). In this case you can look through it by scrolling the Mathcad-worksheet window itself.
And if a bulky matrix has to be printed out (unfortunately, a sheet of paper has not scrolling bars), then we can recommend to print this matrix-table several times, having preliminarily displayed its separate regions with the help of the runner — see the figure displaying the matrix M three times so that all its elements become visible.
Tip 9. Do not pass multiplication sign
09 Do not pass
multiplication sign
The
Mathcad environment does not require a multiplication
sign to multiply a constant by a variable. This method seemingly brings
Mathcad-worksheets more closer to documents written in handwriting: 2 à, not 2 × à, 5 kg, not 5 × kg and so on.
But
here are three reasons against this technique that runs counter to the
programmer tradition to put a multiplication sign (asterisk) between any
factors: 2 * à, 5 * kg and so on.
Firstly; design 2 × à is more or less clear: two is multiplied by the variable a. And expression 2 a carries some ambiguity: this is both the product of 2 by a and a postfix operator named a, which operand is a constant (two).
Secondly; the behavior of expression 2 a is not known in the case
when this construction is applied in symbolical mathematics.
The figure illustrates the third «contra» argument. If two is reduced by unit (see the bottom of the figure), then it should remain unit, but by no means two and furthermore with minus? The secret of this «focus» appears to be evident, but this simplicity is, indeed, «worth than larceny». The point is that not unit is subtracted from two, but two is multiplied by minus unit[5] in the given example. This service of Mathcad (programmer does not have to enter a multiplication sign) has turned out to be an «ill turn» (involuntary pun). There is another ambiguity (see the item 2 in the figure): what is k a — is that the variable k, multiplied by the variable a, or is that a variable, with a name of three symbols, where the second one is a space (see the pieces of Tip 14 and 18). Another interpretation: k a — is a postfix operator with the operand k or a prefix operator with the operand a.
Tip 10. Global assignment at the system solution
10 Global assignment
at the system solution
The Mathcad environment offers the built-in function Find for solution of the system of algebraic equations and inequalities. A system is commonly solved this way: the first approximation to the system root is being set[6] with the help of assignment statements «:=». Then the key word Given is being written, a system of equations and/or inequalities is being set (it is not allowed to use only Boolean operator «¹») and, at last, function Find is being called. This function returns values of its arguments in the form of a column-vector, converting a system of equations into a system of identities. In a number of cases in order to solve the system the standard values of built-in variables TOL and CTOL should to be changed. These variables influence on the accuracy of calculations. Very often this technique requires successive approximations to the right answer. In this situation it is advisable to change the above-mentioned sequence of operators (but not sequence of their satisfaction) — to «pull down» the first approximation and other initial settings under the function Find by means of the «º» operator. If the system is bulky and display can not find room for it, then the shown sequence of operators will obviate the necessity for scrolling a Mathcad-worksheet to adjust input data and to browse the answer.
Tip 11. Superfluous brackets
11 Superfluous
brackets
Many Mathcad functions have a vector or a matrix (array) as their argument. Vectors and matrices represent a collection of scalar values contoured with brackets. On the other hand, brackets also are an attribute of the function enclosing the function argument. This feature leads to appearance of double brackets at the conventional call of a function with a vector-argument (see the first operator in the figure). Double brackets can embarrass a user who may try to remove a «superfluous» pare of brackets not understanding a reason why it is impossible. The way out — lies in the call of a «matrix» function (which argument — is an array) not in function form, but in the form of a prefix operator, which operand does not need a «brackets’» design (the second operator in the figure).
The figure depicts the dialog boxes creating the three prime objects of Mathcad:
· The built-in function insertion — Insert Function; the order of the mean function input to a Mathcad-worksheet is shown. Function mean returns an arithmetic mean of the array elements.
· Insertion of an array — Insert Matrix; there is displayed the order of the row-vector input — that is a one rowed (Rows) matrix with several columns (Columns).
· The prefix operator insertion through the fx button in the Evaluation window.
Input of the first two objects is also possible without call of the dialog boxes, shown on the figure[7]. The third object is required not so frequently, as the two first ones, that is why it can be inserted only through the Evaluation dialog box.
Tip 12. Set of three Boolean functions
12. Set of
three Boolean functions
The
Mathcad environment does not present built-in Boolean (logic) functions And, Or and Not (logic negation), which are an integral
part of all programming languages and which support implementation of all sorts
of logic constructions, loops and alternatives. And what is to be done here!?
Firstly, it is necessary to recollect, that «logic AND» and «logic OR» have also such names as «logic multiplication» and «logic addition»: the given functions are realized through customary multiplication and addition operators (see the item 1 in the figure[8]): the Mathcad environment recognizes logic «not» — as zero, and logic «and» — as a non-zero number.
Secondly, logic «and» and logic «or» functions can be defined (see the item 2 in the figure). In so doing there can be defined any (variable) quantity of Boolean arguments — that is elements of associated vectors of multiplication and addition operators. (Not numbered piece of Tip. It is worth to define two pairs of functions with various names È-And — ÈËÈ-Or to satisfy both Slavophiles and Westerners.)
Thirdly, the Mathcad environment already has
built-in functions that are well suited for logic tasks, — they are min and max (see the item 3 in the
figure). And furthermore, they are also convenient for the following reasons:
·
It is not necessary to
«litter» a Mathcad-worksheet with supplementary built-in functions AND, OR and
so on;
·
Functions min and max accept both matrix, column-vector
and row-vector arguments that allows more compact notations;
·
Functions min and max accept not only logic expressions (yes – no, 1 – 0) as
their arguments (array elements), but also real
expressions, varying smoothly from zero to unit. Thus it is possible to
operate not only with precise but also with fuzzy
logic.
Not numbered Tip. If it is necessary to hasten the program operation, then it is worth to use the multiplication operator instead of the min (And) function, having placed, where possible, an expression most often assuming the value zero on the first place. The point is that the multiplication operator returns zero if the first factor equals zero in order to hasten computations, while the function min (And) is too pedantic: it sorts out all elements of a matrix or a vector — its argument.
Tip 13. The built-in function redefinition
13 The built-in
function redefinition
Mathcad
offers the user a unique opportunity of renaming
and redefinition of built-in
functions. Renaming seems more or less intelligible (the old substance in the
new frame): a new user function is being defined which is completely or
partially coincident with one of the built-in functions. What is the reason for
that? The items 1 and 2 of the figure illustrate two examples of the built-in
function renaming. First, it is possible to replace a short name of the
function (identity) with a more integrated one (Unit square
matrix — see the item 1). We obtain a
certain hybrid of a function name and a comment. The renamed function in the
item 1 is called as a prefix operator (fx), rather than as a function, that makes possible to get
rid of brackets and to make the operator, creating a new matrix, absolutely
natural (refer also to the Tip 11). This effect is also aided by three spaces
in a new function name (see the pieces of Tip 14-16).
In the item 2 the name of the function reproducing
the built-in one (augment - «horizontal» confluence of two matrices) is abbreviated to one symbol. It allows drastic reduction of the program size because this program use the
renamed function repeatedly. Besides, a word is replaced with the «contraction»
symbol, which is more easy to remember.
The item 3 demonstrates redefinition (the old substance in the new frame) of the built-in function stack to visualize its essence. The function augment (see the item 2 in the figure) accomplishes the «horizontal» confluence of two matrices – its arguments are positioned also «horizontally». Function stack is dedicated to the «vertical» confluence of two matrices but its arguments are positioned still horizontally (see the item 3.1), that is not quite relevant. Redefinition of the function stack – replacement of two arguments by one column-vector with two elements (see the item 3.2) – restores «geometrical» logic of the matrix confluence functions.
The note. It is impossible to redefine the built-in functions WRITEPRN and APPENDPRN. Their operation is analyzed in the Tip 57.
Tip 14. Special symbols in the name of variables and functions
14 Special
symbols in the name of variables and functions
Programming environments always place certain restrictions on the variable and
function names. As compared to conventional languages Mathcad has less restrictions:
the variable name in Mathcad can include Greek letters (Cyrillic letters
deserve particular investigation – see the Tip ?) and subscripts. All these
peculiarities of Mathcad enable names assigned to different variables long ago
in mathematics (p, for instance, rather than Pl), physics (tin), chemistry and
other branches of science. On the other hand, the Mathcad-variables have their
own restrictions associated, in a manner of speaking, with a bad «heredity»
preventing the variable name from insertion of some symbols: «$», «&», «?»
etc. The point is that these and other symbols (keyboard keys) are loaded with
a purpose of input in a Mathcad-worksheet of sum operators («$»), integration
operators («&»), differentiation («?»), product («#») etc. If to push these
buttons at input of the variable name, there will not appear an indicated
symbol, but a blank of the corresponding operator. But, if you are eager for
that, it is possible to enter special symbols and spaces into a name of user’s
function.
This technique of the «conventional» restriction removal from the variable names has its own history. The 30-th Tip recommends to use string variables — series of symbols framed by inverted commas — for comments of the Mathcad-program operators. Such series may include spaces and other symbols («$», «?», «&» and so on). Sometimes when editing such Mathcad-program there may happen «a glitch», that will result in a loss of inverted commas by string constants. Constants will «turn red» indicating an error situation. But every cloud has a silver lining — it is possible to take advantage of this situation (transformation of a string constant into the variable name), and that is described in the figure. The «ablush» series of symbols should be copied and subsequently utilized as a name of a variable or a function.
The technique of the insertion of
special symbols and spaces into the Mathcad names (and many people find this
method a pure sedition) has such «historical» ground. The DOS environment and Windows 3.* did not allow spaces, Cyrillic letters and other special symbols in
the file names. And the Windows 95/98 environment has
lifted such restriction.
Tip 15. Enumerated variable names
15 Enumerated variable names
The previous Tip shows how to lift some restrictions from the Mathcad variable names — how to insert special symbols and spaces into a name. But some restrictions on the variable names still remain. In particular, it is impossible to see a point in the Mathcad-variable name: you can insert a point but afterwards (when cursor is taken away) the point disappears, halving the variable name – to the name itself and a subscript text: there was t.in, it has become t in. (A subscript text is not entirely desirable for the Mathcad-worksheets, since it can be confused with the operator of the vector element indication — numerical subscript). If to continue the analogy started in the previous Tip, it is possible to note, that the point is also slightly ambiguous in the file name of the operational system Windows 95/98: by tradition the point separates the file name and its extension (its «surname»). If the full file name containing more than one point it is not always clear, where the name finishes and the extension begins. The second restriction on the variable name is in that it is impossible to start a name with a number. But… It is meant an Arabic number (by default).And there also exist Roman numbers apart from Arabic ones. The figure illustrates solution of a conventional linear programming task: a furniture factory can produce two models of chairs at the cost of 70 and 108 relative units[9]. On this order human and material resources are allocated. Besides, quantity of boards, cloth and time for producing of each chair is known:
|
Chair |
Board consumption, meters |
Cloth consumption, square meters |
Time consumption, man-hours |
|
The first model |
2 |
0.5 |
2 |
|
The second model |
4 |
0.25 |
2.5 |
|
Resources |
440 |
65 |
320 |
The question is to project
manufacture of chairs so that their cost is highest possible.
There are two unknown
variables in this task which are
enumerated and have special symbols within their names (minus sign and
space): I-st chair and II-nd chair.
Tip 16. User's physical units
16 User's physical
units
Mathcad presents quite a lot of system (predefined) variables keeping the built-in physical units of various systems (see appendix 6). But the user has a right to complement this list and put into calculation user’s units of measurement (see the figure), combining them with the built-in ones.
The user can be ignorant of
units he really needs for calculation at once. These units may be put into
calculation as necessary: for example, one needs to input time in the week
duration units — then it is sufficient to return to the «head» of calculation
entering week:= 7 × day and so on.
Operator ì: = m is also convenient for the following needs. It is possible to be lost in
reverie and to assign a new value to the variable m, which is very popular (see Tip
1). This operation can break the orderly system of physical units stored in
Mathcad. Operator ì: = m liberates the system variable m to store something else.
Tip 17. Nonphysical units of measurement.
17 Nonphysical units
of measurement
We did not use units of
measurement in the 15-th Tip considering the linear programming task — the
problem on the chair manufacture plan. But these units were mentioned in the problem
description (see the table of the Tip 15). Along with physical units (meters,
square meters and hours) the given task employs countable (chairs, people) and
financial units.
The Mathcad program package has
the built-in system variables (refer to appendix 6) keeping physical units of
measurement. The figure solves the problem on the chair output plan by
application of the all quantity
measurement unit.
The American currency unit ($US) is put
into calculation on the principle «Time is money»: the given calculations does
not use a physical quantity «time» and it can be temporarily «loaded» with
another duty. Certainly, it is easier to write down: $US := 1, and pieces := 1 (our pieces are
«attached» to gram-molecules, which are also not utilized in the given
calculations), but it is fraught with such a mistake:
$US + pieces = 2
There should appear not a numerical answer (two),
but an error message «Incompatible dimensions».Physical and other quantities
are used in the described calculation not only for making it more «readable»
but also for the dimension conformity control in the inserted formulas.
Tip 18. Once more about special symbols in the variable names
18 Once more about special symbols in the variable
names
The authors have published their technique of
insertion spaces and special symbols into the variable names, described in the
15-th Tip, on the Collaboratory site (http://webserve.mathsoft.com/mathcad/collab) of Mathsoft (Mathcad designer company). That was
quite intriguing:
I have spaces in var-name := 10
Can you do it? := 2
I have spaces in var-name + Can you do it? = 12
There were three kinds of responses to this quiz:
1.
And what’s the big idea?!
2.
How do you contrive to do
that?
3.
I have been inserted special
symbols into the variable names for a long time, but I’ve never known, that it
is possible to enter spaces too.
The third response leans upon the fact, that one may use special chord (Shift+Ctrl+k) for names of functions and variables to block the Mathcad system perception of special symbols (space, $ and so on) as the command of the name envelopment with cursor (space) or the sum operator ($). The figure displays definition of the variable with two spaces and «$» symbol within its name (Cost in $US). And the second variable with a space inside (Russian ruble) is obtained by editing the first one. It is impossible to insert spaces and special symbols at editing. To do that the above mentioned chord is required (the Tip 54 describes the third technique of creating nonstandard variable and function names).
Tip 19. The invisible function and function-werewolf
19 The invisible
function and function-werewolf
The
Tip 15 recommends to write down not Arabic (it is impossible) but Roman numbers
starting the variable name to enumerate variables. (But, true, one can write The first chair, The second chair,
substituting Arabic number with words).
But many people can count Roman only up to twelve,
recollecting the clock-dial. But we face Roman numbers not only checking the
time, but also keeping financial payments: amount of money — is an Arabic
number, and its cash fragmentation — a Roman one. The program shown on the figure
will prove useful to the automatic teller operation, for example.
But it is not the very purpose to cite this
program. The object is to give the reader another piece of Tip.
A lot of Mathcad built-in functions and operators
have their inverse analogues: sin ¾ arcsin, åx ¾ ln(x), differentiation — integration etc. Producing a pare of user functions
(direct — inverse), one may act in a such way: produce the first function, and define
the second (inverse) function employing the first one. If the function has a
real argument then the task is reduced to the equation root search, and if
argument is integer, the task may be solved through selection.
The figure illustrates operation of such
«double-sided» function. This function returns a Roman number, if its argument
is an Arabic one, and, vice versa, it returns an Arabic number, if its argument
is a Roman one. Single function incorporates both direct and inverse
algorithms. The given function determines the direction of conversion
identifying argument’s type. If (see
operator if) it
is of the string type (the built-in Boolean function IsString is used here), then Arabic number is searched,
otherwise (otherwise)
— a Roman one.
Apotheosis of the technique of insertion special
symbols and spaces into the variable names (see the pieces of Tip 15 and 18 —
is the user function, described on the figure, with a name of a single space. The question is, what
can this function prove useful to.
The anonymous function allows not only
recalculation without overloading a Mathcad-worksheet with unnecessary words
and symbols ("XXV" =25 is clear without any comments), but also simulation of
calculations (subtraction, root extraction, sine computation) with Roman
numbers (see the end of the figure). The focus is that our function is called
not conventionally, but in the form of prefix and postfix operators: there are
no brackets framing an argument (or, rather, operand), and the symbol of
operator itself is invisible (see the Tip 11). Thus it leads to the absolute
illusion of Mathcad’s ability to operate with Roman numbers without any
supplementary tools — operators and functions[10].
There follow some pieces of Tip from the figure:
1.
In specific cases it is
expedient to combine direct and inverse calculation algorithms into a single
function. The reason is that these algorithms may have the same base (our pairs
of Roman and Arabic numbers, for example): a single program is shorter than two
separate ones.
2.
Sometimes it is expedient
to give an invisible name to an object of a Mathcad-worksheet. There can be a
space or a chain of spaces (another technique — the name is not a space, but
have a white colored font — see the Tip 50). There can exist several different
styled variables or functions with a name of a single space (see the Tip 43).
3.
The function displayed in
the figure can appear useful for encryption of messages. For instance, you may
send such a message to your trade agent: «Minimum price is MM[11],
and maximum — MMCD». If you apprehend that the competitor, you protect the
message by encryption, is acquainted with Roman numbers, then another pair of
numbers may fit: the Arabic numbers analogue are, for example, Fibonacci
numbers, (1, 2, 3, 5, 8, 13 and so on indefinitely), and their Roman analogues
—are their serial numbers («1», «2», 3», «4», «5», «6» and so on indefinitely).
Tip 20. Adding superfluous brackets
20 Adding superfluous
brackets
The
reader has likely already noticed that some pieces of Tip in the book interfere
each other. Tip 11 has recommendations to
exclude superfluous brackets from Mathcad-expressions. And now we will
discuss insertion of auxiliary brackets.
The 1-st item in the figure once more time notes
that the Mathcad environment, as well as all other programming environments and
mathematics on the whole, value involution operator «superior» to
multiplication operator, which in its turn, is valued «superior» to addition
operator. These brackets once again accentuate the given hierarchy of
operators, which can be familiar not to everyone (see examples in the figure of
the Tip 34).
The item 2 by means of the function (treelike
operator) augment
simulates a not-reshuffled pack of cards — a row-vector with 52 elements (cards
— «two», «three»… «ace»). Brackets look far from superfluous in the pack
simulating «tree» (compare items 2.1 and 2.2).
In the 3-rd item the Boolean function Verdict is formed, which returns
results of voting. The participant votes (1 — pro, 0 — contra) are stored in a
vector-argument X:
verdict is approved when more than two thirds of voters vote «pro» (function mean returns an arithmetic
mean of the array elements). Brackets here also stress an algorithm of function
operation and the satisfaction sequence of operators «>» and «:=». In the
first case the function mean is called in the form of a prefix operator, and in the
second — in the form of a function.
Inserting supplementary brackets one should
remember that this method improving the Mathcad-worksheet «readability» may
reduce the speed of its execution.
Tip 21. The variable dimension control
21 The variable
dimension control
Sometimes
the user is necessary to input by means of the «:=» operator not a simple value
of a variable in a certain range (see the Tip 6), but a value with a fixed dimension of a physical quantity.
Input of a dimensionless quantity or a quantity with another dimension should
result in an emergency halt of the Mathcad-worksheet execution. Such scenario
is realized by operators of the figure: user should set a length into the variable d (its style is User 1 — see the Tip 43), i.e. he should set a constant
multiplied by either a built-in or user unit of length (see the Tip 43). An
input dimension is checked as follows: variable d is firstly raised by and immediately reduced by a unit meter (m – the system variable). The value of the new variable d (its style is already not
User 1,
as the first variable d style, but Variables) does not suffer any variations but, in the case
that the length is added by anything else, there occurs an error (see the upper
half of the figure). The focus of the displayed Mathcad-worksheet is in that it
contains the two same named but
differently styled variables d. The right value was entered only after the third try.
(To be continued in the Tip 45).
Tip 22. Graphical solution of the linear programming task
22 Graphical solution
of the linear programming task
The
Mathcad-worksheets in the figures of the pieces of Tip 15 and 17 solves the
linear programming task. The Mathcad environment does not offer special visualization facilities for solution of
such tasks. But with a certain skill this problem is solvable by means of a
conventional Cartesian graph – see the figure. The core of the problem is
described in the Tip 15.
To solve the linear programming task graphically
there are defined two functions Constr (constraints) and CF (criterion function) with one argument Chair1 (quantity of chairs of
the first model[12]).
Application of symbolical mathematics at the function output is dictated rather
not by needs of inequality solution (solve) — that is easy manually[13],
it serves for the graphical solution core illustration: straight line of the of
criterion function slides upwards (animation
of Mathcad is very pertinent here, when the system variable FRAME (the animation frame
counter) is associated with the value of the criterion function). Solution will
be achieved at the moment when the straight line (criterion function) «takes
leave» of a polygon (the region of the constraint satisfaction). The graphical
solution in question both suits for an
integer task (a region bounded with a polygonal line is substituted by a
multitude of points) and for a nonlinear
task (line segments are replaced with portions of a curve: convex or
concave).
The note — not numbered Tip. The item 2 illustrates two attempts to solve the criterion function relative to the variable Chair2 (chairs of the second model). The first operator solve (with a small font of its expression) defines the Euro to dollar rate as 1.2, and the second — 12/10. This substitution, equivalent on the first sight, changes the form of the answer factors (see also the Tip 2).
Tip 23. Color Cartesian graph
23 Color Cartesian
graph
The
given piece of Tip extends the subject of the
Mathcad non-typical graphics, started in the previous piece of Tip.
Mathcad offers the three-dimensional graph coloring facilities: for example, surface vertexes can be colored more «warmly» while a concave remains more «cold». Unfortunately, this principle cannot be applied to a two-dimensional diagram. But if something impossible is very desirable, then it becomes possible. The figure exhibits a single argument function diagram, so that — see comments to the right of the diagram. Unfortunately, this book is not colored, but the reader can trust the author, that this line on the diagram is displayed with all rainbow hues: maximums are toned more warmly, and minimums — more coldly. The trick of this figure lies in that it presents not a Cartesian graph but a colored surface, turned to a spectator in a special way (with one «side»). It is worth to hide the operators and the forming matrix Ì, which elements — are values of the function under analysis (they are recurring over the columns), by locking up an appropriate area (see the top of the figure and also the Tip 62). This operation produces an absolute illusion of a colored Cartesian graph.
Not numbered Tip. It is expedient to «pull»
operators, the forming user function y(x) and diversity limits of the diagram (x íà÷-x êîí) up to the diagram. Operator «º» (global assignment) proves useful here, it spreads its
influence not only downwards that is executed by operator «:=» but also
upwards.
Tip 24. Graphical solution of the set of two algebraic equations
24 Graphical
solution of the set of two algebraic equations
To solve the problem of this Tip heading one should graph two Cartesian graphs and verify a point of their intersection. But this task may appear arduous for systems are not consistently represented in a suitable for graphing form: y1(x):=…, y2(x):= …. Equation sets are generally assigned in a common way — f1(x, y)=0, f2(x, y)=0 — variables in this notation are very seldom dividable to reduce this task to a straightforward one:
|
x
+ y = 5 |
|
y1(x):=
5 – x |
|
|
® |
|
|
y
= x2 |
|
y2(x):=
x2 |
The figure depicts this
problem solution: matrix M is filled up by means of two loops with
a parameter (x value and y value). The matrix elements display
status of equations on the x-y plane: if selection of x and y
values with the augment of dx and dy shows any expression to
reverse its sign in the given point[14],
then corresponding element of matrix M takes the value unit, and
zero — otherwise. We are scanning the diagram area as though as «popping up»
(oscillographing — on the level line
diagram, specially formatted) our curves (at the center of the Bernoulli
lemniscate («entwined with ribbons»). How long will this «scanning» take — this
is another question[15].
Constant 300 (order of the matrix M) allows compromise settlement
between speed of graphing and its quality.
The technique shown in the figure is suitable also for the single equation diagram construction in the form of: f(x, y)=0.
Tip 25. Radar diagram
25 Radar diagram
To visualize tabular data one may find convenient a special
business-graphics — radar diagram:
rays are being drawn from the diagram center, then cuts are being made on these
rays at a distance proportional to the elements of the visualized vector. Later
the given cuts are being connected by straight lines. A polygon is being
obtained, representing a certain phenomenon or a certain object by its area and
shape.
The figure illustrates results
of the matrix M data visualization through the radar diagram. The
crux of the problem is: nine experts
are engaged to estimate any ten
characteristics of any object.
Experts put down marks between 2 (unsatisfactory) and 5 (excellent). A regular
polygon on the radar diagram (and this is nothing more than a polar plot of Mathcad) displays an average
mark put by experts, while an irregular polygon illustrates an average mark put
for individual characteristics: «weaknesses» and «strong spots» of the examinee
become clear at once.
Tip 26. Start with a symbol and finish with a number
26 Start with a symbol
and finish with a number
In the Tip 24 we have tried to solve graphically a system of two algebraic
equations — we have defined the range of the system real root existence and by
means of numerical methods we have
found values of the unknown variables for the first approximation of the
solution search (this is implemented in the Tip 44). But it is also possible to
wedge a «symbol» in the «graph» — «number» line. The main point of the Tip. It
is worth to try to apply symbolical mathematics of Mathcad[16]
to a task before its numerical
solution. The figure illustrates analytical solution of our system of two
nonlinear algebraic equations with the help of the Mathcad-operator solve. All the 24 system roots are
found, of which the four real ones are depicted on the diagram of the Tip 24.
The figure displays only five points in mantissa (see the key float, 5). There are displayed 20 points by default (without the float key), and the maximum amount is — 250 points. (The same named command from the symbolical mathematics menu (the solve command) is capable of returning up to 4000 signs in mantissa. But this command operates only with equations, rather than with sets of equations).
The roots can be displayed in
the decimal fraction form with a floating point (0.043778), instead of a
«scientific» form (4.377810-2). To do that one should enclose an expression
with cursor and push the «=» button. That makes possible a «numerical» answer
to be formatted as needed (see also the Tip 3).
Tip 27. Commit, but audit
27 Commit, but audit
This Tip, actually, concerns all
software intended for the mathematical task solution, not only Mathcad. So,
having obtained the answer, don’t be too lazy to check it up. In the figure the system roots are substituted into
the equations. For this purpose the matrix (from the Mathcad-worksheet of the Tip
26) containing system roots is entered in the variable Roots. Then the pairs of value from
the Roots matrix rows are inserted in the x and y variables, which in their turn are substituted
into the left-hand members of the algebraic equations of the system under
control. Analyzing discrepancy the
solution quality is estimated.
Tip 28. Three methods to preset the three-dimensional graph existence range
28 Three methods to
preset the three-dimensional graph existence range
The surface construction in the Mathcad environment is anticipated by filling of a matrix, which elements — are the values of the two variable function under analysis in the orthogonal area. In this case the matrix row numbers are being related with the values of the first argument, and column numbers — of the second one.
The figure in the item 1
demonstrates the standard technique of the given matrix filling, described in the
Mathcad program package documentation: the control points are the row and
column numbers of the analyzed function matrix of the values.
The items 2 and 3 present alternative and somewhat more convenient
techniques for the graph construction: the control points are the corner point coordinates of the
orthogonal area (the item 2) or the center of the square area, which side is
also set by the user. Moreover, if to «let down» operator of the analyzed
function coordinate input (operator «º») close to the graph, then
operations with it become very easy. The third technique is handy for
animation: by means of the variable FRAME it is possible to «fly» over a
surface varying one of the coordinates of the square center; one may move near
or away from a surface varying the D value, observing the surface
through the square «illuminator».
Tip 29. Plane graph in two ticks
29 Plane graph in two
ticks
Here is the standard, «three-step» technology of the
Cartesian graph construction in the Mathcad environment (see the top of the
Figure):
• The 1-st step: there is
assigned the style of the function being analyzed;
• The 2-nd step: there is entered
the Range Variable (we have set it as x), assigning the graph
dispersion over the ordinate axis (x init and x fin) and the number of points on
the graph;
• The 3-rd step: the command is
given to plot the graph, which is subsequently formatted if some parameters do
not suit the user, for this purpose corresponding dialog boxes are called; and
the graph size is altered prolonging its outline (ordinary «window» technology
in the Windows environment).
The Mathcad-environment,
beginning with the 7-th version, allows quick (Quick Graph – see the second
graph in the figure) technology of the «plane» graph (Cartesian and polar) construction:
• The 1-st step: see above, but
the function under analysis is not necessary to be defined as user’s function,
it is sufficient to insert it at y axes;
• The 2-nd step: the command is given to plot the graph; there appear another two defaults: the graph is constructed using a variable number of points depending on the graph size in the range between -10 and 10[17]; as a rule the first default satisfies the user (a compromise settlement between the graph quality and speed of its construction takes place here). And the user has a right to break the second default setting another dispersion range of an argument.
The second technique has one
more advantage (the main point of the Tip): the variable arg stays free from any numerical value, which afterwards can interfere
with, for example, an analytical (symbolical) solution of a task.
Tip 30. Comments in a Mathcad-program
30 Comments in a
Mathcad-program
This Tip and some of the following ones are illustrated (for
instance, the pieces of Tip 31 and 35 — 37) using a simple program[18],
that formulates user’s function named V, returning the volume of a
regular (right) cone depending on the diameter of its basis (d) and the length of its element (l — [el][19]).
The programming language Mathcad has no special facilities (operators) for program commenting. It is thought, that it is possible to comment one or another program operator placing text insets to the right (or around) of the program (see the item 1 on the figure). Furthermore, distinctions in the fonts of formulas and text insets make it difficult to insert a comment where needed.
We have investigated disadvantages of such technology in the Tip 5: it is possible to lose operators little by little «on the way» at dragging.
It is possible to insert comments into a program in the form of separate operators — text constants (the first line of the item 2 program on the figure). And one may act in a different way (see the 2-nd and the 3-rd lines): to record in a program a row-vector with two elements, first of which is a comment, and second one — is the program operator.
Comment by a matrix, which first element — is a text constant, displays a substantial disadvantage: the function ceases «knowing» how to operate with dimensional arguments. This limitation is due to that the Mathcad array can not store quantities of different dimension.
Tip 31. Comments in the form of the assignment statement
31 Comments in the
form of the assignment statement
The Mathcad-program commenting techniques, described in
the previous Tip, are not depicted in Mathcad’s help and documentation.
Therefore they cannot be considered legal
— the behavior of hollow text constants in the future versions of Mathcad is
not known. The Mathcad designers not even always provide the complete
compatibility of the version regarding to the documented techniques. And what
to say of not documented techniques?! Hence, taking into account the fact that
the Mathcad PLUS 6.0 environment (and this is the starting version of the
Mathcad programming[20])
does not offer text constants, there can be recommended the legal technique of
the program commenting – see the figure: outside the program there is assigned
something (we have it as zero) to the variable Rem, and then, within the program,
this «something» is assigned to the variables which names — are comments
themselves. If you will it is possible to insert spaces in these names (see the
Tip 15) — we have «dropped» dash instead.
Tip 32. «Language» comments in Mathcad
32 «Language» comments
in Mathcad
The previous Tip has recommendations to comment the
Mathcad-program operators, by shaping notes as it is accepted in the environment
of the programming language BASIC. Though, this language places the keyword Rem before, rather than after a comment:
Rem This is the comment
The adherents of other
programming languages can shape Mathcad-comments as displayed on the figure.
Tip 33. Several operators on one line of a Mathcad-program
33 Several operators
on one line of the Mathcad-program
The 5-th Tip recommends to position operators in columns
when writing a Mathcad-worksheet. This order of operators is obtained of
itself, if to push Enter after input of each operator. After input of the
complete semantic fragment it is expedient to position some operators on one
line to make the Mathcad-worksheet more compact. This is how almost all the
examples in the figures of the book are arranged and how the
Mathcad-documentation recommends to act.
Having a particular desire it is
possible to impart «the second dimension» to a Mathcad-program too, which operators
will be satisfied not only from top to bottom but also from left to right. The
Mathcad toolbar has the button Add Line, but there is no button to create some
free space on a line for a new operator. But this is practicable by other
means.
In the Mathcad 7 Pro environment
(see the first function on the figure), in order to add a new operator to the
right of an operator on the program line, one should enclose this operator with
cursor (not a part of the operator, but the whole operator!) and press the key «,». Whereupon
to the right of the enclosed operator there will appear comma and a
placeholder, where a new operator is inserted in. Everything was perfect –
there were allowable rather bulky programs arranged as a whole on one display,
on a single standard sheet of paper (À4). But here Mathcad 8 Pro has come into
the world and «two-dimensional» programs, containing on one line several
separated with commas operators, has appeared inactive in the environment of
the new Mathcad version. This phenomenon once more time accentuates the fact
that the technique of two and more operators on one line is not documented and
can bug at any moment. The program serviceability was restored after proceeding
to one dimension – i.e. leaving only one operator on a line (that was achieved
by inserting new lines to the worksheet and «superfluous» operators were
carried over there).
But the technique of several
operators situated on one program line still remain very attractive. One may
try to rescue such programs and make them operating in the environment of
Mathcad 8 Pro. If to erase commas separating operators of a line (to bring
cursor to comma and press Del or Back Space), the adjacent operators will be
enclosed in brackets and the program will start working. Brackets which have
appeared during the program reanimation can be erased (see the second function
in the figure), but it is also possible to keep them. In so doing one should
not forget that several operators – still remain the not documented technique.
Such two or three operators on one line represent a delayed mine ready to
«explode» at any time. (Truth to tell, the Mathcad environment has also
«documented» mines. It happens very frequently that even a not sophisticated
Mathcad-worksheet, formulated in due form, halts and needs modification after
carrying it to a new version. One of the authors has a number of programs,
which main destination – is to solve sets of the nonlinear algebraic equations.
The block Given-Find perfectly operates in the fifth
version of Mathcad, but in the sixth version it has started returning the error
message instead of the answer. Unfortunately, such troubles are quite frequent
at the transition to a newer version of Mathcad: «The best – is the enemy of
the good»).
Two lines with a single operator on each one can be converted into one line with two operators in the following way: the upper operator is enclosed by cursor
|
a ß 1½ b ß
2 c ß
a + b |
and then Del is pressed, that results in such transformation:
|
a ß
1 b ß
2 c ß
a + b |
Furthermore, it is possible to enclose operator b ß 2 with cursor, press Del and obtain a line with three operators. There is nothing unusual in such operations: every text editor allows transformations like that by placing cursor to the end of a line and pressing Del; two lines (two paragraphs) turn into a single line (a single paragraph).
If the reader, in this or in any other book or in the journal article, will come across a Mathcad-program with several operators on a line, he should perceive it as a «stuffed» program which needs to be «unstuffed» – every line should keep only one operator.
Several operators on a line are fraught with one more trouble:
A := a ¬ 2 b ¬ 3 c ¬ 4
The question is, what value will be assigned to the variable A, after satisfaction of such «horizontal» program. In principle – it should equal four: Mathcad-program has to return the last value it has operated with (ñ ¬ 4). But it is not true. The value of the variable A will equal twenty four – the product of numbers appearing in the given program line. In the environment of Mathcad 7 Pro the value of A, returned by our program, equals four, that conforms to the program logic. There is your compatibility of versions! Apparently, the Mathcad 8 Pro compiler recognizes spaces between the operators – as missing multiplication signs (see the Tip 9). The program will return the correct answer, if to arrange it «vertically» again or even if to write the operator ñ ¬ 4 on a separate line.
From this the resume follows! If the reader will take the risk of writing several operators on one program line in the Mathcad-environment, he ought to remember that the last line of the program (or operand of the operator return) should keep only one operator.
The Mathcad programming language – is relatively «young». All programming languages (BASIC, Pascal, C, fortran etc.) allowed only one operator on a line in their early versions. Then («at a mature age») this restriction was lifted to make the programmer decide himself whether to write several or only one operator on a line. Let's hope, that in the future Mathcad versions the Add Line button on the toolbar will be supplemented with the Add Operator button and the described technique of making a program more compact will become legal.
Tip 34. Simulation of conventional programming languages in the Mathcad environment
34 Simulation of
conventional programming languages in the Mathcad environment
Many Mathcad users, having operated with conventional programming languages, find the «multi-stored» technique of formula recording quite natural beyond the program body, and rather «wild» within the program. Such users have got used to the text («one-stored», single-level) formula recording technique. To leave habits alone (and they are, as is known — the second nature) and to make programs more compact one may remember the technique of the built-in function redefinition and renaming, described in the 20 Tip. But it is also possible to redefine the basic built-in operators. For the greater program compactness and for the further indulgence to the user habits the «multi-stored» Mathcad operators can be recorded like «single-stored» ones: integration, differentiation, summation, product etc. The figure formulates the function V, returning the value of the cone volume using «basic-like» operators and functions (sqr, «^» and so on). The ordinary way to formulate the function V is presented on the figure of the Tip 30. The figure in this Tip creates functions with non-standard names (+, -, *, /, ^, è **[21]) by means of the technology of the non-standard symbol insertion, described in the pieces of Tip 14 and 16.
Tip 35. Operators in the programless mode
35 Operators in the
programless mode
Before creating a program it is desirable to «sound»
operators it should contain. The item 1 in the figure presents operators
computing the volume of our cone by measured parameters — the foot diameter and
the element length. The 2-nd item comprises these operators in the program body
where variables r, S and h already become local and
inaccessible for monitoring (inaccessible, if to be not acquainted with the
following Tip).
If there is a loop put into a program (and, in fact, loops are the very reason requiring programming in the Mathcad environment), then, outside the program body, one may try to realize it through successive approximations, which peculiarities are outlined in the Tip 46.
Tips 36-99 in translating