Tip 49. Formula-incognito
Fig. 49.
Formula-incognito
When we
watch the pictures of arrest of the offenders on TV we can see that as rule the
faces of the official staff are closed by black boxes, which move within the
screen synchronously with the motion of the policemen[1]. It is possible to lay the same
boxes on the operators of Mathcad-document if the author does not want to open
his secrets (see the fig. 49, where the mechanism of
the forming of the matrix Ì, which elements are the centres of
the surface[2] net, is black-out). Similar boxes
are laid by the change of color of background of the formulas[3] from white color to black one.
This
tip was written for the seventh version of Mathcad, where it is impossible to slam
the singled out areas of the document (see tip 50 and 62).
But in eighth and later versions (in these versions it is possible to slam an
area) the tip can be useful: it is possible not to slam the area with the
operators-incognito but to protect from editing. This way we intrigue future
user, for example, if we show that enough difficult calculation realizes by few
number of operators. (As a matter of fact user can change the color of the
variables and the constants of the whole document from black to white and… he
tears the mask of the formulas-incognito.)
Tip 50. Protection of the areas of Mathcad-document
Fig. 50. Protection
of areas of Mathcad-document
It is
worth to protect the areas of the Mathcad-document, which are not meant for
editing by the author or the other users, by the password (see the
fig. 50, where we slam the prepared upper part of the document with the
formulas -incognito – see the tip 49). The problem of use of any password
is that we can forget it. The author knows how to unlock the password in the
Mathcad-document however instead of telling this secrete he suggests one more
tip.
As
we beware of attracting attention not quite honest people the author asks
people who have had the password but forgotten it to send the lock document to
the address ochkov@twt.mpei.ac.ru, where it will be unlocked and
forwarded. Payment for the service is new tip in the piggy bank of the author.
The password (password) is
inputted by the general technology (see the dialogue window Lock Area in the
fig. 50): user inputs a password (here the password is two symbols, which
do not print but change to the stars in order to avoid the risk that somebody
could spies the password from the shoulder of «the document’s master»[4]); then it is necessary to duplicate
the password in the field Reenter password. It is possible to slam the area at
the same time of input the password (see the tick in the field Collapse when
locked). The area, which you have slammed, becomes the line (see the beginning
of the fig. 62) where you can write the date of hiding the area (see the
tick in the field Show lock timestamp). But it is possible to remove this line,
inscriptions on it and the other
«tracks» at all that they have not been seen on the screen of the
display. Tip without number: if you look through a new unknown
Mathcad-document it is worth to switch over to conditions of color extraction
of the operators (the switch Regions in menu View – see the fig. 1). In this case all invisible and
slammed areas will be visible (they will be singled out by white color on the
grey background).
Tip 51. Dimensionalities and symbolic mathematics
Fig. 51. Dimensionalities and symbolic mathematics
When reader familiarized with the tips
16 and 17 he made sure that the system of physical and the other values, which
is built-in in Mathcad, allows to call this package not only mathematical one
but also physical and mathematical package. Unfortunately units do not work in
symbolic mathematics of Mathcad: as since it «does not know », that a meter has
hundred centimeters, an hour has 60 minutes and etc. But still it is possible
to solve the problems analytically with dimensional values in Mathcad. At that
the variables, which mean some built-in or user’s[5] units will «behave» as general («ordinary») variables in the
expressions where you apply one or another operations of symbolic mathematics
(simplification, root search, differentiation, integration and etc.).
For example, here is the solution of
a little modified famous problem from the Chehov story «Coach», which is solved
by tools of symbolic mathematics of Mathcad. «A merchant bought 138 arshines
of black and blue cloth and he paid for it 250 thousand old rubles, 50 new
(denominate[6]) rubles and 12 dollars. It is asked how many arshines did he buy of one
and another cloths if the blue one cost 25 euro for one arshine but the black
cloth cost 3 rubles?»
In the fig. 51
the problem is solved by two «symbolic» methods using the function Find and
using the operator solve. In this case units are simple general variables,
which along with the other variables take part in symbolic (analytical)
transformations.
Tip 52. Separate the variables of the
program
Fig. 52. Separate the variables of the program
In the fig. 52
there is soft created user’s function, which returns density (r) of some matter (âåù) depending on temperature (Ò) and pressure (p). Two-dimensional spline
interpolation is laid in the calculation: the kernel of the program is tabular
values of the density (kg/m3) depending on discrete values of the
temperature (50, 100, 150, 200 and 300 degrees by the Celsius scale) and on the
pressure (1, 3, 4, 5 and 6 physical atmospheres)[7].
We
try to «obtain» from the program not one but a few tips.
Tip 1
Two-dimensional
spline interpolation, laid in Mathcad, has in mind that tabular values of the
functions have to be kept in a separate square matrix, but their corresponding
argument’s values have to be kept in a rectangular matrix, which has two
columns: the first column keeps discrete values of the first argument, but the
second column keeps discrete values of the second one. But we choose more
visual method: we note all figures in the matrix Ì. At that the
stub of the matrix Ì (without upper element) keeps the
value of the first argument, but the headline (without left element) keeps the
value of the second argument. We write the commentary in the upper left corner
of the matrix (in «nine», if we use the language of footballers) for free space
was not empty. The rest of cells of the matrix keep the values of the function
(in our case it is the values of matter’s density).
The
notation in the tables with the title and stub that we choose for initial data
allows to edit them easily or simply to use for the inquiry: you look at the
table-matrix and see at once that, for example, if t = 200 °Ñ and p = 5 atm then the density of our matter is 365 kg/m3. Before the interpolation the
matrix Ì «is laid» out two independent matrixes with
the help of built-in Mathcad-function submatrix – ÒÐ (with two columns, which keep the
vectors of the arguments) and V (square matrix, which keep the tabular values for the function).
If
user wants to increase the speed of call of the function r, then it is worth to refuse the
matrix Ì and he have to input tabular values in the
matrix ÒÐ at once (the name of this matrix reflects its
structure: the first column is the discrete values of the temperature, the
second one is the discrete values of the pressure) and V (density)[8].
Tip 2
The
method, which is considered in this tip, is present at all tips of the book
where we have to refer to the numbers of lines and columns of the matrix (see
also tip 70
and others). So it is never worth to rely on the concrete value of the system
variable ORIGIN
(generally it is 0 or 1). It is better to make a link to this variable in the
program. For example, in our function it is possible to write the following
way:
ÒÐ<0>¬ submatrix(Ì, 1,
5, 0, 0)
But not:
ÒÐ<ORIGIN> ¬ submatrix(Ì, ORIGIN+1,
rows(M), ORIGIN, ORIGIN)
As a matter of fact this «simplicity» (see the
first operator: ÒÐ<0>¬…) can turn out «worse then
stealing». If our simplified function is found in Mathcad-document, where the
default of the variable ORIGIN is broken (it may be equal not zero but some another whole number),
then the bug will appear. The mistake will appear if we edit the matrix Ì – if we input
new data in it.
Tip 3
Mathcad
gives unique opportunity to change the type (Times, Arial, Courier and others),
the size (8, 10, 12 points and etc.) and the style (bold, italic, underline) of
the print of the variables, functions, and constants. It is impossible
not to use this possibility. In our program the variables and the functions are
divided by groups, which are united by the structure of data. Using the
principle of continuous of the form and the contents we can understand the
features of programming in Mathcad:
·
predefined
variables and built-in functions have bold type Arial Cyr size of 12 points
(here it is possible to pick out the separate group of built-in and users units
of physical values);
·
local variables
of a program – as well type is Arial Cyr, but it is underline and less in size
– 10 points;
·
user’s
functions and variables, which are visible in whole Mathcad-document – normal
type (Variables);
·
formal
variables (arguments of the functions – T and p) – italic;
·
as
well numerical and textual constants can have different type (Constants and
«Small constant»), which underlines the different groups of the constants in
the matrix of table.
It
is marked in the fig. 52 that the different groups of variables «are
attached» to different styles of variables, which are given the
different names[9]: «Local variables», «System
variables» and etc.
We
can change Ó èìåí ïåðåìåííûõ, ôóíêöèè è êîíñòàíò ìîæíî òàêæå ìåíÿòü color of the type of names of the variables, the functions and the
constants. This theme will be considered in the sketch «Color in programs» (http://twt.mpei.ac.ru/ochkov/Color_in_Program/index.htm).
Tip 4
The
advantage of user’s function in the fig. 52 in comparison to its analogue,
which is written in the environments of the traditional programming languages,
is that our function has dimensional arguments (temperature and pressure) and
returns dimensional value (density see tip 16, 17, 21, 45 and others).
Unfortunately arrays of Mathcad can not keep the elements with different
dimension. The way out may be the following: at the begging of the program we
have to deprive the arguments (practically parameters) the dimension, reduced
them to that non-dimensional value, which is used in the stub and title of the
table, but at the end of the program we have to add the dimension of the
tabular data to the outputting value of the function.
Tip 5
In the
tips of the firm Mathsoft,
Inc.(«Tips of day» – see point 74) partial solution of the problem of use
degrees Celsius in the calculations, which are not built-in in Mathcad by means
definition of user’s function is shown. In our case you define the function
with the name °Ñ too, which is called as the postfix
operator (150 °Ñ). This
function converts non-dimensional degrees Celsius to the dimensional Kelvin
(this method is more detailed described in tip 64). In the calculation we have
two definitions of the function °Ñ – point is that they are different functions (the main point of the
tip): in one function Latin letter Ñ is written, but in the other
function Russian letter is written. It is done for user, that he has not
puzzled over on what case to work during solution the problem and «has not got
stuck» on unintelligible mistakes – when instead of Russian letter is inputted
Latin one and vice versa. This curious thing we very often meet with the letter
Ñ, as since Latin and
Russian variants are on the same key of the keyboard.
Tip 6
It is
worth to fill empty space in a matrix up a textual constant, which explains its
contents (see the left upper corner of the matrix M). It is possible to do without any mistakes as
since numerical constants are non-dimensional in the matrix.
Tip 7
If the
initial matrix of tabular data, on which sline-interpolation has to be
conducted, is not squarte one but it is the rectangular one, then it is
possible to recommend to divide this matrix by two cross squarte ones. So we
can conduct interpolation using one of the matrixes depending on the concrete
value of the «long» argument – the argument that is fixed on the long side of
the rectangle.
Tips
may be more, but… seven is beautiful number …
Tip 53. Method of step-by step approximations
Fig. 53. Method
of step-by-step approximations
Well-known
problem about the evolution of an epidemic[10] is solved in the fig. 53:
in a town, where 20 thousand people live (Healthy), unknown number of sick people
appear (Sick), it causes the epidemic, which the
simplest model is described by two formulas. It is asked, how many sick people
were on the first day if there were 100 sick people on the thirteenth day.
The
problem is solved by the method of step-by-step approximations: we set the number
of sick people on the first day (this operator is the last in the document in
the fig. 53[11]) and we keep a look out the
variable Sick13, which value prompt to user how to change the variable Sick1 in the next approximation. We can
fix the «history» of the approximations in the matrix, which is reflected at
the end of the fig. 53. For it we have to write or
copy manually the number triples, which are got at the each step of
approximation to the solution, when the deflection (the last line of the matrix
in the fig. 53) becomes equal zero (or rather nearly
zero).
(Tip
without number. The elements of the matrix can be reflected graphically, if
we visualize step-by-step approximations). Continuation of the theme you could
see in tip 91.
Tip 54. Touch up the names of
the variables
Fig. 54. Touch up the names of the variables
As rule the
process of creation of Mathcad-document breaks into two stages. At the
beginning some draft of the document without the commentaries and with
short names of user’s variables and functions is made. The main purpose of this
stage is the input of the formulas and the debugging of the operators. At this
moment user thinks not enough about that it is important not only the momentary
right result but also the openness for optimization of the program, the study
and the perfection by the author and/or the other people in the
Mathcad-document.
At
the second stage we bring in the commentaries, format numbers and graphs,
change short («dumb») names of the variables and the
functions to long («speaker») ones
in the Mathcad-document. For this change it is useful to use the command Replace from the menu Edit.
At that it is possible to bring in the symbols, which are not available at the
direct print of the names in the names of the variables and of the functions at
the same time. This replacement we can consider as the third way of
writing of the non-standard names – another two ways are described in tip 14
and 18. The creation of the variable with the
inferior (bt) and upper (apostrophe, accent) indexes is reflected
in the fig. 54. (The variables «with the accent» are
widely used in the engineering calculations. For example, t' is the in temperature, but t" is the out temperature[12].) It is impossible to obtain the
accent at the «direct» input of the variable as since the
corresponding keystroke results in the rise of the inverted commas.
Here are some tips without number,
ensuing from the foregoing:
1.
It is worth to write manually («on letters ») the
name of the user’s variable or function only one time. When you call
this variable (the function) next times it is worth to copy it but not
to print again. The method is especially useful when the name is long and/or
includes special symbols, which are not available for manually print. More over
it is possible to mix up Russian and Latin letters, which are «similar» by way
of writing: c, a, ð and
etc.
2.
When you copy the functions it is useful to hold not only its name but also its
arguments, then you can touch up them, change the formal variable, which are
framed by the brackets and are shared by the commas, to the real variables or
the constants (about the formal variables you could see in tip 83).
3.
It is necessary to make an effort to input at once minimum commentaries
in the Mathcad-document and not to put off this work later on[13]. Very often this «later on» is
absent: the program has worked and given more or less acceptable result. But alas
the program is closed for study and the subsequent processing as since it did
not obtain good commentaries in due time.
4.
It is not unreasonable to write the extensional Russian-language commentaries
in the environment of some textual editor (in Word, for example) and
only then to transfer them in the Mathcad-document, as since Mathcad has tolls
for only English spelling check[14]. Apart in Word it is possible to
check hand in hand with spelling syntax and punctuation, to sort
out the synonyms for some words and much more that only «real» textual editor
allows to do. It is possible to inculcate Word in Mathcad (or vice
versa) for we do not carry the textual parts «here and there» (from Mathcad in
Word and back).
5.
The spelling mistakes are possible but they are desired in the names of user’s
variables and functions. That’s why it will not be unnecessary to write the
name in Word at the beginning, and then to carry it in Mathcad.
6.
It is worth to input the built-in functions but the call of master function but
not on letters print of their names. It will exclude some mistakes of «Russian-language» user
(you input intersept instead of intercept,
for example).
Tip 55.
How to create
shareware-programs in Mathcad
Fig. 55. How to create shareware programs in
Mathcad
Shareware[15] is so-called free program that is
speared under the motto: «We have tried if you liked it and buy!» The similar
programs are available via Internet (the site www.download.ru,
for example) or via ÑD-ROM
that are enclosed to different computer magazines (ComputerPress, Game.exe,
Games Country and the other). As rule, the work with the similar program is
connected with some limitations, which induce user to connect with the author for
to register or to get full version of the program or the information of
liberalization. Here you can see not only mercantile interests (see the motto
above) but also the other reasons. For example the author, when he lays some
«shackles» on the program[16], can lead the database of users,
attract them for testing and for perfection of the program: «Tried, liked, then
register (praise, criticize, work together with me and so on and etc.)!»
The example of shareware, close to reader is
freeware Mathcad Explorer, which you can at the sites of Internet[17]. This program allows to work with
Mathcad-documents but with some limitations: for example it is impossible to
save on the drive the modified file[18] – for it you have to buy the
commercial version of Mathcad.
One of possible technologies of creation of
shareware-Mathcad-document is reflected in the fig. 55.
With the help of it our old problem about the cone’s volume is solved (see tip 35, tip
36 and tip 37). It clear that we can solve the other,
more difficult program which solution is the intellectual property of the
author that he want to protect via the mechanism shareware.
In
the point 1 in the fig. 55 the variable Attention![19] has the information, which the author of the
program here and further pushes «under the nose» of future user. This way the
author reminds user that he works with the given program as if «on credit ».
Then user is suggested to input initial data of the calculation – geometrical
sizes of the cone, which volume is calculated by the formulas in the frames
(see also tip 62). To distribute ready program it is
necessary to slam these frames with the input of the password (see tip 66). But before it we have to put down the
calculated cone’s volume some «sly» method in the «out» parameter V: in 70 cases from 100 the variable V keeps the write result and in the
other cases it keeps the contents of the variable Attention!
Here are the defects of the shareware-program
that is showed in the point 1 in the fig. 55:
- it
is impossible to use the dimensional physical values in the calculation
(length, square and volume), as since the variable V has by turns either numerical
or textual value (see tip 45);
- it is necessary to change the
evaluation version of the program to full one (registered version; for it
we can use either email or general mail or hand over the program at the
meeting. Here there is no special program. The inconvenience is that user
will have both evaluation and full version of the program, in which it is
possible to involve in.
These
defects are removed in the point 2 in the fig. 55.
In the first place no it is possible to work with dimensional values owing to
that the message of the author is not already textual constant as we have seen
in the point 1, but it is user’s error message: in the similar case the
recreated function Volume of the cone returns the right result in 70% of calls but the error message in 30%
calls. In he second place our shareware becomes the general software after the
author will tell user his registration number. In this case the variables Name and SerialNumber keep something different from that
was written in the point 2 in the fig. 55: for
example “NN” and 00000000. After that user will try the
program and will make sure that the program returns right and necessary result
(as before the calculation is hidden and protected by the password), he deals
with the author and gets his registration number, which is generated by the
function[20] in the point 3 fig. 55.
When you input user name (for example, it was “NN” and becomes “Orlov K.”) and registration number (it was 0000000 and becomes 6796937) the program stops to remind that
he work with the evaluation version. If the author wishes he is able to send by
mail (or email), to let know by the telephone or to hand over at the private
meeting the password as well, with the help of which the calculation was
closed: «to sell» user not only the results of the calculation for different initial
data but also the calculation. Registration number and password can be encoded
in addition before sending.
The
developer of the program can change the power (number 3) to another one in the
function of the point 3. It is acceptably to change the algorithm of generation
of the passwords to some especially secret one. The main point here is that the
unique number is generated by the name (it may be fractional number: 135.834, for example).
Attention!
The mechanism of creation of the
trial function, which is showed in the fig. 55, is
broken if at the beginning of the document we write Mathcad-operator rnd(x) := 0. In Mathcad, as well known (see tip 13) the built-in functions are defined and we can
use it. It is said that it is possible to break any protection. But the
protection, which described in this tip are not the protection per se, but the
mechanism for reminding the user about his guaranties before the author. Any protection
is meant for honest people, who come, for example, to closed door and pull it
out they understand that the way is closed. A swindler begins to break open
this door or even to beat it out and he does not guess that there is another
open door and the way through it is not always longer one but in some cases is
even shorter.
Tip 56.
Speeding-up the
work with three-dimensional graphics[21]
Fig. 56. Speeding-up the work with
three-dimensional graphics
One of the
defects of the work with three-dimensional graphics of Mathcad is too difficult
to format the graph if we have large quantity of points: to turn, to change the
colors and etc. While user wait for next repainting of the graph he asks
himself not once: «Why have I decided to do it?»
This
problem can be solved a few methods. One of these methods is not use
three-dimensional graphics in Mathcad, but the application the other
special-purpose programs for it such as Axum, for example. But the discussion
of this method exceeds the bounds of the given book, that’s why we’ll not
consider it. We try to solve the problem within the environment Mathcad. The
main point of the method for decrease of «the dormancy» of three-dimensional
graphics is the dynamic decrease of the number of displayed points. It may be
done by two ways. In the first place it is acceptably to decrease the number of
calculated points. However at that user can not always guess beforehand the
necessary size of changing of the variables and their changing step: it may
turn out that the built graph will not have all necessary information. As a
result we have to calculate whole the matrix[22] anew. But if the calculation takes
plenty of time, then it may be simpler not to touch the matrix but to format the
graph at once[23]. It is possible to overcome this
defect if we use the second method, which is suggested here as the tip.
The
main point of the tip is that we construct three-dimensional graphs not of the
initial matrix, but of the doubling one, in which we bring in only the
part of points of the initial matrix. The illustration of the mechanism of
construction is showed in the fig. 56. For dynamic
changing of the power of the detail (of the quantity of points in the matrix,
which is used for construction of the surface) is brought in the variable Detail. Then we bring in new index
variables, we calculate the convert coefficient (the variable with the name k) and finally we fill the second
matrix by the elements from the first one with the given detail. As a result
three-dimensional graph of not the whole initial matrix but only of its part is
constructed.
The
advantage of the given method is that the above-mentioned graph reflects the
whale size of changing of the variables, but with the large changing step.
Using the choice of the value of the detail coefficient it is possible to reach
that three-dimensional graph will quickly «react» to change its format. After
it we can easily adapt its appearance and only then we assign the value 100% to
the variable Detail (or place
the reflection of the initial matrix). After it our three-dimension graph will
look like we have thought.
Tip
without number.
Very often the difficult three-dimensional graphics applies the brake
appreciably to the process of calculation by the operators, which are kept in
the Mathcad-document. Quiet often it prevents the review, scrolling of the
document. In this case it is possible to recommend to turn off the
process of the recalculation and the repainting of the graphs for a certain
time with the help of the switch Disabled Calculation. The more cardinal method
of the shutdown of the graphics is the change it to the picture. For it
the mold of the screen with the graphics is transferred to the environment of
some graphical editor, then we finish off it there and new but already «stark»
form is inserted in the Mathcad-document. (The principle of chopping of Gordian
knot – see the sketch «History of one masterpiece» (http://twt.mpei.ac.ru/ochkov/Lace/Lace.htm).
There compromise method for review the unusual three-dimensional graphics of
Mathcad is suggested: at first we are suggested to look at small picture, then
we see enlarged picture on whole display, and only then «viewer » can decide if
it is worth to download the corresponding Mathcad-document and view the
«masterpiece » in all details – to turn the volumetric structure, «to touch»
its settings via the window of format and etc.
Tip 57.
Work with the
files on the drive[24]
Fig. 57. Work with the files on the drive
When you
work with data files in Mathcad sometimes we have the necessity to check up if
the necessary file already exists, if it is necessary to recopy it (it is done by
the call of the built-in function WRITEPRN) or add it (APPENDPRN). The given problem is solved in
the programming languages: if user would like to add non-existent file, it
automatically creates as new one. The function APPENDPRN does not have the similar
properties: the error message will appear if you want to add non-existent file.
The
given problem can be solved using the different ways. But the simplest method
is to address to the firm MathSoft with the request of making over the function
APPENDPRN or of creation new one of similar
nature but with the possibility of the creation of the file in the absence
thereof. The second ay is to write the language Ñ the function, which either would change the
function APPENDPRN or
simply would check up the existence of the file on the drive. However the
considered problem can be solved by means of built-in possibilities of Mathcad:
via the call of the function READPRN with use of the operator on error and the additional array of two elements,
which keep the file names.
The
fragment of the Mathcad-document is showed in the fig. 57,
which is illustrated the suggested tip. The main point of the method is that we
execute the attempt of reading the data from the file before saving it. If we obtain
the beneficial result (if the file with necessary format of data exists) then
the variable, which keeps the file name for addition (the call of the function APPENDPRN) is assigned the initial file name,
but the variable, which keeps the file name for creation of new file (ôóíêöèÿ WRITEPRN) is assigned empty line. Otherwise (if the
file does not exist on the drive) the above-mentioned operations are made in
the reverse order. The control of right reading is made with the help of the
operator on error, which
executes the operations of the left operand in case if the execution of the
operation of the right operand has been the cause of the error.
If the file does not exist on the drive then
the function WRITEPRN will be
called and the following calls will be of the function APPENDPRN[25].
The defect of this method is that every time
the function of reading data from the file READPRN will be called. It may slow down
very much the process of calculations if the file has large quantity of points.
Tip without number. It is possible to use the functions for
saving the files on the drive for the transfer the value of the variable from
the lower part of the Mathcad-document to upper one. It is necessary, for
example, for realization of the step-by-step approximation method (see tip 53).
Tip 58. The function if and the operator if
Fig. 58. The function if and the operator if
Generally
users of Mathcad for realization of the alternatives out of the program
use the built-in function if from Master function, but inside the program the use the operator
if from the programming panel. But it is possible to recommend to
use the operator if out of the program too. (About the inclusion of the function if in the programs – see tip 40.)
The
user’s function, which returns the sum of income tax[26] depending on the sum of «unsavoury»
income[27], is formed in the fig. 58.
In the point 1 the function Tax is formed owing to multiple enclosure of the built-in function if. At that the function Tax becomes very long and difficult for
reading and editing – we have to write this function manually in the column[28] in the fig. 58.
In the point 2 the function Tax is formed by the operators if and otherwise. The
third variant of the program is perfected in the following directions:
·
units
of cost are brought in (see tip 18);
·
work
of the function is hastened owing to use of the operator return;
·
the
operator otherwise is
removed as since all variants are sorted out and there is only single one.