The creation of the matrix. Functions of Matrix Conversion Specific Matrix Conversion

A single Mathcad number is called a scalar. A set of numbers is called a vector, and a rectangular table of numbers is called a matrix. The common term for a vector chi matrix is ​​an array.

There are three ways to create an array:

  • Zapovnyuyuchi an array of empty fields, as discussed with whom to distribute. This technique is suitable for small arrays.
  • Vikoristovuyuchi discrete argument, schob vyznachit elements for yoga help, as discussed in the attacking division. This method is suitable if there is a clear formula for the calculation of elements through their indexes.
  • Reading them from data files.

You can separate the names of matrices, vectors and scalars, vicarious font for their writing. For example, in rich mathematical and engineering books, the names of vectors are written in bold, and the names of scalar variables are in italics.

Vector creation

A vector is an array, or a matrix, to take revenge on one step. To create a vector in Mathcad, follow this:

  • To press at a free mist or on the field.
  • Choose Matrices from the menu Mathematics or press M. It's a dialogue vikno, like a right-handed little one.
  • Enter the number of rows equal to the number of vector elements in the “Row” field. For example, to create a vector from three elements, instruct 3.
  • Instruct 1 at the “Stovptsiv” field, then press “Create”. Mathcad creates a vector with empty fields for filling.

At the next stage, it is necessary to fill these fields with scalar virases. For which one, follow this:

  • Click on the top floor and give a shout 2.
  • Move the frame to the next field. You can either use the key, or click without the middle on the other field.
  • Inflict 3 on the other field. Let's move the frame, which you see, in the third field, and instruct 4 .

As it is necessary to create vectors, you can leave out the dialogue window “Matrices” as a substitute for a farther whistleblower.

As soon as the vector is created, you can win yoga in the calculations exactly like that, like the number. For example, in order to add the next vector to the second vector, it is necessary to include the following:

  • Press [ ] once or click on any arc of the vector. The visual frame now wraps the entire vector. Tse means that the plus sign, which will be an oversight, is brought up to the vector completely, and not to any element.
  • Press the plus key ( + ). Mathcad shows a field for another vector.
  • Tweak the dialogue box “Matrices” to create another vector from three elements.
  • Save this vector by clicking in the skin field and other numbers, showing right-handed. It is also possible to vikoristovuvati, to collapse from one element to another.
  • Click on the sign = , to get the result.

Adding - just one of the Mathcad operations that are assigned to vectors that matrix. Mathcad also has matrix variables, matrix multiplications, scalar twir, integer steps, determinants and many other operators and functions for vector matrices. Additional lists appear in the sections “Vector and matrix operators” and “Vector and matrix functions” below in this section.

The creation of the matrix

To create a matrix, click on the back of the head at the free station or on the floor. Let's do it:

  • Choose Matrices from the menu Mathematics, or press M. To appear in a dialogue box.

  • Enter the number of rows and columns for the required field. For each butt, the matrix has two rows and three columns. Let's press "Create". Mathcad create matrix from empty fields.
  • At the end, fill in the fields as described in the front section for vectors.

You can twist the matrix in the formulas exactly like this, whether it is a number or a vector.

Skrіz at tsomu helper the term vector is introduced to the vector-stovptsya. The column vector is identical to the matrix with one column. You can also create a row vector by creating a matrix with one row and many stacks. Operators of that function, if they take a vector argument, must always check for a vector-sovpets. The stench is not stagnant to vector-rows. To transform a row vector into a row vector, use the transposition operator1.

Changing the size of the matrix

You can change the layout of the matrix, inserting and sorting rows and columns. For whom it is necessary to visconate such:

  • Click on one of the elements of the matrix to put it in the frame that you see. Mathcad starts inserting or removing the th element.

  • Choose Matrices from the menu Mathematics. It's a dialogue window.

  • Override the number of rows of abo stovptsiv, as it is necessary to insert or remove. Let's press "Insert" or "Vidality". For example, in order to see the stovpets, which is to avenge the vibrating element, instruct 1 in the “Stovptsiv” field, 0 in the “Row” field, and click on “Vidality”.

Axis like Mathcad is visible or insert rows or columns, depending on what is added in the dialog window:

  • When rows are inserted, Mathcad creates rows of empty fields below the selected element. When inserting columns, Mathcad creates columns of empty fields right-handed in the selected element.
  • To insert a row higher than the upper row or a left-handed one in the first row, add the matrix to the frame on the back, as you see, clicking in the middle and pressing [ ] sprat once. Let's choose Matrices and continue, like a zavzhd.
  • If the rows of abosts are removed, Mathcad restarts the rows of abosts that are occupied by the selected element. Mathcad sees rows down in the direction of the element and columns - right-handed in the direction of the element.
  • To override 0 in the “Row” field, Mathcad does not insert, does not see the rows. If you want to override 0 in the field “Stovptsіv”, Mathcad does not insert, do not remove the columns.

Take care that when you see rows of machines, Mathcad finds out the information that is hidden in them.

To see the entire matrix or a vector, increase the frame that you see, and select Virizati from the menu Edit.

Rely on the vipadok chi to create the scenario of your life?

Especially I always choose another option, but if you don’t fit me, you can close the sheet.

Lost..?

Let me tell you one terrible mystery!

The more important you are, the more negatively the pictures are painted.

Fear turns on right away: "And by raptom everything is wrong ..." And then your way will helpfully show you what you can drink "not so." Zreshtoy, you wind yourself up so much that you don’t have another way out. And everything seems to be like this, you say: "Well, axis, I know that!" And if you don’t see it, then: “Fuh, it has passed!”

But how many nerves were stained?

You know, for such tributes, fun, like, it would be possible to be the best opportunity for young people, becoming the most stressful for them. Tse "And raptom everything is wrong?" manifests itself in all beauty, and sees "happy" death, if everything ends up.

Aje mi yourself program your future! really!!

I don’t feel like repeating those that you know so well - “material thoughts”.

Wouldn't it be easier to choose the most suitable option for you and don't worry anymore?

I'll go straight to the head.

My closed information collection has one particular method. With the help of yoga, you can choose for yourself, as you develop those other things in your life. To fight for the help of your intuition.

You just take care of the situation in a singing manner, and your brain starts to see you think about those how it will develop. Yes, obviously, deaky thin, but at the very document.

If you can, you can come up with a proponated option or use a different scenario!

Who has the whole trick!

You can only see, it’s easy to see!

Admittedly, you blame the bazhannya climbing uphill with caravans and occupying a graveyard.

If you think at once, how you can reach it, your brain will immediately see you as a solution. For example, - richly and diligently practice, always show yourself from the shortest side. Ale, how much do you have in your power decision?

Surely, here in the depths of the soul you would like to, so that everything became richly easier and simpler. For example, they took you and promoted you just like that, without any visible reason. Come to work, and the order for recognition has already been signed. Unreal? It’s more real, as if choosing the script for the replacement of bagatory practice.

Іsnuє impersonal options for the development of podіy!

And about the skin one of them, your child knows well! All the same, I went along the path of the smallest support, protecting you along the way from the new and unknown.

And how badly would it be to steal from pobutovih situations!

The axis of you is chosen as a guest until you are not in love with your mother-in-law and your mother-in-law ... You already have an approximate scenario in your head, how you can do it (otherwise, you should see it if you come home). You can't drink it. Ale, you can defiantly do better and take away your satisfaction, in order to choose a different scenario!

Axis yak tse pratsyuє in practice.

“I sang it at the door, but I didn’t know where to go…”

“Your secret method was victorious, and right in the brain began to sipatisya options for pіdsvіdomosti (fantastic just!) - Sanatorium, sea, vіdpochinok at the dacha. Half-heartedly I thought about the sanatorium, but suddenly, suddenly, opposite susides began to appear. I didn't fit in. Then I wrote about the sea, the thought was worthy, but in the middle I felt: “No tickets” (as we were talking, there were only upper policemen, but I can’t ride them, I’m afraid of flying). The dacha didn't call out to be suffocated - mosquitoes, suzidi ... Then I became a check for another decision. The thought came like a read - phoned to friends ... Phoned. It appears, the stench of going to the sea in a wet car and for the joy of taking me with you. I buy a bathing suit, and on the road! » Lilya

“Having tried the technique with the girls…”

“To tell you the truth, I have problems. Friends called on weekends to the club. Call the siju at the table and I don’t dance. That kind of respect for me girls do not turn. Ale, I have edited the script. Zrobiv yak vy vchili and pochav chekati. It has come, what a girl has come to me. We sit together and marvel at the dancers. Chi didn't breathe. Then pіdsvіdomіst saw more and more options. I have come up with a scenario, if the girls themselves ask for me and I can’t see them. Having praised the method and forgetting. So if the axis came to the club, I would like to start the table. I didn’t manage to buy a beer, as it started to be a naytsіkavishe ... I went one at a time, then asked a friend to dance, then a third ... There was never such a thing in my life! Respect to you and respect! Tolyan

* Suggestions made without editing.

Now you yourself can choose how your undertakings will develop!

The technique is rather simple, no special tricks are needed for you.

Everything that you will need to work, tse:

1. For the help of a special reception, clear the brain from the deep thoughts and prepare the mind to work.

2. Victory a number of the most difficult triggers, as if you were wondering your opinion and see you different solutions.

3. Let us praise the rank of the best variant of the development of the subdivision.

You will find all the details in the closed archival document “How to Vibudovuvat the Correct Matrix of Pods for Helping Intuition”.

Usyogo - 730 rubles! To quickly take the method, just go for the help and finish it

I GUARANTEE A SPECIAL PIDTRIM!

I always confirm the call with my pre-payers, and I confirm the obov'yazkovo on all requests. What is wrong in the process of practice, otherwise, if you want to clarify, I will always help you. Possibly, I don’t mind, the fact that the leaves of my mail are richer today, but I guarantee that I’m not sure.

My special address: [email protected] website

P.S. Just think about it! It will be easy for you to independently care for your life! You just choose the desired scenario.

With respect, Oleksandr Kling

Mathcad makes it easy to create matrices, which can be a simple everyday life, for the help of one of the introduced functions.

identity (N)- Alone matrix expanded NxN;

diag(v)- diagonal matrix, on the diagonal there are elements of the vector v:

N – whole number;

v is a vector.

View of submatrices

From matrices or vectors, you can see either a submatrix, or a vector, or an element. І back, you can "glue" a sprat of matrices into one.

Part of the matrix is ​​seen as one of the offensive methods.

    seeing one element assigns the operator to the lower index. The operator is entered by pressing the Subscript button (Lower index) with the x n icon on the Matrix panel or by pressing the keys<[> ;

    to view from the matrix column, click the operator to view the Matrix Column button from the images of the bows<>on the Matrix panel, or use the keys +<6>.This operator is called, by analogy with the previous one, the operator of the upper index;

    to see a row from the matrix, stop the same operator<>to the transposed matrix;

    to see the submatrix, use the function submatrix (A, ir, jr, ic, jc) to rotate a part of the matrix A, which is between rows ir, jr and columns ic, jc inclusive.

NOTE 1

You can see one page or a row from the matrix using the additional submatrix function.

NOTE 2

The same operation zastosovnі to matrix-vectors and matrix-rows. Less memory, scho to become Nx1 and 1xN is clear.

Evil Matrix

In order to add two or more matrices to one, Mathcad has added a couple of matrix functions:

    augment (А, B, C, ...) – a matrix, formed by changing matrix-arguments to the right;

    stack (A, B, C, ...) - a matrix, formed by the matrix-arguments in a downward direction:

A, B, C,. .. - vectors or matrices of variable size.

Specific transformation of matrices

Two more built-in Mathcad functions allow you to create matrices based on the same explicit matrix:

    geninv(A) - fold matrix, reverse (levoruch) rectilinear matrix A;

    rref (A) - transformation of the matrix or vector A into the steps of the view:

A is a matrix, composed of real numbers.

NOTE

The layout of the NxM matrix for the geninv function is due to such that N>M.

Often, it is necessary to rearrange the elements of a matrix or a vector, arranging them in the same row or row in the order of growth or decay. For this є kіlka vbudovannyh funktsіy, yakі allow gnuchko cheruvati sorting matrices:

    sort(v) - sorting the elements of the vector in ascending order

    reverse (v) - permutation of the elements of the vector y in reverse order;

    csort (A, i) - sorting the rows in the matrix with the elements of the 1st column in the ascending order;

    rsort (A, i) - sorting the columns of the matrix and sorting the elements of the i-th row in the order of growth:

v is a vector;

A - matrix;

i - number row index.

NOTE

If the elements of a matrix or a vector are complex, then the sorting is carried out correctly, and the obvious part is ignored.

Sorting matrices by row and by column

To remove information about the characteristics of matrices or vectors in the transfer of such functions (appendix 9 and 10 are appropriate):

    rows (A) – number of rows;

    cols (А) – number of columns;

    length (v) – number of vector elements;

    last(v) - index of the remaining element of the vector

    max(A) The largest element of the array A. If A has the most complex elements, rotate the largest speech part plus i, multiplied by the largest visible part.

    min(A) The smallest element of the array A. If A has the most complex elements, rotate the smallest speech part plus i, multiply by the least obvious part.

A is a matrix or a vector;

v is a vector.

NOTE

If matrix indexes are numbered from 1, then the system constant ORIGIN is not (for locking), but 1, then the number of elements of the vector and the index of the th remaining element are increased.

Rozmir matrix

Rosemir vector_v

The film "The Matrix", brothers Vachevsky, is insanely, є cult among cyberpunk, but how real is the prospect of creating a similar "Matrix" really? In order to check the power circuit, the scientists conducted an analysis of the readiness of supercomputers to create virtual reality. And behind the words of Michael McG'yugan from the Brookhaven National Laboratory, this kind of task of creating pieced light, similar to reality, is not such a fantasy, but the future is not far away!

In the 1950s, Alan Turing, who stood in the wake of modern computer science, having propagated an elementary test for piece intelligence, which, having poked at the fact that, for an hour, rozmove two spivrezmovniki, one of them is a machine, a third person tried exactly in figure out who them who.

A variant of this Turing test is the graphic Turing test, which is reminiscent of the fact that a person, spying and mutually with the piece-created world, is guilty of not being sing-songly vіdіzniti її vіd reality. "Under the vzaєmodієyu maєtsya on the vazі vozhіstі mozhlivіst kontrolvatі ob'єkt, for example, wrap yogo, at tsimu vіn is guilty of promalovuvatisya in real time mode", - explains McGyuїgan.

Irrespective of those that have computers, they can create piece scenes with detailed textures, like a building fooling the human eye, it’s more than a year to paint such scenes. Possibility of passing the graphical Turing test helps to confuse the photorealism of the image, with software security, building to render the image to the real hour - with a speed of 30 frames per second.

As the New Scientist says, to be significant, how close modern computer technology is to the creation of virtual reality, McGuygan has become the fastest supercomputer in the world - Blue Gene / L, is promoted at the Brookhaven Laboratory in New York. The supercomputer is composed of 18 racks, the skin of which has 2 yew. standard processors for PCs, with the help of which wines are used in parallel computing modes, which gives Blue Gene its great productivity of 103 teraflops, or 103 trillion. floating point operations per second. For floating point, the default calculator takes 10 floating point operations per second.

The very last one, having built the supercomputer's building, has become a world of light on objects, which is an important component of a virtual world, which claims to be a reliable image of reality.

The results showed that the best program, which is easy to run on Blue Gene, is 822 times faster, lower on a standard computer, similar to those that the program is not optimized for work with parallel processors. This kind of lightness allows a pleasant equalization of natural lighting at the hour.

"It's a good novelty to say that the human eye perceives the promiscuity of these changes, as if it were natural, - it seems McGuigan. tsyuvati significantly more and more realistically ". And yet, to build Blue Gene with the help of yoga swidkistyu and realism, it’s not possible to do a quick painting of an image in a high-rise building in order to pass the Turing graphic test.

Prote doslidnik upevneniya that supercomputers, zdatnі to pass the test, z'yavlyaetsya protyazhchih rokiv. For this reason, we will be able to pass the test, if the productivity of computers reaches the mark of one petaflops, or one thousand teraflops.

Other contributors are aware that the passage of the Turing graphical test is much more important, the graphics are simply photorealistic, which collapses in a real hour. Reality is richly folded, like Paul Richmond and the University of Sheffield, UK. Behind these words, a piece object may look like a correct one, but if the wine does not crumble in a realistic manner, then one cannot confuse it with a correct one. "The main problem is the creation of realistic imitation, as it includes a realistic imitation of behavior," - say wine.

Mark Grandland from the University of Cambridge submits that Turing's graphic test does not specify what the scene of the virtual world can convey. "Yakby is small on the surface, that to pass the test it is necessary to form a sphere, that it shows the pink light on the surface, that it shows the rose light on the surface, then the test buv b has already been passed for a long time - even wine. - Ale Turing did not instantly reveal, wow the idea is so cool become reality."

McGuigan is good for this because realistic animation is a problem in itself. "Modeling something that is constantly changing seems to be a foldable process, - the reporter confirms. - You are to blame for perekonati looking at what happened, what shook the virtual world, maє vag". Prote wines with optimism marvel at the future of the animation program. On the other hand, the creation of virtual reality is handed over to the robot with light and light. At another point, they already got out - on the right behind the move.

The most straightforward way to create a matrix or a vector is by setting the first button of the Matrix toolbar (Matrix) (div. Rez. "Array" ch. 4). However, more vipadkіv, zokrema when programming folding projects, it's easier to create arrays for additional functions.

Designation of matrix elements through a function

  • matrix(M,N,f) - the creation of the matrix MXN, the element i,j is f(i,j) (Listing 9.19);
    • m - number of rows;
    • N - number of columns;
    • f(i, j) is a function.

Listing 9.19. The creation of the matrix

There are two more specific functions for the matrices to be combined, which are to be found, mainly, for the smooth and effective representation of any fallows in seemingly trivi- mer graphs (such as surface or space curves). Uсі їх arguments, krіm first (functions), neobov'yazkovі. Let's take a look at the first of the functions.

  • GeneSpace(F(or f1, f2, f3) , t0, t1, tgrid, fmap) - nested array that represents the x-, y-i z-coordinates of the parametric space curve given by p;
    • F(t) - vector function of three elements, set parametrically for a single argument t;
    • f1(t), f2(t), f3(t) - scalar functions;
    • t0 - lower boundary t (default -5);
    • t1 - upper boundary t (for default 5);
    • tgrid - number of points of the grid of change t (for locking 2o);
    • fmap is a vector function with three arguments that defines the transformation of coordinates.

Read about investing arrays in rozd. "Creation of the tensor" goal. 4.

Rice. 9.4. A variation of the CreateSpace function with a different set of parameters

An example of a variation of the CreateSpace function is shown in fig. 9.4. Respectfully, to encourage the graphics of the spiral, it is not necessary for a supplementary code, but the parameter of the parametric occurrence of the vector-function F!

The function of creating a matrix for the graph of a trivi- mer surface is set in exactly the same way, with the exception that not one, but two changes are needed for a designated surface. Butt of vikoristannya ilustraє fig. 9.5.

Rice. 9.5. A variation of the CreateMesh function with a different set of parameters

  • CreateMesh(F(or g, or f1, f2, f3) , s0, s1, t0, t1, sgrid, tgrid, fmap) - creates a nested array that represents the x-, y- z-coordinates of the parametric surface given by the function F;
    • F(s,t) - vector function of three elements, given parametrically by two arguments s and t;
    • g(s, t) is a scalar function;
    • f1(s,t),f2(s,t),f3(s,t) - scalar functions;
    • s0, t0 – lower between arguments s, t (for capturing -5);
    • s1, t1 – upper interarguments s, t (for capturing 5);
    • sgrid, tgrid - number of grid points behind the change s and t (default 20);
    • fmap is a vector function of three elements with three arguments that defines the transformation of coordinates.

Apply nested arrays that are created by the createMesh and createspace functions shown in Listing 9.20. The skin matrix consists of three nesting matrices that make up the array, defining the x-, y- and z-coordinates of the points on the surface or curved, obviously.

Listing 9.20. The result of the functions CreateMeeh and CreateSpace (Fig. 9.4 - 9.5)

Creation of matrices of a special kind

In Mathcad, it is easy to create matrices of a simple form for the help of one of the introduced functions. Apply a variation of these functions in Listing 9.21.

  • identity (N) – a single matrix to the NXN world;
  • diag(v) - diagonal matrix, on the diagonal there are elements of the vector v;
  • geninv(A) - folded matrix, pivotal (levoruch) matrix A;
  • rref (A) - transformation of the matrix or vector A into steps of the view;
    • N – whole number;
    • v is a vector;
    • A is a matrix of real numbers.

The layout of the NXM matrix for the geninv function is due to such that N>M.

Listing 9.21. Creation of matrices of a special kind

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