The Learning-matrix and matrix fast power of the provincial selection algorithm

0x00 Introduction

Matrix, as the name implies, is a rectangular array composed of numbers

Like this: $\begin{array}{l}\begin{bmatrix}2&3&4\0&7&13\c&\alpha&\sqrt5\end{bmatrix}\\end{. array}$

is a 3*3 matrix.

The use of matrices in informatics and even mathematics is very extensive, the following is to introduce some of its basic knowledge, as well as the commonly used places. In this paper, we also introduce the fast matrix multiplication.

0x01 What is the definition of matrix matrix

In fact, that's the way it is.

Defines an n-row m-column matrix as a rectangular array consisting of the number of n*m, where "number" can be imaginary, real, 01 ...

The definition of a matrix actually originates from linear mappings in linear algebra, but the definition is too complex to define the matrix in general (including high school mathematics) in the form of columns.

Basic operations of matrices

The basic operations of matrices include addition and multiplication

Matrix addition

Matrix addition defined between matrices with the same number of two rows

Two n*m matrices are added to get a n*m matrix in which the elements at each position of the resulting new matrix are equal to the sum of the elements of the original two matrices at the corresponding positions

Cases:

$\begin{bmatrix}2&4\5&3\8&1\end{bmatrix}+\begin{bmatrix}5&3\6&7\9&2\end{bmatrix}=\ begin{bmatrix}7&7\11&10\17&3\end{bmatrix}$

Matrix addition satisfies commutative law and binding law

Matrix subtraction is similar, do not repeat

Multiplication of matrices

The matrix number multiplication is defined between a matrix and a number

A number and a n*m matrix multiply, get a n*m matrix, each element of the new matrix equals the corresponding element of the old matrix multiplied by that number

Cases:

$2\ast\begin{bmatrix}2&4\5&3\8&1\end{bmatrix}=\begin{bmatrix}4&8\10&6\16&2\end{bmatrix }$

Matrix multiplication satisfies the commutative law (exchange between numbers), the binding law (in the case of only one matrix), and the allocation rate (a multiplication of two matrices or a matrix by two numbers can be)

Matrix multiplication

Matrix multiplication is a big topic that deserves to be taken out alone.

The first definition of matrix multiplication is the "product" of two mappings between two linear spaces in linear algebra, since one such linear mapping can be represented by a matrix, so two mappings (that is, a mapping to b,b maps to c) can be multiplied to represent

Of course, this blog will not drag out the linear algebra, this article will only give the matrix multiplication method-because it is very useful in Oi

Two matrices A and B are capable of matrix multiplication (a*b), when and only if the number of columns of a is the same as the number of rows in B (so matrix multiplication does not satisfy the commutative law)

Set A is the matrix of n*m, B is the matrix of m*k, at the same time set A*b=c

So:

The C matrix is n*k, and the value of each element of the C matrix is computed as follows:

$c \left[i\right]\left[j\right]={\textstyle\sum_{p=1}^m}a\left[i\right]\left[p\right]\ast B\left[p\right]\left[j \right]$

That is, the first (I,J) element of the new matrix is equal to the sum of each of the elements in column J of A and B, which is why the matrix multiplication requires the same number of rows

Cases:

$\begin{bmatrix}10&1\end{bmatrix}\ast\begin{bmatrix}a&0\c&1\end{bmatrix}=\begin{bmatrix}10a+c& 1\end{bmatrix}$

Matrix multiplication does not satisfy the commutative law, but satisfies the binding law (continuous matrix multiplication only for a*b*c) and the distribution rate (as long as it is a valid operation)

The main application of matrix multiplication in Oi

In Oi, matrix multiplication is mainly used to solve linear recurrence problems.

Linear recursion is such a class of recursion:

$h _n=\sum_{i=1}^{n-1}a_i h_i +b$

Where $h$ is a recursive function, $b $ is a constant term, $a _i$ is the coefficient (can be 0), and the number of all $h_i$ in the right-hand is a

Then we can use matrix multiplication to simulate the transfer of

such as the Fibonacci sequence $f_{i+2}=f_i+f_{i+1}$

Its recursion can be obtained by multiplying the two matrices:

State Matrix $\begin{bmatrix}f_i&f_{i+1}\end{bmatrix}$

Transfer matrix $\begin{bmatrix}0&1\1&1\end{bmatrix}$

Obviously the matrix product of the above two matrices is $\begin{bmatrix}f_{i+1}&f_{i+2}\end{bmatrix}$

However, there is a problem here: Matrix multiplication is $o\left (nmk\right) $, obviously less efficient than the original time, then why do you do that?

The answer is the matrix fast power

The base operation template of the matrix is here (there is no multiplication and addition part, because the OI is not commonly used and is not often tested, and presumably is very simple)

structma{ll a[ -][ -],n,m; Ma () {memset (A,0,sizeof(a)); n=m=0;}voidClear () {memset (A,0,sizeof(a)); n=m=0;}ConstMaoperator*(ConstMa &b) {ma re;re.n=n;re.m=b.m;ll i,j,k; for(i=1; i<=n;i++) { for(j=1; j<=b.m;j++) { for(k=1; k<=m;k++) {re.a[i][j]+=a[i][k]*b.a[k][j];                Re.a[i][j]%=mod; }            }        }returnRe }Const void operator=(ConstMa &b) {n=b.n;m=b.m;ll i,j; for(i=1; i<=n;i++) for(j=1; j<=m;j++) A[i][j]=b.a[i][j]; }};

0x02 Matrix Fast Power

Fast Power algorithm Everyone must be familiar with: in $o\left (Log_2 n\right) $ in the time complexity of finding a number of n-th square

So, the fast power of the matrix is the same as it

Or consider the Fibonacci sequence recursion, we find that if we set the initial matrix $\begin{bmatrix}f_1&f_2\end{bmatrix}$ to $a$, and the transfer matrix is $b$, then the $f_n$ we ask for is this:

Set the Matrix $c=a\ast b^{n-1}$, then obviously the second column of the first row of the C matrix is the value of the $f_n$

And because matrix multiplication satisfies the binding law, the idea of a fast power can still be used here, so we can use the fast power of thinking, in a very short time to find out the $b^{n-1}$

What advantages does this algorithm have?

Of course there are, for example, I asked you to ask for the $10^{15}$ of the Fibonacci sequence.

At this point the matrix can be quickly $log 10^{15} \ast$ Matrix multiplication complexity of the time to obtain the solution, and the matrix multiplication of this problem because it is 1*2 matrix multiply 2*2, so the complexity is very low

The template for the quick power of matrices is as follows:

ma ppow(ma x,ma y,ll t){    while(t){        if(t&1) x=x*y;        y=y*y;t>>=1;    }    return x;}

Does it look much like a real fast power? Actually, they're both the same.

When can I apply a matrix to a fast power?

In the first part we said that matrix multiplication can be used to solve linear recurrence problems

Then, when we have linear recursion problem, we can easily use matrix fast power.

The Fibonacci sequence above is an example, and there are several examples:

luoguP2044

luoguP1707

luoguP1357

As you can see, the fast power of the matrix can solve the recursion of single recursion, multiple groups of recursion, multiple groups of recursion plus nonlinear constants, even can be done together with DP, as DP optimization

In general, we construct a single row of K-column state matrices and a transfer matrix of K-row K-Columns

Since each lattice of the new matrix is independent of each other in this case, it is more convenient for us to design the transfer matrix

In the fast power of matrix, we just have to determine the state matrix and transfer matrix, then the problem is generally solved.

0x03 Fast Matrix multiplication

Dig Pit + + ... In a week.

The Learning-matrix and matrix fast power of the provincial selection algorithm