MATLAB numerical calculation

Statistical functions

SUM (A) sum
If a is a vector, sum (a) calculates each element's sum
If A is a matrix, sum (a) evaluates the sum of each column element (that is, each column is treated as a vector, and a row vector is returned, resulting in the sum of the elements of each column)
If A is a multidimensional array, sum (A,dim) sums the corresponding position of the specified dimension, that is, the column is summed when Dim=1, and the column is summed at dim=2 time, and so on.
b = SUM (..., ' double ') and B = sum (..., dim, ' double ') This is used to specify the calculation precision, forcing the return type to double
b = SUM (..., ' native ') and B = sum (..., dim, ' native ') This is used to specify the calculation precision, forcing the return type precision to be the same as the right
If the precision ' double ' or ' Natvie ' is not specified above, if a is an integer type, the double is returned by default, if a is a single or double type, the default returns single or double precision

Example 1+2+...+100

Clear
CLC
a=1:100;
SUM (A)

Results

Ans =

        5050

Cases

Clear
CLC
a=magic (3)
sum (a)
sum (sum (a))

Results

A =

     8     1     6
     3     5     7
     4     9     2


ans =

    all


ans =

    45

Cases

Clear
CLC
a=magic (4)
sum (a,1)
sum (a,2)

Results

A =     2     3
     5     8
     9     7 6 (4)    15 1 ans = over-the-up


ans

    =
    34

Example multidimensional vector

Clear
CLC
A (:,:, 1) =[1 0 2 5;4 1 8 7;3 2 6 3];
A (:,:, 2) =[3 5 4 1;2 6 2 1;4 2 3 0];
A
sum (a,1)
sum (a,2)
sum (a,3)

Results

A (:,:, 2) =

     3     5     4     1
     2     6     2     1
     4     2     3     0


ans (:,:, 1) =

     8     3


ans (:,:, 2) =

     9 9 2


ans (:,:, 1) =

     8
    20


ans (:,:, 2) =
     9


ans =

     4     5     6     6 6     7    10     8
     7     4)     9     3

Example accuracy

Clear
CLC
a=int8 (1:20)
b=sum (a)%a is an integer type, when precision is not specified, the default returns double precision, the result
is correct c=sum (a, ' native ')% because A is a 8-bit integer type, The precision is set to native, so the result precision is int8 (8-bit integer type), the integer range of int8 is -128~127, the range is truncated, so the result is 127
class (b)% view B data type
class (C)% View the data type of C

Results

A =

  Columns 1 through

    1    2    3    4    5    6    7    8    9   ten   12   Columns-   through   -


B =

   210


C =

  127


ans =

double


ans =

int8

Mean (A) averages, parameters are similar to sum, but mean cannot specify precision
If a is a vector, mean (a) calculates the average of all elements of a
If A is a matrix, mean (a) calculates the average of each column element (that is, each column is treated as a vector, and a row vector is returned, resulting in the average of each column element)

If A is a multidimensional array, SUM (A,dim) averages the corresponding position of the specified dimension, that is, the average value of the column at Dim=1, the average of the column at dim=2, and the second analogy

Cases

Clear
CLC
% averages a=1:100 for vectors
;
Mean (A)
% of the matrix averaging
b=magic (4)
mean (B)
% to the multidimensional vector averaging
C (:,:, 1) =[3 5 8 6;3 7 9 4;3 4);
C (:,:, 2) =[2 7 4 3;6 3 4;10 7 9 4];
C
mean (c,1)
mean (c,2)
mean (c,3)

Results

Ans =

    8.5000    8.5000    8.5000    8.5000


C (:,:, 1) =

     3     5     8     6
     3     7     9     4
     3     4


C (:,:, 2) =

     2 7 4     3
     6     3    10     4     7     9     4


ans (:,:, 1) =

    3.0000    5.3333    9.6667    8.0000


ans (:,:, 2) =

    6.0000    5.6667    7.6667    3.6667


ans (:,:, 1) =

    5.5000
    5.7500
    8.2500


ans (:,:, 2) =

    4.0000
    5.7500
    7.5000


ans =

    2.5000    6.0000    6.0000    4.5000
    4.5000    5.0000    9.5000    4.0000
    6.5000    5.5000   10.5000    9.0000

Median the median min to find the minimum max to find the maximum value
Prod the product sort sort
The parameters and methods used are basically consistent with the mean averaging.
Sort is sorted by default in ascending order, but it can also be specified in ascending or descending order by parameter
Sort (x,dim,mode), DIM does not specify the default is 1,mode when the value is ' ascend ', in ascending order (default), MODE value is ' descend ' when sorted in descending order

Cases

Clear
CLC
a=[11 6 9 1 1 8 2 9];
Sort (A, ' ascend ')
sort (A, ' descend ')

Results

Ans =

     1     1     2     6     8     9     9    each


ans =    11     9     9     8     6     2     1     1

Another approach to descending is to reverse the output

Clear
CLC
a=[11 6 9 1 1 8 2 9];
B=sort (A)
c=b (end:-1:1)

Results

B =

     1     1     2     6     8     9     9    each


C     =     9 9     8     6     2     1     1

Deviations and correlations
var (x) to find the variance of X
STD (x) to find the standard deviation of X
Range (x) to find the extreme difference of X
CoV (x) to find the covariance matrix of X
CoV (x, y) to find the covariance of X, y two matrices
CORRCOEF (x) to find the autocorrelation matrix of X
Corrcoef (x, y) to find the cross-correlation coefficients of × and y, the result is square

CORR2 (x, y) to find correlation coefficients

Cases

Clear
CLC
a=randn (5,5)% generates a 5x5 standard normal distribution (mean 0, Variance 1) of the random number matrix
b=var (a)% for the variance of a
B2=var (a (:))% of the variance of all numbers
c=std (a)% seeking a standard deviation
C2=STD (A (:))% for the standard deviation of all numbers
d=range (a)% for the difference of a
e=cov (a)% for a covariance matrix
f=corrcoef (a)% for A's autocorrelation array

Results

A = 1.6035-0.1559-1.2507 0.0125 0.9337 1.2347 0.2761-0.9480-3.0292 0.3503-0.2296-0 .2612-0.7411-0.4570-0.0290-1.5062 0.4434-0.5078 1.2424 0.1825-0.4446 0.3919-0.3206-1    .0667-1.5651 B = 1.6320 0.1056 0.1332 2.4728 0.8687 B2 = 1.0286 C = 1.2775 0.3250 0.3650 1.5725 0.9320 C2 = 1.0142 D = 3.1096 0.7046 0.9301 4.2716 2.4988 E = 1.632 0-0.1957-0.4006-1.1446 0.5936-0.1957 0.1056 0.0748-0.0320-0.1390-0.4006 0.0748 0.133 2 0.1051-0.2914-1.1446-0.0320 0.1051 2.4728 0.1939 0.5936-0.1390-0.2914 0.1939 0.868    7 F = 1.0000-0.4713-0.8592-0.5698 0.4985-0.4713 1.0000 0.6307-0.0626-0.4589-0.8592    0.6307 1.0000 0.1830-0.8564-0.5698-0.0626 0.1830 1.0000 0.1323 0.4985-0.4589-0.8564
 0.1323 1.0000

Polynomial calculation

Defining the polynomial
P=[A1 A2 ... an an+1]
Y = P (1) *x^n + P (2) *x^ (N-1) + ... + p (N) *x + P (n+1)
That

Y = a1*x^n + a2*x^ (N-1) + ... + an*x + an+1

Poly to find the characteristic polynomial
Poly (a) when a is a n*n matrix, the poly (a) command evaluates the characteristic polynomial of a, Det (Lambda*eye (Size (a))-a)
Poly (v) when V is a vector, command poly (v) generates a polynomial with V as its root

Root to find the roots of the polynomial
Root (P)

Cases

Clear
CLC
a=[1 2 3;4 5 6;7 8 0];
P=poly (A)% to find the characteristic polynomial |λe-a|
R=roots (p)%, based on the above characteristic polynomial, to find eigenvalues

Results

p =

    1.0000   -6.0000  -72.0000  -27.0000


r =

   12.1229
   -5.7345
   -0.3884

Hand calculation process

Cases

Clear
CLC
a=[1 2 3];
P=poly (A)

Results

p =

     1    -6    -6

Hand calculation process

Help document the algorithm for finding the characteristic polynomial |λe-a|

n = Length (a)
z = Eig (a);% evaluates all eigenvalues of matrix A, forming vector z
c = zeros (n+1,1); C (1) = 1;
For j = 1:n
    C (2:j+1) = C (2:j+1)-Z (j) *c (1:J);
End

Polyval Polynomial evaluation

Polyval (p,x) P is a polynomial, X can be a vector or a matrix, when a vector, the result is a vector, the MXN matrix, each element is evaluated, the result is still the MXN matrix
Example p (x) = 3x^2+2x+1 at x = 5,7, and 9:

Clear
CLC 
p=[3 2 1];
Polyval (p,[5 7 9])

Results

Ans =   162   262

Example p (x) = 4x^2+4x+1 at x = [2 3 4;5 6 7]:

Clear
CLC
p=[4 4 1];
Polyval (p,[2 3 4;5 6 7])

Results

Ans =
   225 bayi 121   169   

Conv polynomial multiplication
Conv (p1,p2) polynomial-P1 and polynomial P2 multiplication
Deconv Polynomial Division
[Q,r]=deconv (P1,P2) polynomial P1 and polynomial P2 divide, Q is quotient, R is the remainder
The derivation of Polyder polynomial
Polyder (p)-derivative of the polynomial P
Polyder (P1,P2) polynomial P1 and polynomial P2 multiply, equivalent to Polyder (CONV (P1,P2))
[Q,d]=polyder (P1,P2) polynomial fraction p1/p2 (i.e., P1 is a molecule, P2 is the denominator), the result is still fractional q/d (that is, Q is a molecule, D is the denominator)

Example U=X^3+2X^2+3X+4,V=10X^2+20X+30, seeking c=u*v,c/u

Clear
CLC
u=[1 2 3 4];
V=[10];
C=conv (u,v)
[Q,r]=deconv (C,u)

Results

c =

    Ten    +


q =

    ten +


r =

     0     0     0     0     0     0

Example a=3x^2+6x+9,b=x^2+2x, finding the derivative of a*b

Clear
CLC
a=[3 6 9];
B=[1 2 0];
K=polyder (A, B)

Results

K =    18

function Extremum and 0 points

Fminbnd Seeking Extremum
X = Fminbnd (fun,x1,x2) to function value fun for minimum value, x1,x2 for interval, x1 < X < X2
If a maximum value is required, the derivative can be taken first

Cases

Clear
CLC
X1 = fminbnd (@cos, 3,4)
X2 = Fminbnd (@ (x) sin (x) +3,2,5)
f = @ (X,c) (X-C). ^2;
c = 1.5;
X3 = Fminbnd (@ (x) f (x,c), 0, 1)

Results

X1 =

    3.1416


X2 =

    4.7124


X3 =

    0.9999

Fzero 0 Points
X = Fzero (fun,x0) 0 points for function fun, X0 is the initial value, fun function can use @

Cases

Clear
CLC
X1 = Fzero (@sin, 3) The                % function is sin x, the initial value is 3
X2 = Fzero (@ (x) sin (3*x), 2) The       function is sin 3x, the initial value is 2
f = @ (x , c) cos (c.*x);             % defines a function with parameters
c = 2;                            The value of the% setting parameter
X3 = Fzero (@ (x) f (x,c), 0.1)
X4 = Fzero (' x.^3-2.^x+1 ', -1) the     % function is x^3-2^x+1, the initial value is-1

Results

X1 =

    3.1416


X2 =

    2.0944


X3 =

    0.7854


X4 =

   -0.7368