3+4

ans =

     7

diary
while
 while
      |
{Error: Expression or statement is incomplete or incorrect.
} 
for i = [1 2 5 pi -3], i^2, end

ans =

     1


ans =

     4


ans =

    25


ans =

    9.8696


ans =

     9

if cond_expression 
< <=, ~= 
 < <=, ~= 
 |
{Error: Unexpected MATLAB operator.
} 
if, if( cond ) <then> else ...
 if, if( cond ) <then> else ...
   |
{Error: Expression or statement is incomplete or incorrect.
} 
if elseif end
 if elseif end
    |
{Error: Illegal use of reserved keyword "elseif".
} 
doc switch 
n=10;  xs = 9*rand(1,n)+1;
ys = .5*xs.^2 + 2*xs + randn(n);
{Error using <a href="matlab:matlab.internal.language.introspective.errorDocCallback('plus')" style="font-weight:bold"> + </a>
Matrix dimensions must agree.
} 
ys = .5*xs.^2 + 2*xs + randn(1,n);
plot(xs,ys,*)
 plot(xs,ys,*)
            |
{Error: Unexpected MATLAB operator.
} 
plot(xs,ys,'*')
n=50;  xs = 9*rand(1,n)+1;
ys = .5*xs.^2 + 2*xs + randn(1,n);
plot(xs,ys,'*')
M = [ xs', ones(n,1) ];
whos('M')
  Name       Size            Bytes  Class     Attributes

  M         50x2               800  double              

coeff = M \ obs
{Undefined function or variable 'obs'.
} 
coeff = M \ ys
{Error using <a href="matlab:matlab.internal.language.introspective.errorDocCallback('mldivide')" style="font-weight:bold"> \ </a>
Matrix dimensions must agree.
} 
coeff = M \ ys'

coeff =

    7.4815
  -11.2324

mod = xs * coeff(1) + coeff(2);
hold on
plot( xs, mod )
size(M) 

ans =

    50     2

M = [ xs'.^2 M ]

M =

   93.2592    9.6571    1.0000
    1.0852    1.0417    1.0000
   63.5878    7.9742    1.0000
   69.8182    8.3557    1.0000
   77.7616    8.8183    1.0000
    3.0973    1.7599    1.0000
   21.1420    4.5980    1.0000
   11.1478    3.3388    1.0000
   67.2501    8.2006    1.0000
   23.8410    4.8827    1.0000
   84.5633    9.1958    1.0000
    6.9518    2.6366    1.0000
   11.3854    3.3742    1.0000
    5.3354    2.3099    1.0000
    4.9489    2.2246    1.0000
   77.8564    8.8236    1.0000
   38.6553    6.2173    1.0000
   35.3875    5.9487    1.0000
    5.3111    2.3046    1.0000
   75.2952    8.6773    1.0000
   43.5402    6.5985    1.0000
   17.2937    4.1586    1.0000
   31.5759    5.6192    1.0000
   21.3100    4.6163    1.0000
    2.8348    1.6837    1.0000
    9.9808    3.1592    1.0000
    4.4516    2.1099    1.0000
    7.0499    2.6552    1.0000
    9.9829    3.1596    1.0000
   22.6139    4.7554    1.0000
    2.0935    1.4469    1.0000
   83.2555    9.1244    1.0000
   90.3086    9.5031    1.0000
   29.3523    5.4178    1.0000
   29.1954    5.4033    1.0000
   16.3174    4.0395    1.0000
   82.8188    9.1005    1.0000
   18.6902    4.3232    1.0000
    4.0033    2.0008    1.0000
   64.3568    8.0223    1.0000
   20.3189    4.5076    1.0000
   10.0820    3.1752    1.0000
   21.4852    4.6352    1.0000
    3.4898    1.8681    1.0000
    4.7863    2.1878    1.0000
   89.8411    9.4785    1.0000
   92.2601    9.6052    1.0000
   38.1538    6.1769    1.0000
    2.3655    1.5380    1.0000
    9.6909    3.1130    1.0000

M(:,1) = [] 

M =

    9.6571    1.0000
    1.0417    1.0000
    7.9742    1.0000
    8.3557    1.0000
    8.8183    1.0000
    1.7599    1.0000
    4.5980    1.0000
    3.3388    1.0000
    8.2006    1.0000
    4.8827    1.0000
    9.1958    1.0000
    2.6366    1.0000
    3.3742    1.0000
    2.3099    1.0000
    2.2246    1.0000
    8.8236    1.0000
    6.2173    1.0000
    5.9487    1.0000
    2.3046    1.0000
    8.6773    1.0000
    6.5985    1.0000
    4.1586    1.0000
    5.6192    1.0000
    4.6163    1.0000
    1.6837    1.0000
    3.1592    1.0000
    2.1099    1.0000
    2.6552    1.0000
    3.1596    1.0000
    4.7554    1.0000
    1.4469    1.0000
    9.1244    1.0000
    9.5031    1.0000
    5.4178    1.0000
    5.4033    1.0000
    4.0395    1.0000
    9.1005    1.0000
    4.3232    1.0000
    2.0008    1.0000
    8.0223    1.0000
    4.5076    1.0000
    3.1752    1.0000
    4.6352    1.0000
    1.8681    1.0000
    2.1878    1.0000
    9.4785    1.0000
    9.6052    1.0000
    6.1769    1.0000
    1.5380    1.0000
    3.1130    1.0000

M = [ xs'.*M , ones(n,1) ]
{Error using <a href="matlab:matlab.internal.language.introspective.errorDocCallback('times')" style="font-weight:bold"> .* </a>
Matrix dimensions must agree.
} 
xs' .* M
{Error using <a href="matlab:matlab.internal.language.introspective.errorDocCallback('times')" style="font-weight:bold"> .* </a>
Matrix dimensions must agree.
} 
xs' .* M(:,1)

ans =

   93.2592
    1.0852
   63.5878
   69.8182
   77.7616
    3.0973
   21.1420
   11.1478
   67.2501
   23.8410
   84.5633
    6.9518
   11.3854
    5.3354
    4.9489
   77.8564
   38.6553
   35.3875
    5.3111
   75.2952
   43.5402
   17.2937
   31.5759
   21.3100
    2.8348
    9.9808
    4.4516
    7.0499
    9.9829
   22.6139
    2.0935
   83.2555
   90.3086
   29.3523
   29.1954
   16.3174
   82.8188
   18.6902
    4.0033
   64.3568
   20.3189
   10.0820
   21.4852
    3.4898
    4.7863
   89.8411
   92.2601
   38.1538
    2.3655
    9.6909

M = [ xs'.^2 M ]

M =

   93.2592    9.6571    1.0000
    1.0852    1.0417    1.0000
   63.5878    7.9742    1.0000
   69.8182    8.3557    1.0000
   77.7616    8.8183    1.0000
    3.0973    1.7599    1.0000
   21.1420    4.5980    1.0000
   11.1478    3.3388    1.0000
   67.2501    8.2006    1.0000
   23.8410    4.8827    1.0000
   84.5633    9.1958    1.0000
    6.9518    2.6366    1.0000
   11.3854    3.3742    1.0000
    5.3354    2.3099    1.0000
    4.9489    2.2246    1.0000
   77.8564    8.8236    1.0000
   38.6553    6.2173    1.0000
   35.3875    5.9487    1.0000
    5.3111    2.3046    1.0000
   75.2952    8.6773    1.0000
   43.5402    6.5985    1.0000
   17.2937    4.1586    1.0000
   31.5759    5.6192    1.0000
   21.3100    4.6163    1.0000
    2.8348    1.6837    1.0000
    9.9808    3.1592    1.0000
    4.4516    2.1099    1.0000
    7.0499    2.6552    1.0000
    9.9829    3.1596    1.0000
   22.6139    4.7554    1.0000
    2.0935    1.4469    1.0000
   83.2555    9.1244    1.0000
   90.3086    9.5031    1.0000
   29.3523    5.4178    1.0000
   29.1954    5.4033    1.0000
   16.3174    4.0395    1.0000
   82.8188    9.1005    1.0000
   18.6902    4.3232    1.0000
    4.0033    2.0008    1.0000
   64.3568    8.0223    1.0000
   20.3189    4.5076    1.0000
   10.0820    3.1752    1.0000
   21.4852    4.6352    1.0000
    3.4898    1.8681    1.0000
    4.7863    2.1878    1.0000
   89.8411    9.4785    1.0000
   92.2601    9.6052    1.0000
   38.1538    6.1769    1.0000
    2.3655    1.5380    1.0000
    9.6909    3.1130    1.0000

coeff = M \ ys'

coeff =

    0.4958
    1.9755
    0.2126

mod = xs.^2 * coeff(1) + xs * coeff(2) + coeff(3);
plot( xs, mod )
close all
plot( xs, ys, '-', xs, mod, '*' )
sort
{Error using <a href="matlab:matlab.internal.language.introspective.errorDocCallback('sort')" style="font-weight:bold">sort</a>
Not enough input arguments.
} 
plot( xs, mod, '*' )
plot( xs, mod, '*', xs, ys, '+' )
xs

xs =

  Columns 1 through 7

    9.6571    1.0417    7.9742    8.3557    8.8183    1.7599    4.5980

  Columns 8 through 14

    3.3388    8.2006    4.8827    9.1958    2.6366    3.3742    2.3099

  Columns 15 through 21

    2.2246    8.8236    6.2173    5.9487    2.3046    8.6773    6.5985

  Columns 22 through 28

    4.1586    5.6192    4.6163    1.6837    3.1592    2.1099    2.6552

  Columns 29 through 35

    3.1596    4.7554    1.4469    9.1244    9.5031    5.4178    5.4033

  Columns 36 through 42

    4.0395    9.1005    4.3232    2.0008    8.0223    4.5076    3.1752

  Columns 43 through 49

    4.6352    1.8681    2.1878    9.4785    9.6052    6.1769    1.5380

  Column 50

    3.1130

close
plot( xs, mod, '*-' )
help norm
 <strong>norm</strong>   Matrix or vector norm.
      <strong>norm</strong>(X,2) returns the 2-norm of X.
 
      <strong>norm</strong>(X) is the same as <strong>norm</strong>(X,2).
 
      <strong>norm</strong>(X,1) returns the 1-norm of X.
 
      <strong>norm</strong>(X,Inf) returns the infinity norm of X.
 
      <strong>norm</strong>(X,'fro') returns the Frobenius norm of X.
 
    In addition, for vectors...
 
      <strong>norm</strong>(V,P) returns the p-norm of V defined as SUM(ABS(V).^P)^(1/P).
 
      <strong>norm</strong>(V,Inf) returns the largest element of ABS(V).
 
      <strong>norm</strong>(V,-Inf) returns the smallest element of ABS(V).
 
    By convention, NaN is returned if X or V contains NaNs.
 
    See also <a href="matlab:help cond">cond</a>, <a href="matlab:help rcond">rcond</a>, <a href="matlab:help condest">condest</a>, <a href="matlab:help normest">normest</a>, <a href="matlab:help hypot">hypot</a>.

    Reference page in Help browser
       <a href="matlab:doc norm">doc norm</a>

close all
pts = linspace(0, 2*pi, 100);
signal = sin(pts) + sin(1.5*pts) + sin(2*pts);
1

ans =

     1

subplot(1,3,1); plot( pts, signal )
subplot(1,2,1); plot( pts, signal )
axes([0,2*pi,-1.5,3])
{Error using <a href="matlab:matlab.internal.language.introspective.errorDocCallback('axes')" style="font-weight:bold">axes</a>
Handles of type Root cannot be made the current Axes.
} 
axis([0,2*pi,-1.5,3])
signal = sin(pts) + sin(1.5*pts) + sin(2*pts) + sin(45*pts)*.05;
subplot(1,2,2); plot( pts, signal )
axis([0,2*pi,-1.5,3])
freq = fft(signal);
freq2 = fftshift(freq);
figure
subplot(1,2,1);
plot(abs(freq2))
size(freq2)

ans =

     1   100

width=10;
freq2=
 freq2=
       |
{Error: Expression or statement is incomplete or incorrect.
} 
freq2(1:width)=0;
freq2(end-width:end)=0;
subplot(1,2,2);
plot(abs(freq2))
ifft(ifftshift(freq2))

ans =

  Columns 1 through 3

   0.0290 + 0.0002i   0.2674 + 0.0002i   0.5759 - 0.0004i

  Columns 4 through 6

   0.8399 + 0.0005i   1.1082 - 0.0004i   1.3655 + 0.0001i

  Columns 7 through 9

   1.6001 + 0.0003i   1.8256 - 0.0005i   2.0256 + 0.0005i

  Columns 10 through 12

   2.2055 - 0.0003i   2.3633 - 0.0000i   2.4927 + 0.0003i

  Columns 13 through 15

   2.5991 - 0.0005i   2.6765 + 0.0004i   2.7264 - 0.0002i

  Columns 16 through 18

   2.7499 - 0.0001i   2.7442 + 0.0004i   2.7130 - 0.0004i

  Columns 19 through 21

   2.6554 + 0.0003i   2.5724 - 0.0001i   2.4673 - 0.0002i

  Columns 22 through 24

   2.3395 + 0.0003i   2.1928 - 0.0003i   2.0290 + 0.0002i

  Columns 25 through 27

   1.8494 + 0.0000i   1.6584 - 0.0002i   1.4570 + 0.0003i

  Columns 28 through 30

   1.2485 - 0.0002i   1.0364 + 0.0001i   0.8216 + 0.0001i

  Columns 31 through 33

   0.6086 - 0.0002i   0.3991 + 0.0002i   0.1950 - 0.0001i

  Columns 34 through 36

   0.0003 - 0.0000i  -0.1850 + 0.0002i  -0.3574 - 0.0002i

  Columns 37 through 39

  -0.5156 + 0.0001i  -0.6591 - 0.0000i  -0.7851 - 0.0001i

  Columns 40 through 42

  -0.8943 + 0.0002i  -0.9854 - 0.0001i  -1.0576 + 0.0000i

  Columns 43 through 45

  -1.1124 + 0.0001i  -1.1481 - 0.0002i  -1.1667 + 0.0001i

  Columns 46 through 48

  -1.1688 - 0.0001i  -1.1542 - 0.0001i  -1.1261 + 0.0002i

  Columns 49 through 51

  -1.0840 - 0.0002i  -1.0304 + 0.0001i  -0.9674 + 0.0001i

  Columns 52 through 54

  -0.8954 - 0.0002i  -0.8179 + 0.0002i  -0.7355 - 0.0002i

  Columns 55 through 57

  -0.6506 + 0.0000i  -0.5655 + 0.0002i  -0.4807 - 0.0003i

  Columns 58 through 60

  -0.3994 + 0.0002i  -0.3224 - 0.0001i  -0.2509 - 0.0001i

  Columns 61 through 63

  -0.1871 + 0.0003i  -0.1311 - 0.0003i  -0.0844 + 0.0002i

  Columns 64 through 66

  -0.0478 - 0.0000i  -0.0210 - 0.0002i  -0.0053 + 0.0004i

  Columns 67 through 69

  -0.0000 - 0.0004i  -0.0051 + 0.0002i  -0.0205 + 0.0001i

  Columns 70 through 72

  -0.0451 - 0.0003i  -0.0784 + 0.0004i  -0.1198 - 0.0004i

  Columns 73 through 75

  -0.1673 + 0.0001i  -0.2208 + 0.0002i  -0.2782 - 0.0004i

  Columns 76 through 78

  -0.3379 + 0.0005i  -0.3999 - 0.0003i  -0.4606 + 0.0000i

  Columns 79 through 81

  -0.5203 + 0.0003i  -0.5773 - 0.0005i  -0.6286 + 0.0005i

  Columns 82 through 84

  -0.6760 - 0.0003i  -0.7150 - 0.0001i  -0.7465 + 0.0004i

  Columns 85 through 87

  -0.7705 - 0.0005i  -0.7823 + 0.0004i  -0.7870 - 0.0002i

  Columns 88 through 90

  -0.7794 - 0.0002i  -0.7607 + 0.0005i  -0.7347 - 0.0005i

  Columns 91 through 93

  -0.6931 + 0.0004i  -0.6465 - 0.0000i  -0.5886 - 0.0003i

  Columns 94 through 96

  -0.5186 + 0.0005i  -0.4503 - 0.0005i  -0.3625 + 0.0003i

  Columns 97 through 99

  -0.2794 + 0.0001i  -0.1944 - 0.0004i  -0.0794 + 0.0005i

  Column 100

  -0.0282 - 0.0005i

figure
plot(real(ifft(ifftshift(freq2))))
signal = sin(pts) + sin(1.5*pts) + sin(2*pts) + sin(45*pts)*.05;
close all
n = 4; M = rand(M); M = M*M'+eye(n); eig(M) 
{Error using <a href="matlab:matlab.internal.language.introspective.errorDocCallback('rand')" style="font-weight:bold">rand</a>
Size vector should be a row vector with real elements.
} 
n = 4; M = rand(n); M = M*M'+eye(n); eig(M)

ans =

    1.0024
    1.0868
    1.2897
    5.3532

n = 40; M = rand(n); M = M*M'+eye(n); eig(M)

ans =

    1.0000
    1.0204
    1.0327
    1.0385
    1.1022
    1.1835
    1.2344
    1.2772
    1.4596
    1.5509
    1.6021
    1.7707
    1.8149
    1.9555
    2.2100
    2.3938
    2.4478
    2.5865
    2.7003
    2.8417
    3.2626
    3.5558
    3.8654
    3.9918
    4.1198
    4.6793
    4.8116
    5.2023
    5.9060
    6.1229
    6.3474
    6.8980
    7.2899
    7.5546
    8.6194
    9.4572
   10.0656
   10.6385
   12.6638
  407.1671

n = 40; M = rand(n); M = M*M'+eye(n); L = chol(M,'lower'); z=L\b; x=L'\z; norm(M*x-b)
{Undefined function or variable 'b'.
} 
n=40; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); z=L\b; x=L'\z; norm(M*x-b)

ans =

   1.7064e-14

n=40; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); z=L\b; x=L'\z; norm(M*x-b)

ans =

   5.2513e-15

n=40; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); z=L\b; x=L'\z; norm(M*x-b)

ans =

   1.8113e-14

n=40; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); z=L\b; x=L'\z; norm(M*x-b)

ans =

   8.0734e-15

n=400; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); z=L\b; x=L'\z; norm(M*x-b)

ans =

   1.1669e-12

n=400; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); z=L\b; x=L'\z; norm(M*x-b)

ans =

   1.6355e-12

n=400; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); z=L\b; x=L'\z; norm(M*x-b)

ans =

   2.3704e-12

n=400; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); z=L\b; x=L'\z; norm(M*x-b)

ans =

   2.1587e-12

n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); tic; z=L\b; x=L'\z; toc
Elapsed time is 0.000450 seconds.
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); tic; z=L\b; x=L'\z; toc
Elapsed time is 0.000313 seconds.
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); tic; z=L\b; x=L'\z; toc
Elapsed time is 0.000266 seconds.
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); L=chol(M,'lower'); tic; z=L\b; x=L'\z; toc
Elapsed time is 0.000296 seconds.
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.000714 seconds.
clc
clc
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.000840 seconds.
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.000701 seconds.
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.000930 seconds.
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.000717 seconds.
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.000813 seconds.
n=100; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.000848 seconds.
n=500; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.001618 seconds.
n=500; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.001796 seconds.
n=500; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.001810 seconds.
n=500; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.001913 seconds.
n=1000; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.006222 seconds.
n=5000; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.372981 seconds.
n=5000; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.392133 seconds.
n=5000; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.383055 seconds.
n=5000; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.388970 seconds.
n=5; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; x=inv(M)*b; norm(M*x-b)

ans =

   4.0030e-16

n=50; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; x=inv(M)*b; norm(M*x-b)

ans =

   2.9350e-13

n=50; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; x=inv(M)*b; norm(M*x-b)

ans =

   2.0913e-13

n=500; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; x=inv(M)*b; norm(M*x-b)

ans =

   1.9015e-09

n=500; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; x=inv(M)*b; norm(M*x-b)

ans =

   1.0975e-09

n=500; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; x=inv(M)*b; norm(M*x-b)

ans =

   2.5665e-10

n=500; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; x=inv(M)*b; toc
Elapsed time is 0.005725 seconds.
n=5000; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; L=chol(M,'lower'); z=L\b; x=L'\z; toc
Elapsed time is 0.384933 seconds.
n=5000; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; x=inv(M)*b; toc
Elapsed time is 2.183554 seconds.
n=5000; M=rand(n); M=M*M'+eye(n); b=rand(n,1); tic; x=M\b; toc
Elapsed time is 0.452595 seconds.
help backslash

backslash not found.

Use the Help browser search field to <a href="matlab:docsearch backslash">search the documentation</a>, or
type "<a href="matlab:help help">help help</a>" for help command options, such as help for methods.

help \
 \   Backslash or left matrix divide.
    A\B is the matrix division of A into B, which is roughly the
    same as INV(A)*B , except it is computed in a different way.
    If A is an N-by-N matrix and B is a column vector with N
    components, or a matrix with several such columns, then
    X = A\B is the solution to the equation A*X = B. A warning 
    message is printed if A is badly scaled or nearly singular.
    A\EYE(SIZE(A)) produces the inverse of A.
 
    If A is an M-by-N matrix with M < or > N and B is a column
    vector with M components, or a matrix with several such columns,
    then X = A\B is the solution in the least squares sense to the
    under- or overdetermined system of equations A*X = B. The
    effective rank, K, of A is determined from the QR decomposition
    with pivoting. A solution X is computed which has at most K
    nonzero components per column. If K < N this will usually not
    be the same solution as PINV(A)*B.  A\EYE(SIZE(A)) produces a
    generalized inverse of A.
 
    C = <strong>mldivide</strong>(A,B) is called for the syntax 'A \ B' when A or B is an
    object.
 
    See also <a href="matlab:help ldivide">ldivide</a>, <a href="matlab:help rdivide">rdivide</a>, <a href="matlab:help mrdivide">mrdivide</a>.

    <a href="matlab:matlab.internal.language.introspective.displayOverloads('mldivide')">Other functions named mldivide</a>

    Reference page in Help browser
       <a href="matlab:doc mldivide">doc mldivide</a>

help ldivide
 .\  Left array divide.
    A.\B denotes element-by-element division.  A and B
    must have the same dimensions unless one is a scalar.
    A scalar can be divided with anything.
 
    C = <strong>ldivide</strong>(A,B) is called for the syntax 'A .\ B' when A or B is an
    object.
 
    See also <a href="matlab:help rdivide">rdivide</a>, <a href="matlab:help mrdivide">mrdivide</a>, <a href="matlab:help mldivide">mldivide</a>.

    <a href="matlab:matlab.internal.language.introspective.displayOverloads('ldivide')">Other functions named ldivide</a>

    Reference page in Help browser
       <a href="matlab:doc ldivide">doc ldivide</a>

help mldivide
 \   Backslash or left matrix divide.
    A\B is the matrix division of A into B, which is roughly the
    same as INV(A)*B , except it is computed in a different way.
    If A is an N-by-N matrix and B is a column vector with N
    components, or a matrix with several such columns, then
    X = A\B is the solution to the equation A*X = B. A warning 
    message is printed if A is badly scaled or nearly singular.
    A\EYE(SIZE(A)) produces the inverse of A.
 
    If A is an M-by-N matrix with M < or > N and B is a column
    vector with M components, or a matrix with several such columns,
    then X = A\B is the solution in the least squares sense to the
    under- or overdetermined system of equations A*X = B. The
    effective rank, K, of A is determined from the QR decomposition
    with pivoting. A solution X is computed which has at most K
    nonzero components per column. If K < N this will usually not
    be the same solution as PINV(A)*B.  A\EYE(SIZE(A)) produces a
    generalized inverse of A.
 
    C = <strong>mldivide</strong>(A,B) is called for the syntax 'A \ B' when A or B is an
    object.
 
    See also <a href="matlab:help ldivide">ldivide</a>, <a href="matlab:help rdivide">rdivide</a>, <a href="matlab:help mrdivide">mrdivide</a>.

    <a href="matlab:matlab.internal.language.introspective.displayOverloads('mldivide')">Other functions named mldivide</a>

    Reference page in Help browser
       <a href="matlab:doc mldivide">doc mldivide</a>

doc mldivide
lu,qr,chol
{Error using <a href="matlab:matlab.internal.language.introspective.errorDocCallback('lu')" style="font-weight:bold">lu</a>
Not enough input arguments.
} 
eig,svd
{Error using <a href="matlab:matlab.internal.language.introspective.errorDocCallback('eig')" style="font-weight:bold">eig</a>
Not enough input arguments.
} 
M = [2 4; -6 8]

M =

     2     4
    -6     8

v([ 4 3 1 2 ]) = [ M(1,:), M(2,:) ] 

v =

    -6     8     4     2

sin(pi)

ans =

   1.2246e-16

2^54+2

ans =

   1.8014e+16

2^54+2==2^54

ans =

     1

t = rand

t =

    0.7587

s=single(t)

s =

    0.7587

d=double(s)

d =

    0.7587

s==d

ans =

     1

clc
sin(pi)

ans =

   1.2246e-16

b = 2

b =

     2

whos('b')
  Name      Size            Bytes  Class     Attributes

  b         1x1                 8  double              

2^54+2==2^54

ans =

     1

2^53+2==2^53

ans =

     0

clc
double(single(pi)) == single(pi)

ans =

     1

diary