1679971140

# Using Singular Value Separation in Python and Numpy (linalg.svd)

In this pythonn - Numpy tutorial we will learn about Numpy linalg.svd: Singular Value Decomposition in Python. In mathematics, a singular value decomposition (SVD) of a matrix refers to the factorization of a matrix into three separate matrices. It is a more generalized version of an eigenvalue decomposition of matrices. It is further related to the polar decompositions.

In Python, it is easy to calculate the singular decomposition of a complex or a real matrix using the numerical python or the numpy library. The numpy library consists of various linear algebraic functions including one for calculating the singular value decomposition of a matrix.

In machine learning models, singular value decomposition is widely used to train models and in neural networks. It helps in improving accuracy and in reducing the noise in data. Singular value decomposition transforms one vector into another without them necessarily having the same dimension. Hence, it makes matrix manipulation in vector spaces easier and efficient. It is also used in regression analysis.

## Syntax of Numpy linalg.svd() function

The function that calculates the singular value decomposition of a matrix in python belongs to the numpy module, named linalg.svd() .

The syntax of the numpy linalg.svd () is as follows:

``````numpy.linalg.svd(A, full_matrices=True, compute_uv=True, hermitian=False)
``````

You can customize the true and false boolean values based on your requirements.

The parameters of the function are given below:

• A->array_like: This is the required matrix whose singular value decomposition is being calculated. It can be real or complex as required. It’s dimension should be >= 2.
• full_matrices->boolean value(optional): If set to true, then the Hermitian transpose of the given matrix is a square, if it’s false then it isn’t.
• compute_uv->boolen value(optional): It determines whether the Hermitian transpose is to be calculated or not in addition to the singular value decomposition.
• hermitian->boolean value(optional): The given matrix is considered hermitian(that is symmetric, with real values) which might provide a more efficient method for computation.

The function returns three types of matrices based on the parameters mentioned above:

• S->array_like: The vector containing the singular values in the descending order with dimensions same as the original matrix.
• u->array_like: This is an optional solution that is returned when compute_uv is set to True. It is a set of vectors with singular values.
• v-> array_like: Set of unitary arrays only returned when compute_uv is set to True.

It raises a LinALgError when the singular values diverse.

## Prerequisites for setup

Before we dive into the examples, make sure you have the numpy module installed in your local system. This is required for using linear algebraic functions like the one discussed in this article. Run the following command in your terminal.

``pip install numpy``

That’s all you need right now, let’s look at how we will implement the code in the next section.

To calculate Singular Value Decomposition (SVD) in Python, use the NumPy library’s linalg.svd() function. Its syntax is numpy.linalg.svd(A, full_matrices=True, compute_uv=True, hermitian=False), where A is the matrix for which SVD is being calculated. It returns three matrices: S, U, and V.

## Example 1: Calculating the Singular Value Decomposition of a 3×3 Matrix

In this first example we will take a 3X3 matrix and compute its singular value decomposition in the following way:

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to create an array
A=np.array([[2,4,6],
[8,10,12],
[14,16,18]])
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

The output will be:

``````the output is=
s(the singular value) =  [3.36962067e+01 2.13673903e+00 8.83684950e-16]
u =  [[-0.21483724  0.88723069  0.40824829]
[-0.52058739  0.24964395 -0.81649658]
[-0.82633754 -0.38794278  0.40824829]]
v =  [[-0.47967118 -0.57236779 -0.66506441]
[-0.77669099 -0.07568647  0.62531805]
[-0.40824829  0.81649658 -0.40824829]]``````

Example 1

## Example 2: Calculating the Singular Value Decomposition of a Random Matrix

In this example, we will be using the numpy.random.randint() function to create a random matrix. Let’s get into it!

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to craete an array
A=np.random.randint(5, 200, size=(3,3))
#display the created matrix
print("The input matrix is=",A)
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

The output will be as follows:

``````The input matrix is= [[ 36  74 101]
[104 129 185]
[139 121 112]]
the output is=
s(the singular value) =  [348.32979681  61.03199722  10.12165841]
u =  [[-0.3635535  -0.48363012 -0.79619769]
[-0.70916514 -0.41054007  0.57318554]
[-0.60408084  0.77301925 -0.19372034]]
v =  [[-0.49036384 -0.54970618 -0.67628871]
[ 0.77570499  0.0784348  -0.62620264]
[ 0.39727203 -0.83166766  0.38794824]]``````

Example 2

## Wrapping Up

In this article, we explored the concept of singular value decomposition in mathematics and how to calculate it using Python’s numpy module. We used the linalg.svd() function to compute the singular value decomposition of both given and random matrices. Numpy provides an efficient and easy-to-use method for performing linear algebra operations, making it highly valuable in machine learning, neural networks, and regression analysis. Keep exploring other linear algebraic functions in numpy to enhance your mathematical toolset in Python.

1679971140

## Using Singular Value Separation in Python and Numpy (linalg.svd)

In this pythonn - Numpy tutorial we will learn about Numpy linalg.svd: Singular Value Decomposition in Python. In mathematics, a singular value decomposition (SVD) of a matrix refers to the factorization of a matrix into three separate matrices. It is a more generalized version of an eigenvalue decomposition of matrices. It is further related to the polar decompositions.

In Python, it is easy to calculate the singular decomposition of a complex or a real matrix using the numerical python or the numpy library. The numpy library consists of various linear algebraic functions including one for calculating the singular value decomposition of a matrix.

In machine learning models, singular value decomposition is widely used to train models and in neural networks. It helps in improving accuracy and in reducing the noise in data. Singular value decomposition transforms one vector into another without them necessarily having the same dimension. Hence, it makes matrix manipulation in vector spaces easier and efficient. It is also used in regression analysis.

## Syntax of Numpy linalg.svd() function

The function that calculates the singular value decomposition of a matrix in python belongs to the numpy module, named linalg.svd() .

The syntax of the numpy linalg.svd () is as follows:

``````numpy.linalg.svd(A, full_matrices=True, compute_uv=True, hermitian=False)
``````

You can customize the true and false boolean values based on your requirements.

The parameters of the function are given below:

• A->array_like: This is the required matrix whose singular value decomposition is being calculated. It can be real or complex as required. It’s dimension should be >= 2.
• full_matrices->boolean value(optional): If set to true, then the Hermitian transpose of the given matrix is a square, if it’s false then it isn’t.
• compute_uv->boolen value(optional): It determines whether the Hermitian transpose is to be calculated or not in addition to the singular value decomposition.
• hermitian->boolean value(optional): The given matrix is considered hermitian(that is symmetric, with real values) which might provide a more efficient method for computation.

The function returns three types of matrices based on the parameters mentioned above:

• S->array_like: The vector containing the singular values in the descending order with dimensions same as the original matrix.
• u->array_like: This is an optional solution that is returned when compute_uv is set to True. It is a set of vectors with singular values.
• v-> array_like: Set of unitary arrays only returned when compute_uv is set to True.

It raises a LinALgError when the singular values diverse.

## Prerequisites for setup

Before we dive into the examples, make sure you have the numpy module installed in your local system. This is required for using linear algebraic functions like the one discussed in this article. Run the following command in your terminal.

``pip install numpy``

That’s all you need right now, let’s look at how we will implement the code in the next section.

To calculate Singular Value Decomposition (SVD) in Python, use the NumPy library’s linalg.svd() function. Its syntax is numpy.linalg.svd(A, full_matrices=True, compute_uv=True, hermitian=False), where A is the matrix for which SVD is being calculated. It returns three matrices: S, U, and V.

## Example 1: Calculating the Singular Value Decomposition of a 3×3 Matrix

In this first example we will take a 3X3 matrix and compute its singular value decomposition in the following way:

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to create an array
A=np.array([[2,4,6],
[8,10,12],
[14,16,18]])
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

The output will be:

``````the output is=
s(the singular value) =  [3.36962067e+01 2.13673903e+00 8.83684950e-16]
u =  [[-0.21483724  0.88723069  0.40824829]
[-0.52058739  0.24964395 -0.81649658]
[-0.82633754 -0.38794278  0.40824829]]
v =  [[-0.47967118 -0.57236779 -0.66506441]
[-0.77669099 -0.07568647  0.62531805]
[-0.40824829  0.81649658 -0.40824829]]``````

Example 1

## Example 2: Calculating the Singular Value Decomposition of a Random Matrix

In this example, we will be using the numpy.random.randint() function to create a random matrix. Let’s get into it!

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to craete an array
A=np.random.randint(5, 200, size=(3,3))
#display the created matrix
print("The input matrix is=",A)
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

The output will be as follows:

``````The input matrix is= [[ 36  74 101]
[104 129 185]
[139 121 112]]
the output is=
s(the singular value) =  [348.32979681  61.03199722  10.12165841]
u =  [[-0.3635535  -0.48363012 -0.79619769]
[-0.70916514 -0.41054007  0.57318554]
[-0.60408084  0.77301925 -0.19372034]]
v =  [[-0.49036384 -0.54970618 -0.67628871]
[ 0.77570499  0.0784348  -0.62620264]
[ 0.39727203 -0.83166766  0.38794824]]``````

Example 2

## Wrapping Up

In this article, we explored the concept of singular value decomposition in mathematics and how to calculate it using Python’s numpy module. We used the linalg.svd() function to compute the singular value decomposition of both given and random matrices. Numpy provides an efficient and easy-to-use method for performing linear algebra operations, making it highly valuable in machine learning, neural networks, and regression analysis. Keep exploring other linear algebraic functions in numpy to enhance your mathematical toolset in Python.

1667425440

## pdf2gerb

Perl script converts PDF files to Gerber format

Pdf2Gerb generates Gerber 274X photoplotting and Excellon drill files from PDFs of a PCB. Up to three PDFs are used: the top copper layer, the bottom copper layer (for 2-sided PCBs), and an optional silk screen layer. The PDFs can be created directly from any PDF drawing software, or a PDF print driver can be used to capture the Print output if the drawing software does not directly support output to PDF.

The general workflow is as follows:

2. Print the top and bottom copper and top silk screen layers to a PDF file.
3. Run Pdf2Gerb on the PDFs to create Gerber and Excellon files.
4. Use a Gerber viewer to double-check the output against the original PCB design.
6. Submit the files to a PCB manufacturer.

Please note that Pdf2Gerb does NOT perform DRC (Design Rule Checks), as these will vary according to individual PCB manufacturer conventions and capabilities. Also note that Pdf2Gerb is not perfect, so the output files must always be checked before submitting them. As of version 1.6, Pdf2Gerb supports most PCB elements, such as round and square pads, round holes, traces, SMD pads, ground planes, no-fill areas, and panelization. However, because it interprets the graphical output of a Print function, there are limitations in what it can recognize (or there may be bugs).

See docs/Pdf2Gerb.pdf for install/setup, config, usage, and other info.

## pdf2gerb_cfg.pm

``````#Pdf2Gerb config settings:
#Put this file in same folder/directory as pdf2gerb.pl itself (global settings),
#or copy to another folder/directory with PDFs if you want PCB-specific settings.
#There is only one user of this file, so we don't need a custom package or namespace.
#NOTE: all constants defined in here will be added to main namespace.
#package pdf2gerb_cfg;

use strict; #trap undef vars (easier debug)
use warnings; #other useful info (easier debug)

##############################################################################################
#configurable settings:
#change values here instead of in main pfg2gerb.pl file

use constant WANT_COLORS => (\$^O !~ m/Win/); #ANSI colors no worky on Windows? this must be set < first DebugPrint() call

#just a little warning; set realistic expectations:
#DebugPrint("\${\(CYAN)}Pdf2Gerb.pl \${\(VERSION)}, \$^O O/S\n\${\(YELLOW)}\${\(BOLD)}\${\(ITALIC)}This is EXPERIMENTAL software.  \nGerber files MAY CONTAIN ERRORS.  Please CHECK them before fabrication!\${\(RESET)}", 0); #if WANT_DEBUG

use constant METRIC => FALSE; #set to TRUE for metric units (only affect final numbers in output files, not internal arithmetic)
use constant APERTURE_LIMIT => 0; #34; #max #apertures to use; generate warnings if too many apertures are used (0 to not check)
use constant DRILL_FMT => '2.4'; #'2.3'; #'2.4' is the default for PCB fab; change to '2.3' for CNC

use constant WANT_DEBUG => 0; #10; #level of debug wanted; higher == more, lower == less, 0 == none
use constant GERBER_DEBUG => 0; #level of debug to include in Gerber file; DON'T USE FOR FABRICATION
use constant WANT_STREAMS => FALSE; #TRUE; #save decompressed streams to files (for debug)
use constant WANT_ALLINPUT => FALSE; #TRUE; #save entire input stream (for debug ONLY)

#DebugPrint(sprintf("\${\(CYAN)}DEBUG: stdout %d, gerber %d, want streams? %d, all input? %d, O/S: \$^O, Perl: \$]\${\(RESET)}\n", WANT_DEBUG, GERBER_DEBUG, WANT_STREAMS, WANT_ALLINPUT), 1);
#DebugPrint(sprintf("max int = %d, min int = %d\n", MAXINT, MININT), 1);

#define standard trace and pad sizes to reduce scaling or PDF rendering errors:
#This avoids weird aperture settings and replaces them with more standardized values.
#(I'm not sure how photoplotters handle strange sizes).
#Fewer choices here gives more accurate mapping in the final Gerber files.
#units are in inches
use constant TOOL_SIZES => #add more as desired
(
#round or square pads (> 0) and drills (< 0):
.010, -.001,  #tiny pads for SMD; dummy drill size (too small for practical use, but needed so StandardTool will use this entry)
.031, -.014,  #used for vias
.041, -.020,  #smallest non-filled plated hole
.051, -.025,
.056, -.029,  #useful for IC pins
.070, -.033,
#    .090, -.043,  #NOTE: 600 dpi is not high enough resolution to reliably distinguish between .043" and .046", so choose 1 of the 2 here
.100, -.046,
.115, -.052,
.130, -.061,
.140, -.067,
.150, -.079,
.175, -.088,
.190, -.093,
.200, -.100,
.220, -.110,
.160, -.125,  #useful for mounting holes
#some additional pad sizes without holes (repeat a previous hole size if you just want the pad size):
.090, -.040,  #want a .090 pad option, but use dummy hole size
.065, -.040, #.065 x .065 rect pad
.035, -.040, #.035 x .065 rect pad
#traces:
.001,  #too thin for real traces; use only for board outlines
.006,  #minimum real trace width; mainly used for text
.008,  #mainly used for mid-sized text, not traces
.010,  #minimum recommended trace width for low-current signals
.012,
.015,  #moderate low-voltage current
.020,  #heavier trace for power, ground (even if a lighter one is adequate)
.025,
.030,  #heavy-current traces; be careful with these ones!
.040,
.050,
.060,
.080,
.100,
.120,
);
#Areas larger than the values below will be filled with parallel lines:
#This cuts down on the number of aperture sizes used.
#Set to 0 to always use an aperture or drill, regardless of size.
use constant { MAX_APERTURE => max((TOOL_SIZES)) + .004, MAX_DRILL => -min((TOOL_SIZES)) + .004 }; #max aperture and drill sizes (plus a little tolerance)
#DebugPrint(sprintf("using %d standard tool sizes: %s, max aper %.3f, max drill %.3f\n", scalar((TOOL_SIZES)), join(", ", (TOOL_SIZES)), MAX_APERTURE, MAX_DRILL), 1);

#NOTE: Compare the PDF to the original CAD file to check the accuracy of the PDF rendering and parsing!
#for example, the CAD software I used generated the following circles for holes:
#CAD hole size:   parsed PDF diameter:      error:
#  .014                .016                +.002
#  .020                .02267              +.00267
#  .025                .026                +.001
#  .029                .03167              +.00267
#  .033                .036                +.003
#  .040                .04267              +.00267
#This was usually ~ .002" - .003" too big compared to the hole as displayed in the CAD software.
#To compensate for PDF rendering errors (either during CAD Print function or PDF parsing logic), adjust the values below as needed.
#units are pixels; for example, a value of 2.4 at 600 dpi = .0004 inch, 2 at 600 dpi = .0033"
use constant
{
HOLE_ADJUST => -0.004 * 600, #-2.6, #holes seemed to be slightly oversized (by .002" - .004"), so shrink them a little
RNDPAD_ADJUST => -0.003 * 600, #-2, #-2.4, #round pads seemed to be slightly oversized, so shrink them a little
SQRPAD_ADJUST => +0.001 * 600, #+.5, #square pads are sometimes too small by .00067, so bump them up a little
TRACE_ADJUST => 0, #(pixels) traces seemed to be okay?
REDUCE_TOLERANCE => .001, #(inches) allow this much variation when reducing circles and rects
};

#Also, my CAD's Print function or the PDF print driver I used was a little off for circles, so define some additional adjustment values here:
#Values are added to X/Y coordinates; units are pixels; for example, a value of 1 at 600 dpi would be ~= .002 inch
use constant
{
CIRCLE_ADJUST_MINY => -0.001 * 600, #-1, #circles were a little too high, so nudge them a little lower
CIRCLE_ADJUST_MAXX => +0.001 * 600, #+1, #circles were a little too far to the left, so nudge them a little to the right
SUBST_CIRCLE_CLIPRECT => FALSE, #generate circle and substitute for clip rects (to compensate for the way some CAD software draws circles)
WANT_CLIPRECT => TRUE, #FALSE, #AI doesn't need clip rect at all? should be on normally?
RECT_COMPLETION => FALSE, #TRUE, #fill in 4th side of rect when 3 sides found
};

use constant SOLDER_MARGIN => +.012; #units are inches

#line join/cap styles:
use constant
{
CAP_NONE => 0, #butt (none); line is exact length
CAP_ROUND => 1, #round cap/join; line overhangs by a semi-circle at either end
CAP_SQUARE => 2, #square cap/join; line overhangs by a half square on either end
CAP_OVERRIDE => FALSE, #cap style overrides drawing logic
};

#number of elements in each shape type:
use constant
{
RECT_SHAPELEN => 6, #x0, y0, x1, y1, count, "rect" (start, end corners)
LINE_SHAPELEN => 6, #x0, y0, x1, y1, count, "line" (line seg)
CURVE_SHAPELEN => 10, #xstart, ystart, x0, y0, x1, y1, xend, yend, count, "curve" (bezier 2 points)
CIRCLE_SHAPELEN => 5, #x, y, 5, count, "circle" (center + radius)
};
#const my %SHAPELEN =
our %SHAPELEN =
(
rect => RECT_SHAPELEN,
line => LINE_SHAPELEN,
curve => CURVE_SHAPELEN,
circle => CIRCLE_SHAPELEN,
);

#panelization:
#This will repeat the entire body the number of times indicated along the X or Y axes (files grow accordingly).
#Display elements that overhang PCB boundary can be squashed or left as-is (typically text or other silk screen markings).
#Set "overhangs" TRUE to allow overhangs, FALSE to truncate them.
use constant PANELIZE => {'x' => 1, 'y' => 1, 'xpad' => 0, 'ypad' => 0, 'overhangs' => TRUE}; #number of times to repeat in X and Y directions

# Set this to 1 if you need TurboCAD support.
#\$turboCAD = FALSE; #is this still needed as an option?

#CIRCAD pad generation uses an appropriate aperture, then moves it (stroke) "a little" - we use this to find pads and distinguish them from PCB holes.
use constant PAD_STROKE => 0.3; #0.0005 * 600; #units are pixels
#convert very short traces to pads or holes:
use constant TRACE_MINLEN => .001; #units are inches
#use constant ALWAYS_XY => TRUE; #FALSE; #force XY even if X or Y doesn't change; NOTE: needs to be TRUE for all pads to show in FlatCAM and ViewPlot
use constant REMOVE_POLARITY => FALSE; #TRUE; #set to remove subtractive (negative) polarity; NOTE: must be FALSE for ground planes

#PDF uses "points", each point = 1/72 inch
#combined with a PDF scale factor of .12, this gives 600 dpi resolution (1/72 * .12 = 600 dpi)
use constant INCHES_PER_POINT => 1/72; #0.0138888889; #multiply point-size by this to get inches

# The precision used when computing a bezier curve. Higher numbers are more precise but slower (and generate larger files).
#\$bezierPrecision = 100;
use constant BEZIER_PRECISION => 36; #100; #use const; reduced for faster rendering (mainly used for silk screen and thermal pads)

# Ground planes and silk screen or larger copper rectangles or circles are filled line-by-line using this resolution.
use constant FILL_WIDTH => .01; #fill at most 0.01 inch at a time

# The max number of characters to read into memory
use constant MAX_BYTES => 10 * M; #bumped up to 10 MB, use const

use constant DUP_DRILL1 => TRUE; #FALSE; #kludge: ViewPlot doesn't load drill files that are too small so duplicate first tool

my \$runtime = time(); #Time::HiRes::gettimeofday(); #measure my execution time

print STDERR "Loaded config settings from '\${\(__FILE__)}'.\n";
1; #last value must be truthful to indicate successful load

#############################################################################################
#junk/experiment:

#use Package::Constants;
#use Exporter qw(import); #https://perldoc.perl.org/Exporter.html

#my \$caller = "pdf2gerb::";

#sub cfg
#{
#    my \$proto = shift;
#    my \$class = ref(\$proto) || \$proto;
#    my \$settings =
#    {
#        \$WANT_DEBUG => 990, #10; #level of debug wanted; higher == more, lower == less, 0 == none
#    };
#    bless(\$settings, \$class);
#    return \$settings;
#}

#use constant HELLO => "hi there2"; #"main::HELLO" => "hi there";
#use constant GOODBYE => 14; #"main::GOODBYE" => 12;

#our @EXPORT_OK = Package::Constants->list(__PACKAGE__); #https://www.perlmonks.org/?node_id=1072691; NOTE: "_OK" skips short/common names

#print STDERR scalar(@EXPORT_OK) . " consts exported:\n";
#foreach(@EXPORT_OK) { print STDERR "\$_\n"; }
#my \$val = main::thing("xyz");
#print STDERR "caller gave me \$val\n";
#foreach my \$arg (@ARGV) { print STDERR "arg \$arg\n"; }``````

Author: swannman
Source Code: https://github.com/swannman/pdf2gerb

1680061020

## Usando separação de valor singular em Python e Numpy (linalg.svd)

Neste tutorial pythonn - Numpy, aprenderemos sobre Numpy linalg.svd: Decomposição de valor singular em Python. Em matemática, uma decomposição de valor singular (SVD) de uma matriz refere-se à fatoração de uma matriz em três matrizes separadas. É uma versão mais generalizada de uma decomposição de valores próprios de matrizes. Está ainda relacionado com as decomposições polares.

Em Python, é fácil calcular a decomposição singular de uma matriz complexa ou real usando o python numérico ou a biblioteca numpy. A biblioteca numpy consiste em várias funções algébricas lineares, incluindo uma para calcular a decomposição do valor singular de uma matriz.

Em modelos de aprendizado de máquina , a decomposição de valor singular é amplamente utilizada para treinar modelos e em redes neurais. Ajuda a melhorar a precisão e a reduzir o ruído nos dados. A decomposição em valor singular transforma um vetor em outro sem que eles tenham necessariamente a mesma dimensão. Portanto, torna a manipulação de matrizes em espaços vetoriais mais fácil e eficiente. Também é usado na análise de regressão .

## Sintaxe da função Numpy linalg.svd()

A função que calcula a decomposição do valor singular de uma matriz em python pertence ao módulo numpy, chamado linalg.svd() .

A sintaxe do numpy linalg.svd () é a seguinte:

``````numpy.linalg.svd(A, full_matrices=True, compute_uv=True, hermitian=False)
``````

Você pode personalizar os valores booleanos verdadeiro e falso com base em seus requisitos.

Os parâmetros da função são dados a seguir:

• A->array_like: Esta é a matriz necessária cuja decomposição de valor singular está sendo calculada. Pode ser real ou complexo, conforme necessário. Sua dimensão deve ser >= 2.
• full_matrices->boolean value(opcional): Se definido como true, então a transposta Hermitiana da matriz dada é um quadrado, se for false então não é.
• compute_uv->boolen value(opcional): Determina se a transposição hermitiana deve ser calculada ou não além da decomposição do valor singular.
• hermitian->valor booleano (opcional): A matriz fornecida é considerada hermitiana (ou seja, simétrica, com valores reais), o que pode fornecer um método de cálculo mais eficiente.

A função retorna três tipos de matrizes com base nos parâmetros mencionados acima:

• S->array_like : O vetor contendo os valores singulares na ordem decrescente com as mesmas dimensões da matriz original.
• u->array_like : Esta é uma solução opcional que é retornada quando compute_uv é definido como True. É um conjunto de vetores com valores singulares.
• v-> array_like : Conjunto de arrays unitários retornados apenas quando compute_uv é definido como True.

Gera um LinALgError quando os valores singulares são diversos.

## Pré-requisitos para configuração

Antes de mergulharmos nos exemplos, certifique-se de ter o módulo numpy instalado em seu sistema local. Isso é necessário para usar funções algébricas lineares como a discutida neste artigo. Execute o seguinte comando em seu terminal.

``pip install numpy``

Isso é tudo que você precisa agora, vamos ver como vamos implementar o código na próxima seção.

Para calcular a Decomposição de Valor Singular (SVD) em Python, use a função linalg.svd() da biblioteca NumPy. Sua sintaxe é numpy.linalg.svd(A, full_matrices=True, compute_uv=True, hermitian=False), onde A é a matriz para a qual SVD está sendo calculado. Ele retorna três matrizes: S, U e V.

## Exemplo 1: Calculando a Decomposição de Valor Singular de uma Matriz 3 × 3

Neste primeiro exemplo, pegaremos uma matriz 3X3 e calcularemos sua decomposição de valor singular da seguinte maneira:

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to create an array
A=np.array([[2,4,6],
[8,10,12],
[14,16,18]])
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

A saída será:

``````the output is=
s(the singular value) =  [3.36962067e+01 2.13673903e+00 8.83684950e-16]
u =  [[-0.21483724  0.88723069  0.40824829]
[-0.52058739  0.24964395 -0.81649658]
[-0.82633754 -0.38794278  0.40824829]]
v =  [[-0.47967118 -0.57236779 -0.66506441]
[-0.77669099 -0.07568647  0.62531805]
[-0.40824829  0.81649658 -0.40824829]]``````

Exemplo 1

## Exemplo 2: Calculando a Decomposição de Valor Singular de uma Matriz Aleatória

Neste exemplo, usaremos a função numpy.random.randint() para criar uma matriz aleatória. Vamos entrar nisso!

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to craete an array
A=np.random.randint(5, 200, size=(3,3))
#display the created matrix
print("The input matrix is=",A)
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

A saída será a seguinte:

``````The input matrix is= [[ 36  74 101]
[104 129 185]
[139 121 112]]
the output is=
s(the singular value) =  [348.32979681  61.03199722  10.12165841]
u =  [[-0.3635535  -0.48363012 -0.79619769]
[-0.70916514 -0.41054007  0.57318554]
[-0.60408084  0.77301925 -0.19372034]]
v =  [[-0.49036384 -0.54970618 -0.67628871]
[ 0.77570499  0.0784348  -0.62620264]
[ 0.39727203 -0.83166766  0.38794824]]``````

Exemplo 2

## Empacotando

Neste artigo, exploramos o conceito de decomposição de valor singular em matemática e como calculá-la usando o módulo numpy do Python. Usamos a função linalg.svd() para calcular a decomposição de valor singular de matrizes fornecidas e aleatórias. O Numpy fornece um método eficiente e fácil de usar para realizar operações de álgebra linear, tornando-o altamente valioso em aprendizado de máquina, redes neurais e análise de regressão. Continue explorando outras funções algébricas lineares em numpy para aprimorar seu conjunto de ferramentas matemáticas em Python.

1680070980

## Numpy linalg.svd() 函数的语法

python中计算矩阵奇异值分解的函数属于numpy模块，名为linalg.svd()。

numpy linalg.svd() 的语法如下：

``````numpy.linalg.svd(A, full_matrices=True, compute_uv=True, hermitian=False)
``````

• A->array_like：这是需要计算奇异值分解的矩阵。根据需要，它可以是真实的或复杂的。它的维度应该> = 2。
• full_matrices->boolean value（可选）：如果设置为 true，则给定矩阵的 Hermitian 转置为正方形，如果为 false，则不是。
• compute_uv->boolen value(optional)：决定是否在奇异值分解的基础上计算Hermitian转置。
• hermitian->boolean value（可选）：给定的矩阵被认为是 hermitian（即对称的，具有实数值），这可能提供更有效的计算方法。

• S->array_like：包含奇异值的向量，按降序排列，维度与原始矩阵相同。
• u->array_like：这是一个可选的解决方案，当 compute_uv 设置为 True 时返回。它是一组具有奇异值的向量。
• v-> array_like：单一数组集仅在 compute_uv 设置为 True 时返回。

## 设置的先决条件

``pip install numpy``

## 示例 1：计算 3×3 矩阵的奇异值分解

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to create an array
A=np.array([[2,4,6],
[8,10,12],
[14,16,18]])
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

``````the output is=
s(the singular value) =  [3.36962067e+01 2.13673903e+00 8.83684950e-16]
u =  [[-0.21483724  0.88723069  0.40824829]
[-0.52058739  0.24964395 -0.81649658]
[-0.82633754 -0.38794278  0.40824829]]
v =  [[-0.47967118 -0.57236779 -0.66506441]
[-0.77669099 -0.07568647  0.62531805]
[-0.40824829  0.81649658 -0.40824829]]``````

## 示例 2：计算随机矩阵的奇异值分解

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to craete an array
A=np.random.randint(5, 200, size=(3,3))
#display the created matrix
print("The input matrix is=",A)
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

``````The input matrix is= [[ 36  74 101]
[104 129 185]
[139 121 112]]
the output is=
s(the singular value) =  [348.32979681  61.03199722  10.12165841]
u =  [[-0.3635535  -0.48363012 -0.79619769]
[-0.70916514 -0.41054007  0.57318554]
[-0.60408084  0.77301925 -0.19372034]]
v =  [[-0.49036384 -0.54970618 -0.67628871]
[ 0.77570499  0.0784348  -0.62620264]
[ 0.39727203 -0.83166766  0.38794824]]``````

1680066180

## Разделение единственного числа в Python и Numpy (linalg.svd)

В этом руководстве по pythonn — Numpy мы узнаем о Numpy linalg.svd: разложение по единственному значению в Python. В математике разложение матрицы по сингулярным числам (SVD) относится к разложению матрицы на три отдельные матрицы. Это более обобщенная версия разложения матриц по собственным значениям. Это также связано с полярными разложениями.

В Python легко вычислить сингулярное разложение сложной или вещественной матрицы, используя числовой python или библиотеку numpy. Библиотека numpy состоит из различных линейных алгебраических функций, включая функцию для вычисления разложения матрицы по сингулярным числам.

В моделях машинного обучения разложение по сингулярным числам широко используется для обучения моделей и в нейронных сетях. Это помогает повысить точность и уменьшить шум в данных. Разложение по сингулярным значениям преобразует один вектор в другой, при этом они не обязательно имеют одинаковую размерность. Следовательно, это делает матричные операции в векторных пространствах более простыми и эффективными. Он также используется в регрессионном анализе .

## Синтаксис функции Numpy linalg.svd()

Функция, которая вычисляет разложение матрицы по сингулярным числам в python, принадлежит модулю numpy с именем linalg.svd() .

Синтаксис numpy linalg.svd() следующий:

``````numpy.linalg.svd(A, full_matrices=True, compute_uv=True, hermitian=False)
``````

Вы можете настроить истинные и ложные логические значения в соответствии с вашими требованиями.

Параметры функции приведены ниже:

• A->array_like: это требуемая матрица, для которой вычисляется разложение по сингулярным числам. Он может быть реальным или сложным по мере необходимости. Его размер должен быть >= 2.
• full_matrices->boolean value(необязательно): если установлено значение true, то эрмитовское транспонирование данной матрицы является квадратом, если оно false, то это не так.
• calculate_uv->boolen value (необязательно): определяет, следует ли вычислять эрмитову транспонирование в дополнение к разложению по сингулярным значениям.
• hermitian->boolean value(необязательно): Данная матрица считается эрмитовой (то есть симметричной, с действительными значениями), что может обеспечить более эффективный метод вычислений.

Функция возвращает три типа матриц на основе указанных выше параметров:

• S->array_like : вектор, содержащий сингулярные значения в порядке убывания с размерами, такими же, как исходная матрица.
• u->array_like : это необязательное решение, которое возвращается, когда для параметра calculate_uv установлено значение True. Это набор векторов с сингулярными значениями.
• v-> array_like : Набор унитарных массивов возвращается только в том случае, если для параметра calculate_uv установлено значение True.

Он вызывает LinALgError , когда сингулярные значения различаются.

## Предварительные условия для настройки

Прежде чем мы углубимся в примеры, убедитесь, что в вашей локальной системе установлен модуль numpy. Это необходимо для использования линейных алгебраических функций, подобных той, что обсуждается в этой статье. Запустите следующую команду в своем терминале.

``pip install numpy``

Это все, что вам нужно прямо сейчас, давайте посмотрим, как мы будем реализовывать код в следующем разделе.

Чтобы вычислить разложение по сингулярным значениям (SVD) в Python, используйте функцию linalg.svd() из библиотеки NumPy. Его синтаксис таков: numpy.linalg.svd(A, full_matrices=True, calculate_uv=True, hermitian=False), где A — матрица, для которой вычисляется SVD. Он возвращает три матрицы: S, U и V.

## Пример 1. Вычисление сингулярного разложения матрицы 3×3

В этом первом примере мы возьмем матрицу 3X3 и вычислим ее разложение по сингулярным числам следующим образом:

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to create an array
A=np.array([[2,4,6],
[8,10,12],
[14,16,18]])
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

Вывод будет:

``````the output is=
s(the singular value) =  [3.36962067e+01 2.13673903e+00 8.83684950e-16]
u =  [[-0.21483724  0.88723069  0.40824829]
[-0.52058739  0.24964395 -0.81649658]
[-0.82633754 -0.38794278  0.40824829]]
v =  [[-0.47967118 -0.57236779 -0.66506441]
[-0.77669099 -0.07568647  0.62531805]
[-0.40824829  0.81649658 -0.40824829]]``````

Пример 1

## Пример 2. Вычисление сингулярного разложения случайной матрицы

В этом примере мы будем использовать функцию numpy.random.randint() для создания случайной матрицы. Давайте погрузимся в это!

``````#importing the numpy module
import numpy as np
#using the numpy.array() function to craete an array
A=np.random.randint(5, 200, size=(3,3))
#display the created matrix
print("The input matrix is=",A)
#calculatin all three matrices for the output
#using the numpy linalg.svd function
u,s,v=np.linalg.svd(A, compute_uv=True)
#displaying the result
print("the output is=")
print('s(the singular value) = ',s)
print('u = ',u)
print('v = ',v)``````

Вывод будет следующим:

``````The input matrix is= [[ 36  74 101]
[104 129 185]
[139 121 112]]
the output is=
s(the singular value) =  [348.32979681  61.03199722  10.12165841]
u =  [[-0.3635535  -0.48363012 -0.79619769]
[-0.70916514 -0.41054007  0.57318554]
[-0.60408084  0.77301925 -0.19372034]]
v =  [[-0.49036384 -0.54970618 -0.67628871]
[ 0.77570499  0.0784348  -0.62620264]
[ 0.39727203 -0.83166766  0.38794824]]``````

Пример 2

## Подведение итогов

В этой статье мы рассмотрели концепцию разложения по сингулярным числам в математике и способы ее вычисления с помощью модуля Python numpy. Мы использовали функцию linalg.svd() для вычисления разложения по сингулярным числам как заданных, так и случайных матриц. Numpy предоставляет эффективный и простой в использовании метод выполнения операций линейной алгебры, что делает его очень ценным для машинного обучения, нейронных сетей и регрессионного анализа. Продолжайте изучать другие линейные алгебраические функции в numpy, чтобы расширить свой набор математических инструментов в Python.