线性求解器Ax=b的验证

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已于 2023-03-16 14:19:46 修改 · 553 阅读

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#matlab #c++ #IO

于 2023-03-11 11:40:39 首次发布

文章介绍了如何使用MatrixMarket网站获取不同性质的矩阵数据来验证线性求解器的性能。Matlab的mmread函数用于读取MatrixMarket格式的文件，mmwrite函数则用于写入数据。此外，还提供了C++的IO操作示例。文章提供了一个测试案例，涉及读取、写入矩阵数据并比较写入后数据的准确性。

文章目录

前言

一般情况集成了一个线性求解器（Ax=b），我们需要验证其性能和精度，这时需要大量数据来做验证，尤其是有不同性质的矩阵 $A$ 的数据，例如：稀疏性，对称性，正定性，对角占优等。

Matrix Market

Matrix Market
Matrix Market 网站提供了友好的接口和数据，方便我们验证求解器的精度和性能，尤其是对各种性质的矩阵 $A$ 都有很多数据，包含不同维数。

Matlab IO

Read data

function  [A,rows,cols,entries,rep,field,symm] = mmread(filename)
%
% function  [A] = mmread(filename)
%
% function  [A,rows,cols,entries,rep,field,symm] = mmread(filename)
%
%      Reads the contents of the Matrix Market file 'filename'
%      into the matrix 'A'.  'A' will be either sparse or full,
%      depending on the Matrix Market format indicated by
%      'coordinate' (coordinate sparse storage), or
%      'array' (dense array storage).  The data will be duplicated
%      as appropriate if symmetry is indicated in the header.
%
%      Optionally, size information about the matrix can be 
%      obtained by using the return values rows, cols, and
%      entries, where entries is the number of nonzero entries
%      in the final matrix. Type information can also be retrieved
%      using the optional return values rep (representation), field,
%      and symm (symmetry).
%

mmfile = fopen(filename,'r');
if ( mmfile == -1 )
 disp(filename);
 error('File not found');
end;

header = fgets(mmfile);
if (header == -1 )
  error('Empty file.')
end

% NOTE: If using a version of Matlab for which strtok is not
%       defined, substitute 'gettok' for 'strtok' in the 
%       following lines, and download gettok.m from the
%       Matrix Market site.    
[head0,header]   = strtok(header);  % see note above
[head1,header]   = strtok(header);
[rep,header]     = strtok(header);
[field,header]   = strtok(header);
[symm,header]    = strtok(header);
head1 = lower(head1);
rep   = lower(rep);
field = lower(field);
symm  = lower(symm);
if ( length(symm) == 0 )
   disp(['Not enough words in header line of file ',filename]) 
   disp('Recognized format: ')
   disp('%%MatrixMarket matrix representation field symmetry')
   error('Check header line.')
end
if ( ~ strcmp(head0,'%%MatrixMarket') )
   error('Not a valid MatrixMarket header.')
end
if (  ~ strcmp(head1,'matrix') )
   disp(['This seems to be a MatrixMarket ',head1,' file.']);
   disp('This function only knows how to read MatrixMarket matrix files.');
   disp('  ');
   error('  ');
end

% Read through comments, ignoring them

commentline = fgets(mmfile);
while length(commentline) > 0 & commentline(1) == '%',
  commentline = fgets(mmfile);
end

% Read size information, then branch according to
% sparse or dense format

if ( strcmp(rep,'coordinate')) %  read matrix given in sparse 
                              %  coordinate matrix format

  [sizeinfo,count] = sscanf(commentline,'%d%d%d');
  while ( count == 0 )
     commentline =  fgets(mmfile);
     if (commentline == -1 )
       error('End-of-file reached before size information was found.')
     end
     [sizeinfo,count] = sscanf(commentline,'%d%d%d');
     if ( count > 0 & count ~= 3 )
       error('Invalid size specification line.')
     end
  end
  rows = sizeinfo(1);
  cols = sizeinfo(2);
  entries = sizeinfo(3);
  
  if  ( strcmp(field,'real') )               % real valued entries:
  
    [T,count] = fscanf(mmfile,'%f',3);
    T = [T; fscanf(mmfile,'%f')];
    if ( size(T) ~= 3*entries )
       message = ...
       str2mat('Data file does not contain expected amount of data.',...
               'Check that number of data lines matches nonzero count.');
       disp(message);
       error('Invalid data.');
    end
    T = reshape(T,3,entries)';
    A = sparse(T(:,1), T(:,2), T(:,3), rows , cols);
  
  elseif   ( strcmp(field,'complex'))            % complex valued entries:
  
    T = fscanf(mmfile,'%f',4);
    T = [T; fscanf(mmfile,'%f')];
    if ( size(T) ~= 4*entries )
       message = ...
       str2mat('Data file does not contain expected amount of data.',...
               'Check that number of data lines matches nonzero count.');
       disp(message);
       error('Invalid data.');
    end
    T = reshape(T,4,entries)';
    A = sparse(T(:,1), T(:,2), T(:,3) + T(:,4)*sqrt(-1), rows , cols)