eigen 笔记2

4. Block operations

1) Using block operations

Block of size(p, q), starting at (i, j)

dynamic-size block expression: matrix.block(i, j, p, q);

fixed-size block expression: matrix.block<p, q>(i, j);

int main()
{
  Eigen::MatrixXf m(,);
  m <<  , , , ,
        , , , ,
        ,,,,
       ,,,;
  cout << "Block in the middle" << endl;
  cout << m.block<,>(,) << endl << endl;
  for (int i = ; i <= ; ++i)
  {
    cout << "Block of size " << i << "x" << i << endl;
    cout << m.block(,,i,i) << endl << endl;
  }
}

Output:

Block in the middle
 
Block of size 1x1
 
Block of size 2x2
 
Block of size 3x3

.block() 的返回值可以是左值, 也可以是右值.

2) Columns and rows

matrix.row(i), matrix.col(j)

int main()
{
  Eigen::MatrixXf m(,);
  m << ,,,
       ,,,
       ,,;
  cout << "Here is the matrix m:" << endl << m << endl;
  cout << "2nd Row: " << m.row() << endl;
  m.col() +=  * m.col();
  cout << "After adding 3 times the first column into the third column, the matrix m is:\n";
  cout << m << endl;
}

3) Corner-related operations

太复杂, 没兴趣.

4) Block operations for vectors

同上, 用时再查.

5. Advanced initialization

1) The comma initializer

RowVectorXd vec1();
vec1 << , , ;
 
RowVectorXd vec2();
vec2 << , , , ;
 
RowVectorXd joined();
joined << vec1, vec2;
 
MatrixXf matA(, );
matA << , , , ;
 
MatrixXf matB(, );
matB << matA, matA/, matA/, matA;
 
Matrix3f m;
m.row() << , , ;
m.block(,,,) << , , , ;
m.col().tail() << , ;   

2) Special matrices and arrays

Zero(), Constant(value), Constant(rows, cols, value), Random(), Identity(), LinSpaced(size, low, high),

Identity() only for matrix;

LinSpaced only for vectors and one-dimensional arrays; high也包含在数组中. 默认都是列向量.

std::cout << "A fixed-size array:\n";
Array33f a1 = Array33f::Zero();
 
std::cout << "A one-dimensional dynamic-size array:\n";
ArrayXf a2 = ArrayXf::Zero();
 
std::cout << "A two-dimensional dynamic-size array:\n";ArrayXXf a3 = ArrayXXf::Zero(, );

ArrayXXf table(, );
table.col() = ArrayXf::LinSpaced(, , );
table.col() = M_PI /  * table.col();
table.col() = table.col().sin();
table.col() = table.col().cos();
 
Output:
  Degrees   Radians      Sine    Cosine
 
            0.175     0.174     0.985
            0.349     0.342      0.94
            0.524       0.5     0.866
            0.698     0.643     0.766
            0.873     0.766     0.643
             1.05     0.866       0.5
             1.22      0.94     0.342
              1.4     0.985     0.174
             1.57          -4.37e-08

类似的, 有 setZero(), MatrixBase::setIdentity(), DenseBase::setLinSpaced()

3) Usage as temporary objects

int main()
{
  MatrixXd m = MatrixXd::Random(,);
  m = (m + MatrixXd::Constant(,,1.2)) * ;
  cout << "m =" << endl << m << endl;
  VectorXd v();
  v << , , ;
  cout << "m * v =" << endl << m * v << endl;
}

comma-initializer

MatrixXf mat = MatrixXf::Random(, );
std::cout << mat << std::endl << std::endl;
mat = (MatrixXf(,) << , , , ).finished() * mat;

注意到, .finished() 是必须的.

6. Reductions, visitors and broadcasting

1) Reducations

sum(), prod(), mean(), minCoeff(), maxCoeff(), trace()

2) Norm computations

应该用不到.

3) Boolean reductions

all(), any(), count()

int main()
{
  ArrayXXf a(,);
 
  a << ,,
       ,;
  cout << "(a > 0).all()   = " << (a > ).all() << endl;
  cout << "(a > 0).any()   = " << (a > ).any() << endl;
  cout << "(a > 0).count() = " << (a > ).count() << endl;
  cout << endl;
  cout << "(a > 2).all()   = " << (a > ).all() << endl;
  cout << "(a > 2).any()   = " << (a > ).any() << endl;
  cout << "(a > 2).count() = " << (a > ).count() << endl;
}
 
Output:
(a > ).all()   =
(a > ).any()   =
(a > ).count() = 
 
(a > ).all()   =
(a > ).any()   =
(a > ).count() = 

4) Visitors

maxCoeff(&x, &y), minCoeff(&x, &y)

x, y 即为行索引和列索引, MatrixXf::Index, 注意, 不是 int

  //get location of maximum
  MatrixXf::Index maxRow, maxCol;
  float max = m.maxCoeff(&maxRow, &maxCol);
 
  //get location of minimum
  MatrixXf::Index minRow, minCol;
  float min = m.minCoeff(&minRow, &minCol);

5) Partial reductions

colwise(), rowwise()

int main() {
    Eigen::MatrixXf mat(, );
    mat << , , , ,
           , , , ;
    cout << "Column's maximum: " << endl
        << mat.colwise().maxCoeff() << endl;
    cout << "Row's maximum: " << endl
        << mat.rowwise().maxCoeff() << endl;
}
 
Output:
Column's maximum:
 
Row's maximum:

combining partial reductions with other operations

int main() {
    MatrixXf mat(, );
    mat << , , , ,
           , , , ;
    MatrixXf::Index maxIndex;
    float maxNorm = mat.colwise().sum().maxCoeff(&maxIndex);
 
    cout << "Maximum sum at position: " << maxIndex << endl;
    cout << "The corresponding vector is: " << endl;
    cout << mat.col(maxIndex) << endl;
    cout << "And its sum is: " << maxNorm << endl;
}
 
Output:
Maximum sum at position
The corresponding vector is: 
 
And its sum is is: 

6) Broadcasting

int main() {
    MatrixXf mat(, );
    VectorXf v();
 
    mat << , , , ,
           , , , ;
    v << , ;
    w << , , , ;
 
    mat.colwise() += v;
 
    cout << "Broadcasting result: " << endl;
    cout << mat << endl;
 
    mat.rowwise() += w.transpose();
    cout << "Broadcasting result: " << endl;
    cout << mat << endl;
}
 
Output:

Broadcasting result:
1 2 6 9
4 2 8 3

Broadcasting result: 

7) Combining broadcasting with other operations

略.

7. Interfacing with raw buffers: the Map class

1) Map types and declaring Map variables

Map<Matrix<typename Scalar, int RowsAtCompileTime, int ColsAtCompileTime> >

RowsAtCompileTime 和 ColsAtCompileTime 是生成矩阵的行数和列数

Map<MatrixXf> mf(pf, rows, columns);

pf 指向一个定义 coefficient array 的 region 的起始, 这里注意 pf 是一个指针, 指向的是一个 coefficient, rows 和 columns 是该 mf 的行和列, 默认全部.

这里有一个问题, Map 模板参数指定的 RowsAtCompileTime 和 ColsAtCompileTime 与括号中的 rows 和 columns 有什么关系?

a. Matrix

Dynamic

Map<Matrix<int, Dynamic, Dynamic> > 即 Map<MatrixXi>, 此时生成 matrix 的行和列由 rows 和 columns 指定

这里还要注意一下, pf 的属性, 如果 pf 是指向一个内存区域, 内存区域的数字怎么获取, 要看这个区域存储的是什么.

存储 array, 假设 array 长度为 15, 要转换成3行4列矩阵, 那么取前12个数

存储 matrix, 注意 matrix 默认是按列存储的, 所以会取按列排布的前12个数, 也就是前4列

fixed-size

Map<Matrix<int, 3, 4> > , 此时生成 Matrix 的行和列已定, rows 和 columns 要与 3, 4 保持一致, 或者省略否则报错.

b. Vector

Dynamic

Map<Matrix<int, 1, Dynamic> > or Map<Matrix<int, Dynamic, 1> >

Vector 有一个特别地方, rows 和 columns 可以只写一个, 比如 Map<VectorXi> m2map(p, m2.size()), 也可以全写, 但是维度要与 Map 中的一致.

fixed-size

注意维度一致.

Map<const Vector4i> mi(pi);

声明一个 fixed-size read-only vector.

Map 还有几个可选的模板参数

Map<typename MatrixType, int MapOptions, typename StrideType>

MapOptions: Aligned or Unaligned, 默认 Unaligned

StrideType: 决定 array 的排布, 使用 Stride class

int array[];
for(int i = ; i < ; ++i) array[i] = i;
cout << "Column-major:\n" << Map<Matrix<int,,> >(array) << endl;
cout << "Row-major:\n" << Map<Matrix<int,,,RowMajor> >(array) << endl;
cout << "Row-major using stride:\n" <<
  Map<Matrix<int,,>, Unaligned, Stride<,> >(array) << endl;
 
Output:
 
Column-major:
 
Row-major:
 
Row-major using stride:

2) Using Map variables

注意: a Map type is not identical to its Dense equivalent.

typedef Matrix<float,,Dynamic> MatrixType;
typedef Map<MatrixType> MapType;
typedef Map<const MatrixType> MapTypeConst;   // a read-only map
const int n_dims = ;
 
MatrixType m1(n_dims), m2(n_dims);
m1.setRandom();
m2.setRandom();
float *p = &m2();  // get the address storing the data for m2
MapType m2map(p,m2.size());   // m2map shares data with m2
MapTypeConst m2mapconst(p,m2.size());  // a read-only accessor for m2
cout << "m1: " << m1 << endl;
cout << "m2: " << m2 << endl;
cout << "Squared euclidean distance: " << (m1-m2).squaredNorm() << endl;
cout << "Squared euclidean distance, using map: " <<
  (m1-m2map).squaredNorm() << endl;
m2map() = ;   // this will change m2, since they share the same array
cout << "Updated m2: " << m2 << endl;
cout << "m2 coefficient 2, constant accessor: " << m2mapconst() << endl;
/* m2mapconst(2) = 5; */   // this yields a compile-time error
 
Output:
 
m1:   0.68 -0.211  0.566  0.597  0.823
m2: -0.605  -0.33  0.536 -0.444  0.108
Squared euclidean distance: 3.26
Squared euclidean distance, using map: 3.26
Updated m2: -0.605  -0.33  0.536        0.108
m2 coefficient , constant accessor: 0.536

这里注意一下, m2map, m2mapconst 与 m2 共享内存空间, 所以对 m2map 的修改, 会同步到 m2 和 m2mapconst.

但是, 如果如果 *p 指向的是 array 中的地址, 那么 map 转化出的 matrix 不会与 array 共享内存空间.

3) Changing the mapped array

用 new 可以对声明后的 Map object 进行修改

int data[] = {,,,,,,,,};
Map<RowVectorXi> v(data,);
cout << "The mapped vector v is: " << v << "\n";
new (&v) Map<RowVectorXi>(data+,);
cout << "Now v is: " << v << "\n";
 
Output:
The mapped vector v is:
Now v is:     

所以也可以声明未知 array 的 map object

Map<Matrix3f> A(NULL);  // don't try to use this matrix yet!
VectorXf b(n_matrices);
for (int i = ; i < n_matrices; i++)
{
  new (&A) Map<Matrix3f>(get_matrix_pointer(i));
  b(i) = A.trace();
}

8. Reshape and Slicing

1) Reshape

MatrixXf M1(,);    // Column-major storage
M1 << , , ,
      , , ,
      , , ;
Map<RowVectorXf> v1(M1.data(), M1.size());
cout << "v1:" << endl << v1 << endl;
Matrix<float,Dynamic,Dynamic,RowMajor> M2(M1);
Map<RowVectorXf> v2(M2.data(), M2.size());
cout << "v2:" << endl << v2 << endl;
 
Output:
v1:
 
v2:
        

MatrixXf M1(,);    // Column-major storage
M1 << , , ,  ,  ,  ,
      , , , , , ;
Map<MatrixXf> M2(M1.data(), ,);
cout << "M2:" << endl << M2 << endl;
 
Output:
M2:

注意这里, 存储方式的不同, 会导致输出不同.

2) Slicing

RowVectorXf v = RowVectorXf::LinSpaced(,,);
cout << "Input:" << endl << v << endl;
Map<RowVectorXf,,InnerStride<> > v2(v.data(), v.size()/);
cout << "Even:" << v2 << endl;
 
Output:
Input:
 
Even:              

MatrixXf M1 = MatrixXf::Random(,);
cout << "Column major input:" << endl << M1 << "\n";
Map<MatrixXf,,OuterStride<> > M2(M1.data(), M1.rows(), (M1.cols()+)/, OuterStride<>(M1.outerStride()*));
cout << "1 column over 3:" << endl << M2 << "\n";
typedef Matrix<float,Dynamic,Dynamic,RowMajor> RowMajorMatrixXf;
RowMajorMatrixXf M3(M1);
cout << "Row major input:" << endl << M3 << "\n";
Map<RowMajorMatrixXf,,Stride<Dynamic,> > M4(M3.data(), M3.rows(), (M3.cols()+)/,
                                              Stride<Dynamic,>(M3.outerStride(),));
cout << "1 column over 3:" << endl << M4 << "\n";
 
Output:
Column major input:
   0.68   0.597   -0.33   0.108   -0.27   0.832  -0.717  -0.514
 -0.211   0.823   0.536 -0.0452  0.0268   0.271   0.214  -0.726
  0.566  -0.605  -0.444   0.258   0.904   0.435  -0.967   0.608
 column over :
   0.68   0.108  -0.717
 -0.211 -0.0452   0.214
  0.566   0.258  -0.967
Row major input:
   0.68   0.597   -0.33   0.108   -0.27   0.832  -0.717  -0.514
 -0.211   0.823   0.536 -0.0452  0.0268   0.271   0.214  -0.726
  0.566  -0.605  -0.444   0.258   0.904   0.435  -0.967   0.608
 column over :
   0.68   0.108  -0.717
 -0.211 -0.0452   0.214
  0.566   0.258  -0.967

注意 M1.data(), 返回一个指针.

9. Aliasing

m = m.transpose() 会造成 aliasing

a) m = m.transpose().eval();

b) m = m.transposeInPlace();

component-wise operations is safe.

mat = 2 * mat;

mat = mat - MatrixXf::Identity(2, 2);

matB = matA * matA; // Simple bu not quite as efficient

matB.noalias() = matA * matA; // Use this

10. Storage orders

默认 column-major, 也可以选择 row-major

11. Quick reference guide

Eigen/LU    Inverse, determinant, LU decompositions with solver (FullPivLU, PartialPivLU)

Eigen/SVD    SVD decompositions with least-squares solver (JacobiSVD, BDCSVD)

Eigen/QR    decomposition with solver (HouseholderQR, ColPivHouseholderQR, FullPivHouseholderQR)

...

Finished.