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MQL5 ReferansıMatris ve Vektör MetotlarıOpenBLASEigen ValuesGeneral MatricesEigenSolver2X 

EigenSolver2X

Compute generalized eigenvalues and eigenvectors for a pair of regular square matrices in Expert mode, i.e. with the ability to influence the computation algorithm and the ability to obtain accompanying computation data (lapack function GGEVX). Both matrices must be the same size.

Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV).

Computing for type matrix<double>

bool  matrix::EigenSolver2X(
   matrix&               B,                       // second matrix in the pair
   ENUM_EIG_BALANCE      balance,                 // input matrix balancing method
   ENUM_EIG_VECTORS      jobv,                    // determines computation of right and left eigenvectors
   ENUM_EIG_SENSE        sense,                   // determines computation of reciprocal condition numbers
   vectorc&              alpha,                   // vector of computed eigenvalues
   vector&               beta,                    // vector of eigenvalue divisors
   matrix&               left_eigenvectors,       // matrix of computed left vectors
   matrix&               right_eigenvectors       // matrix of computed right vectors  
   matrix&               shur_matrix1,            // the first part of the real Schur form of the "balanced" versions of the input A and B
   matrix&               shur_matrix2,            // the second part of the real Schur form of the "balanced" versions of the input A and B
   long&                 ilo,                     // subscript of balanced matrix
   long&                 ihi,                     // superscript of balanced matrix
   vector&               lscale,                  // details of the permutations and scaling factors applied to the left side of A and B
   vector&               rscale,                  // details of the permutations and scaling factors applied to the right side of A and B
   double&               ab_norm,                 // one-norm of balanced input matrix
   double&               bb_norm,                 // one-norm of balanced second matrix B
   vector&               rconde,                  // vector of reciprocal condition numbers for each eigenvalue
   vector&               rcondv                   // vector of reciprocal condition numbers for each eigenvector
   );

Computing for type matrix<float>

bool  matrixf::EigenSolver2X(
   matrixf&              B,                       // second matrix in the pair
   ENUM_EIG_BALANCE      balance,                 // input matrix balancing method
   ENUM_EIG_VECTORS      jobv,                    // determines computation of right and left eigenvectors
   ENUM_EIG_SENSE        sense,                   // determines computation of reciprocal condition numbers
   vectorcf&             alpha,                   // vector of computed eigenvalues
   vectorf&              beta,                    // vector of eigenvalue divisors
   matrixf&              left_eigenvectors,       // matrix of computed left vectors
   matrixf&              right_eigenvectors       // matrix of computed right vectors  
   matrixf&              shur_matrix1,            // the first part of the real Schur form of the "balanced" versions of the input A and B
   matrixf&              shur_matrix2,            // the second part of the real Schur form of the "balanced" versions of the input A and B
   long&                 ilo,                     // subscript of balanced matrix
   long&                 ihi,                     // superscript of balanced matrix
   vectorf&              lscale,                  // details of the permutations and scaling factors applied to the left side of A and B
   vectorf&              rscale,                  // details of the permutations and scaling factors applied to the right side of A and B
   float&                ab_norm,                 // one-norm of balanced input matrix
   float&                bb_norm,                 // one-norm of balanced second matrix B
   vectorf&              rconde,                  // vector of reciprocal condition numbers for each eigenvalue
   vectorf&              rcondv                   // vector of reciprocal condition numbers for each eigenvector
   );

Computing for type matrix<complex>

bool  matrixc::EigenSolver2X(
   matrixc&              B,                       // second matrix in the pair
   ENUM_EIG_BALANCE      balance,                 // input matrix balancing method
   ENUM_EIG_VECTORS      jobv,                    // determines computation of right and left eigenvectors
   ENUM_EIG_SENSE        sense,                   // determines computation of reciprocal condition numbers
   vectorc&              alpha,                   // vector of computed eigenvalues
   vectorc&              beta,                    // vector of eigenvalue divisors
   matrixc&              left_eigenvectors,       // matrix of computed left vectors
   matrixc&              right_eigenvectors       // matrix of computed right vectors  
   matrixc&              shur_matrix1,            // the first part of the real Schur form of the "balanced" versions of the input A and B
   matrixc&              shur_matrix2,            // the second part of the real Schur form of the "balanced" versions of the input A and B
   long&                 ilo,                     // subscript of balanced matrix
   long&                 ihi,                     // superscript of balanced matrix
   vector&               lscale,                  // details of the permutations and scaling factors applied to the left side of A and B
   vector&               rscale,                  // details of the permutations and scaling factors applied to the right side of A and B
   double&               ab_norm,                 // one-norm of balanced input matrix
   double&               bb_norm,                 // one-norm of balanced second matrix B
   vector&               rconde,                  // vector of reciprocal condition numbers for each eigenvalue
   vector&               rcondv                   // vector of reciprocal condition numbers for each eigenvector 
   );

Computing for type matrix<complexf>

bool  matrixcf::EigenSolver2X(
   matrixcf&             B,                       // second matrix in the pair
   ENUM_EIG_BALANCE      balance,                 // input matrix balancing method
   ENUM_EIG_VECTORS      jobv,                    // determines computation of right and left eigenvectors
   ENUM_EIG_SENSE        sense,                   // determines computation of reciprocal condition numbers
   vectorcf&             alpha,                   // vector of computed eigenvalues
   vectorf&              beta,                    // vector of eigenvalue divisors
   matrixcf&             left_eigenvectors,       // matrix of computed left vectors
   matrixcf&             right_eigenvectors       // matrix of computed right vectors  
   matrixcf&             shur_matrix1,            // the first part of the real Schur form of the "balanced" versions of the input A and B
   matrixcf&             shur_matrix2,            // the second part of the real Schur form of the "balanced" versions of the input A and B
   long&                 ilo,                     // subscript of balanced matrix
   long&                 ihi,                     // superscript of balanced matrix
   vectorf&              lscale,                  // details of the permutations and scaling factors applied to the left side of A and B
   vectorf&              rscale,                  // details of the permutations and scaling factors applied to the right side of A and B
   float&                ab_norm,                 // one-norm of balanced input matrix
   float&                bb_norm,                 // one-norm of balanced second matrix B
   vectorf&              rconde,                  // vector of reciprocal condition numbers for each eigenvalue
   vectorf&              rcondv                   // vector of reciprocal condition numbers for each eigenvector 
   );

Parameters

B

[in]  The second matrix in the pair.

balance

[in]  Value from the ENUM_EIG_BALANCE enumeration which determines the need and method for balancing the input matrix; it is used to improve the conditioning of the eigenvalues and eigenvectors.

jobv

[in]   Value from the ENUM_EIG_VECTORS enumeration which determines the method for computing left and right eigenvectors.

sense

[in]   Value from the ENUM_EIG_SENSE enumeration which determines the need to compute reciprocal condition numbers.

eigen_values

[out] Vector of eigenvalues.

left_eigenvectors

[out] Matrix of left eigenvectors.

right_eigenvectors

[out] Matrix of right eigenvectors.

shur_matrix1, shur_matrix2

[out]  2 parts of balanced matrix in Schur form; the matrix is not filled if neither left nor right eigenvectors are computed.

ilo

[out]  Subscript of the balanced matrix; the matrix is not filled if no balancing is applied.

ihi

[out]  Superscript of the balanced matrix; the matrix is not filled if no balancing is applied.

lscale

[out]  Vector contains details of the permutations and scaling factors applied to the left side of A and B.

If PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then

lscale(j) = PL(j), for j = 1,..., ilo-1

= DL(j), for j = ilo,...,ihi

= PL(j) for j = ihi+1,..., n.

The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.

rscale

[out]  Vector contains details of the permutations and scaling factors applied to the right side of A and B.

If PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then

rscale(j) = PR(j), for j = 1,..., ilo-1

= DR(j), for j = ilo,...,ihi

= PR(j) for j = ihi+1,..., n.

The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.

ab_norm

[out]  One-norm of the balanced input matrix (the maximum of the sum of absolute values of elements in any of the matrix columns).

bb_norm

[out]  One-norm of balanced second matrix B.

rconde

[out]  Vector of reciprocal condition numbers for each eigenvalue; it is computed if the 'sense' parameter is set to 'E' or 'B'.

rcondv

[out]  Vector of reciprocal condition numbers for each eigenvector; it is computed if the 'sense' parameter is set to 'V' or 'B'.

Return Value

The function returns 'true' on success or 'false' if an error occurs.

Note

Input matrices balancing depends on the value of the balance parameter.

ENUM_EIG_BALANCE

An enumeration that specifies whether the matrices should be balanced.

ID

Description

EIGBALANCE_N

Do not diagonally scale or permute.

EIGBALANCE_P

Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.

EIGBALANCE_S

Diagonally scale the matrix. Do not permute.

EIGBALANCE_B      

Both diagonally scale and permute.

ENUM_EIG_VECTORS

An enumeration defining the need to compute eigenvectors.

ID

Description

EIGVECTORS_N

Only eigenvalues are computed, without vectors.

EIGVECTORS_L

Only left eigenvectors are computed.

EIGVECTORS_R

Only right eigenvectors are computed.

EIGVECTORS_LR

Left and right eigenvectors are computed, eigenvalues are always computed.

 

ENUM_EIG_SENSE

An enumeration determining the need to compute reciprocal condition numbers.

ID

Description

EIGSENSE_N

None of reciprocal condition numbers are computed.

EIGSENSE_E

Computed for eigenvalues only.

EIGSENSE_V

Computed for right eigenvectors only.

EIGSENSE_B      

Computed for eigenvalues and right eigenvectors.