Lapack Utilities

tgsyl!(A, B, C, D, E, F) -> (C, F, scale)

Solve the Sylvester system of matrix equations

  AX - YB = scale*C
  DX - YE = scale*F ,

where X and Y are unknown matrices, the pairs (A, D), (B, E) and (C, F) have the same sizes, and the pairs (A, D) and (B, E) are in generalized (real) Schur canonical form, i.e. A, B are upper quasi triangular and D, E are upper triangular. Returns X (overwriting C), Y (overwriting F) and scale.

tgsyl!(trans, A, B, C, D, E, F) -> (C, F, scale)

Solve for trans = 'T' and real matrices or for trans = 'C' and complex matrices, the (adjoint) Sylvester system of matrix equations

  A'X + D'Y = scale*C
  XB' + YE' = scale*(-F) .

tgsyl!('N', A, B, C, D, E, F) corresponds to the call tgsyl!(A, B, C, D, E, F).

Interface to the LAPACK subroutines DTGSYL/STGSYL/ZTGSYL/CTGSYL.

lanv2(A, B, C, D) -> (RT1R, RT1I, RT2R, RT2I, CS, SN)

Compute the Schur factorization of a real 2-by-2 nonsymmetric matrix [A,B;C,D] in standard form. A, B, C, D are overwritten on output by the corresponding elements of the standardised Schur form. RT1R+im*RT1I and RT2R+im*RT2I are the resulting eigenvalues. CS and SN are the parameters of the rotation matrix. Interface to the LAPACK subroutines DLANV2/SLANV2.

lag2(A, B, SAFMIN) -> (SCALE1, SCALE2, WR1, WR2, WI)

Compute the eigenvalues of a 2-by-2 generalized real eigenvalue problem for the matrix pair (A,B), with scaling as necessary to avoid over-/underflow. SAFMIN is the smallest positive number s.t. 1/SAFMIN does not overflow. If WI = 0, WR1/SCALE1 and WR2/SCALE2 are the resulting real eigenvalues, while if WI <> 0, then (WR1+/-im*WI)/SCALE1 are the resulting complex eigenvalues. Interface to the LAPACK subroutines DLAG2/SLAG2.

ladiv(A, B, C, D) -> (P, Q)

Perform the complex division in real arithmetic

$P + iQ = \displaystyle\frac{A+iB}{C+iD}$

by avoiding unnecessary overflow. Interface to the LAPACK subroutines DLADIV/SLADIV.

lacn2!(V, X, ISGN, EST, KASE, ISAVE ) -> (EST, KASE )

Estimate the 1-norm of a real linear operator A, using reverse communication by applying the operator or its transpose/adjoint to a vector. KASE is a parameter to control the norm evaluation process as follows. On the initial call, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether the real vector X should be overwritten by A * X or A' * X at the next call. On the final return, KASE will again be 0 and EST is an estimate (a lower bound) for the 1-norm of A. V is a real work vector, ISGN is an integer work vector and ISAVE is a 3-dimensional integer vector used to save information between the calls. Interface to the LAPACK subroutines DLACN2/SLACN2.

lacn2!(V, X, EST, KASE, ISAVE ) -> (EST, KASE )

Estimate the 1-norm of a complex linear operator A, using reverse communication by applying the operator or its adjoint to a vector. KASE is a parameter to control the norm evaluation process as follows. On the initial call, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether the complex vector X should be overwritten by A * X or A' * X at the next call. On the final return, KASE will again be 0 and EST is an estimate (a lower bound) for the 1-norm of A. V is a complex work vector and ISAVE is a 3-dimensional integer vector used to save information between the calls. Interface to the LAPACK subroutines ZLACN2/CLACN2.