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eq.ijs
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eq.ijs
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NB. Eigenvalues and Schur form
NB.
NB. hgexxe Eigenvalues of pair of structured matrices
NB. hgexxs Eigenvalues and the Schur form of pair of
NB. structured matrices
NB.
NB. testhgeq Test hgexxxxx by square matrices
NB. testeq Adv. to make verb to test hgexxxxx by matrices
NB. of generator and shape given
NB.
NB. Copyright 2010,2011,2013,2017,2018,2020,2021,2023,2024
NB. Igor Zhuravlov
NB.
NB. This file is part of mt
NB.
NB. mt is free software: you can redistribute it and/or
NB. modify it under the terms of the GNU Lesser General
NB. Public License as published by the Free Software
NB. Foundation, either version 3 of the License, or (at your
NB. option) any later version.
NB.
NB. mt is distributed in the hope that it will be useful, but
NB. WITHOUT ANY WARRANTY; without even the implied warranty
NB. of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
NB. See the GNU Lesser General Public License for more
NB. details.
NB.
NB. You should have received a copy of the GNU Lesser General
NB. Public License along with mt. If not, see
NB. <http://www.gnu.org/licenses/>.
NB. =========================================================
NB. Configuration
coclass 'mt'
NB. =========================================================
NB. Local definitions
NB. ---------------------------------------------------------
NB. laqr1
NB.
NB. Description:
NB. 1st column of (H-s1*I)*(H-s2*I)
NB.
NB. Syntax:
NB. vK=. (s1,s2) laqr1 H
NB. where
NB. H - 2×2-matrix or 3×3-matrix
NB.
NB. Notes:
NB. - implements LAPACK's xLAQR1
NB.
NB. TODO:
NB. - tacit
laqr1=: 4 : 0
's1 s2'=. x
((-&s1 upddiag ]) (mp (% norm1t)) ((-&s2 updl 0) ({."1) ])) y
)
NB. ---------------------------------------------------------
NB. hgexxeo
NB.
NB. Description:
NB. Compute generalized eigenvalues of hs-segment
NB.
NB. Syntax:
NB. 'HTupd signbc'=. hs hgexxeo H ,: T
NB. where
NB. hs - 2-vector of integers (h,s) 'head' and 'size',
NB. defines eigenvalues range
NB. H - n×n-matrix, either HSL or HSU inside the
NB. submatrix H[h:h+s-1,h:h+s-1], and L or U
NB. outside
NB. T - n×n-matrix, either L or U
NB. HTupd -:Hupd ,: Tupd
NB. Hupd - H with hs-segment of diagonal replaced by
NB. alpha (see hgexx)
NB. Tupd - T with hs-segment of diagonal replaced by
NB. beta (see hgexx)
NB. signbc - s-vector, scaling factors to form Q,Z later
hgexxeo=: 4 : 0
'Hd Td'=. (0 , x) diag"2 y
absb=. | Td
'signbc Td'=. (,:~ FP_SFMIN < absb)} 1 0 2 |: (1 ,: + * Td) ,: 0 ,: absb
((((Hd * signbc) ,: Td) (;"1) 0 , x) setdiag"1 2 y) ; signbc
)
NB. ---------------------------------------------------------
NB. hgezqso
NB. hgeqzso
NB.
NB. Description:
NB. Compute generalized eigenvalues of hs-segment and
NB. reduce corresponding rows (hgezqso) or columns
NB. (hgeqzso) to generalized Schur form
NB.
NB. Syntax:
NB. 'HTupd signbc'=. hs hgexxso H ,: T
NB. where
NB. hs - 2-vector of integers (h,s) 'head' and 'size',
NB. defines eigenvalues range
NB. H - n×n-matrix, either lower (hgezqso) or upper
NB. (hgeqzso) Hessenberg inside the submatrix
NB. H[h:h+s-1,h:h+s-1], and lower (hgezqso) or
NB. upper (hgeqzso) triangular outside
NB. T - n×n-matrix, either lower (hgezqso) or upper
NB. (hgeqzso) triangular
NB. HTupd -:Hupd ,: Tupd
NB. Hupd - H with rows (hgezqso) or columns (hgeqzso)
NB. from hs-segment transformed to Shur form
NB. Tupd - T with rows (hgezqso) or columns (hgeqzso)
NB. from hs-segment transformed to Shur form
NB. signbc - s-vector, scaling factors to form Q,Z later
hgezqso=: 4 : 0
liso=. liso4dhs x
'y signbc'=. x hgexxeo y
subHT=. liso ({"2) y
((((,:~) liso >/ (i. c y))} subHT ,: subHT (*"2) signbc) liso}"2 y) ; signbc
)
hgeqzso=: 4 : 0
liso=. liso4dhs x
'y signbc'=. x hgexxeo y
subHT=. liso ({"1) y
((((,:~) (i. c y) </ liso)} subHT ,: subHT (*"1) signbc) liso}"1 y) ; signbc
)
NB. ---------------------------------------------------------
NB. hgezq
NB. hgeqz
NB.
NB. Description:
NB. Adv. to make dyad to find eigenvalues of either lower
NB. (hgezq) or upper (hgeqz) Hessenberg-triangular pair
NB. (H,T) and, optionally, to reduce this pair to
NB. generalized Schur form
NB.
NB. Syntax:
NB. 'HTupd dQ1 dZ1'=. hs (hgexxxo`init`reset`step hgexx) HT
NB. where
NB. hgexxxo - monad to compute generalized eigenvalues of
NB. hs-segment and, optionally, to reduce
NB. corresponding columns to generalized Schur
NB. form, is either hgezqso (hgezq only), hgeqzso
NB. (hgeqz only) or hgexxeo, is called as:
NB. 'HTupd signbc'=. hgexxxo hs ; HT
NB. init - dyad to initialize counters, is called as:
NB. 'ifrstm ilastm'=. (h , ilast) init (0 , n-1)
NB. reset - dyad to reset counters optionally, is called
NB. as:
NB. 'ifrstm ilastm'=. (h , ilast) reset (ifrstm , ilastm)
NB. step - monad to change counter optionally, is
NB. called as:
NB. ifrstm=. ifirst step ifrstm
NB.
NB. Notes:
NB. - refer to hgeqzxxxx below for parameters description
NB. - non-converged eigenvalues are set to NaN
hgezq=: 1 : 0
:
'`hgezqxo init reset step'=. m
e=. +/ 'h s'=. x
dQ1=. dZ1=. ((0 4 $ 0))
abnorm=. (0 2 ,. ,.~ x) norms"2;.0 y
'atol btol'=. abtol=. FP_SFMIN >. FP_PREC * abnorm
'ascale bscale'=. abscale=. % FP_SFMIN >. abnorm
'y signbc'=. ((c y) (] , -) e) hgezqxo y NB. process eigenvalues (columns) h+s:n-1
dZ1=. dZ1 , 4 ({."1) signbc ,. (c y) liso4th e
NB. Eigenvalues h+s:n-1 have been found.
NB. Initialize dynamic indices
ilast=. <: e
'ifrstm ilastm'=. (h , ilast) init (0 , <: c y)
NB. ifrstm - the column of the last
NB. splitting column to the left
NB. of the column ilast, this is
NB. always at least h
iiter=. 0 NB. counts iterations since the last
NB. eigenvalue was found, to tell when to
NB. use an extraordinary shift
eshift=. 0
maxit=. 30 * s NB. the maximum number of ZQ sweep allowed
jiter=. 0
NB. Main ZQ iteration loop
NB. Row operations modify columns ifrstm:*
NB. Column operations modify rows *:ilastm
while. jiter < maxit do.
goto60=. 1 NB. set default branching
NB. split the matrix if possible, by two tests:
NB. 1. H[j-1,j]=0 OR j=h
NB. 2. T[j,j]=0
if. ilast ~: h do.
if. atol < sorim (< 0 , ilast - 1 0) { y do.
if. btol < | (< 1 , ,~ ilast) { y do.
NB. general case: j < ilast
j=. <: ilast
while. j >: h do.
NB. test 1: H[j-1,j]=0 OR j=h
if. j = h do.
ilazro=. 1
elseif. atol >: sorim (< 0 , j - 1 0) { y do.
y=. 0 (< 0 , j - 1 0)} y
ilazro=. 1
else.
ilazro=. 0
end.
NB. test 2: T[j,j]=0
if. btol > | (< 1 , ,~ j) { y do.
y=. 0 (< 1 , ,~ j)} y
NB. test 2a: check for 2 consecutive small
NB. superdiagonals in H
ilazr2=. 0
if. -. ilazro do.
'Hj1j Hjj1 Hjj'=. sorim (< 0 ,. ((_1 0,0 1,:0 0)) + j) { y
if. 0 >: (Hj1j , Hjj) mp (ascale * (Hjj1 , -atol)) do.
ilazr2=. 1
end.
end.
NB. If both tests (1 & 2) pass, i.e., the
NB. leading diagonal element of T in the block
NB. is zero, then split a 1x1 block off at the
NB. left, i.e. at the j-th row/column. The
NB. leading diagonal element of the remainder
NB. can also be zero, so this may have to be
NB. done repeatedly.
if. ilazro +. ilazr2 do.
jch=. j
liso=. (>: ilastm) liso4th j
while. jch < ilast do.
'y cs'=. rot rotga y ; (< 0 ; liso ; (jch + 0 1)) ; 0
liso=. }. liso
y=. cs&rot&.((< 1 ; liso ; (jch + 0 1))&{) y
dZ1=. dZ1 , (+ cs) , jch + 0 1
if. ilazr2 do.
y=. *&({. cs)&.((< 0 , jch - 1 0)&{) y
ilazr2=. 0
end.
if. btol <: sorim (< 1 , jch + 1 1) { y do.
if. ilast > >: jch do.
ifirst=. >: jch
goto60=. 0
end.
goto_l60.
end.
y=. 0 (< 1 , jch + 1 1)} y
jch=. >: jch
end.
else.
NB. Only test 2 passed - chase the zero to
NB. T[ilast,ilast], then process as in the
NB. case T[ilast,ilast]=0
jch=. j
lisor=. (>: ilastm) liso4th <: j
lisoc=. (2 + j) liso4th ifrstm
while. jch < ilast do.
'y cs'=. rot rotga y ; (< 1 ; (2 }. lisor) ; (jch + 0 1)) ; 0
y=. cs&rot&.((< 0 ; lisor ; (jch + 0 1))&{) y
dZ1=. dZ1 , (+ cs) , jch + 0 1
'y cs'=. rot&.|: rotga y ; (< 0 ; (jch - 0 1) ; lisoc) ; ((< < a: ; _1))
y=. cs&(rot&.|:)&.((< 1 ; (jch - 0 1) ; (_2 }. lisoc))&{) y
dQ1=. dQ1 , cs , jch - 0 1
lisor=. }. lisor
lisoc=. lisoc , 2 + jch
jch=. >: jch
end.
end.
goto_l50.
elseif. ilazro do.
ifirst=. j
goto60=. 0
goto_l60.
end.
j=. <: j
end.
NB. drop-through is impossible
((_. ; '') setdiag"2 y) ; ,~ a: NB. set all eigenvalues to NaN
return.
else.
y=. 0 (< 1 , ,~ ilast)} y
end.
label_l50.
NB. T[ilast,ilast]=0 - clear H[ilast-1,ilast] to
NB. split off a 1x1 block
liso=. (>: ilast) liso4th ifrstm
'y cs'=. rot&.|: rotga y ; (< 0 ; (ilast - 0 1) ; liso) ; ((< < a: ; _1))
y=. cs&(rot&.|:)&.((< 1 ; (ilast - 0 1) ; (}: liso))&{) y
dQ1=. dQ1 , cs , ilast - 0 1
else.
y=. 0 (< 0 , ilast - 1 0)} y
end.
end.
label_l60.
if. goto60 do.
NB. H[ilast-1,ilast]=0 - standartize B, set alpha and
NB. beta
'y signbc'=. (ilast , 1) hgezqxo y NB. process ilast-th eigenvalue (column)
dQ1=. dQ1 , 4 ({."1) signbc ,. ilast
NB. goto next block - exit if finished
ilast=. <: ilast
if. ilast < h do.
NB. normal exit
'y signbc'=. (0 , 0 >. <: h) hgezqxo y NB. process eigenvalues (columns) 0:h-1
dQ1=. dQ1 , 4 ({."1) signbc ,. i. 0 >. <: h
y ; dQ1 ; dZ1
return.
end.
NB. reset counters
iiter=. 0
eshift=. 0
'ifrstm ilastm'=. (h , ilast) reset (ifrstm , ilastm)
else.
NB. ZQ step
NB. This iteration only involves rows/columns
NB. ifirst:ilast. We assume ifirst<ilast, and that the
NB. diagonal of B is non-zero
iiter=. >: iiter
ifrstm=. ifirst step ifrstm
NB. Compute the shift.
NB. At this point, ifirst<ilast, and the diagonal
NB. elements of T[ifirst:ilast,ifirst:ilast] are larger
NB. than btol in magnitude
if. 10 | iiter do.
NB. The Wilkinson shift, i.e., the eigenvalues of the
NB. bottom-right 2x2 block of T^_1*H which is nearest
NB. to the bottom-right element.
NB. We factor T as D*L, where L is unit lower
NB. triangular, and compute L^_1*(D^_1*H)
'L21 DA11 DA12 DA21 DA22'=. %/ ((6 0 1 2 3 ,: 7 4 4 7 7)) ({,) abscale * ((< a: ; ;~ ilast - 1 0) { y)
IBA22=. DA22 - L21 * DA12
t1=. -: DA11 + IBA22
rtdisc=. %: (t1 , DA21 , -DA11) mp (t1 , DA12 , DA22)
temp=. +/!.0 */ +. rtdisc , t1 - IBA22
shift=. t1 - temp negneg rtdisc
else.
NB. Exceptional shift. Chosen for no paticularly good
NB. reason
eshift=. eshift + + %/ abscale * (;/ 0 1 ,. ((0 _1 ,: _1 _1)) + ilast) { y
shift=. eshift
end.
NB. now check for two consecutive small subdiagonals
HTd=. (0 1 (,"0 1) ilast (] , -) ifirst) diag"1 2/ y
ctemp=. (- (shift&*))/ abscale * {. HTd
temp=. (sorim }."1 ctemp) ,: ascale * sorim ((< 1 ; 0 ; <<0)) { HTd
tempr=. >./ temp
temp=. temp %"1 ((0 , 1 - FP_EPS) I. tempr)} 1 , tempr ,: 1
'istart ctemp'=. (+&ifirst , {&ctemp) (ilast - ifirst) | >: (>:/ temp * atol ,: sorim ((< 1 ; 0 ; <<_1)) { HTd) i: 1
NB. do an implicit-shift ZQ sweep
NB. initial Z
cs=. lartg ctemp , ascale * (< 0 , istart + 0 1) { y
NB. sweep
j=. istart
lisor=. (>: ilastm) liso4th j
lisoc=. (j + 2) liso4th ifrstm
while. j < ilast do.
liso=. j + 0 1
NB. is a first iteration?
if. j = istart do.
y=. cs&rot"2&.((< a: ; lisor ; liso)&{) y
else.
'y cs'=. rot rotga y ; (< 0 ; lisor ; liso) ; 0
lisor=. }. lisor
y=. cs&rot&.((< 1 ; lisor ; liso)&{) y
end.
dZ1=. dZ1 , (+ cs) , liso
liso=. j + 1 0
'y cs'=. rot&.|: rotga y ; (< 1 ; liso ; lisoc) ; ((< < a: ; _1))
NB. isn't a last iteration?
if. j < <: ilast do.
lisoc=. lisoc , j + 2
end.
y=. cs&(rot&.|:)&.((< 0 ; liso ; lisoc)&{) y
dQ1=. dQ1 , cs , liso
j=. >: j
end.
end.
jiter=. >: jiter
end.
NB. drop-through means non-convergence, set incorrect eigenvalues 0:ilast to NaN
((_. ; 0 0 , >: ilast) setdiag"2 y) ; dQ1 ; dZ1
)
hgeqz=: 1 : 0
:
'`hgeqzxo init reset step'=. m
e=. +/ 'h s'=. x
dQ1=. dZ1=. ((0 4 $ 0))
abnorm=. (0 2 ,. ,.~ x) norms"2;.0 y
'atol btol'=. abtol=. FP_SFMIN >. FP_PREC * abnorm
'ascale bscale'=. abscale=. % FP_SFMIN >. abnorm
'y signbc'=. ((c y) (] , -) e) hgeqzxo y NB. process eigenvalues (columns) h+s:n-1
dZ1=. dZ1 , 4 ({."1) signbc ,. (c y) liso4th e
NB. Eigenvalues h+s:n-1 have been found.
NB. Initialize dynamic indices
ilast=. <: e
'ifrstm ilastm'=. (h , ilast) init (0 , <: c y)
NB. ifrstm - the row of the last splitting
NB. row above row ilast, this is
NB. always at least h
iiter=. 0 NB. counts iterations since the last
NB. eigenvalue was found, to tell when to
NB. use an extraordinary shift
eshift=. 0
maxit=. 30 * s NB. the maximum number of QZ sweep allowed
jiter=. 0
NB. Main QZ iteration loop
NB. Column operations modify rows ifrstm:*
NB. Row operations modify columns *:ilastm
while. jiter < maxit do.
goto60=. 1 NB. set default branching
NB. split the matrix if possible, by to tests:
NB. 1. H[j,j-1]=0 OR j=h
NB. 2. T[j,j]=0
if. ilast ~: h do.
if. atol < sorim (< 0 , ilast - 0 1) { y do.
if. btol < | (< 1 , ,~ ilast) { y do.
NB. general case: j < ilast
j=. <: ilast
while. j >: h do.
NB. test 1: H[j,j-1]=0 OR j=h
if. j = h do.
ilazro=. 1
elseif. atol >: sorim (< 0 , j - 0 1) { y do.
y=. 0 (< 0 , j - 0 1)} y
ilazro=. 1
else.
ilazro=. 0
end.
NB. test 2: T[j,j]=0
if. btol > | (< 1 , ,~ j) { y do.
y=. 0 (< 1 , ,~ j)} y
NB. test 2a: check for 2 consecutive small
NB. subdiagonals in H
ilazr2=. 0
if. -. ilazro do.
'Hjj1 Hj1j Hjj'=. sorim (< 0 ,. ((0 _1,1 0,:0 0)) + j) { y
if. 0 >: (Hjj1 , Hjj) mp (ascale * (Hj1j , -atol)) do.
ilazr2=. 1
end.
end.
NB. If both tests (1 & 2) pass, i.e., the
NB. leading diagonal element of T in the block
NB. is zero, then split a 1x1 block off at the
NB. top, i.e. at the j-th row/column. The
NB. leading diagonal element of the remainder
NB. can also be zero, so this may have to be
NB. done repeatedly.
if. ilazro +. ilazr2 do.
jch=. j
liso=. (>: ilastm) liso4th j
while. jch < ilast do.
'y cs'=. rot&.|: rotga y ; (< 0 ; (jch + 0 1) ; liso) ; ((< < a: ; 0))
liso=. }. liso
y=. cs&(rot&.|:)&.((< 1 ; (jch + 0 1) ; liso)&{) y
dQ1=. dQ1 , (+ cs) , jch + 0 1
if. ilazr2 do.
y=. *&({. cs)&.((< 0 , jch - 0 1)&{) y
ilazr2=. 0
end.
if. btol <: sorim (< 1 , jch + 1 1) { y do.
if. ilast > >: jch do.
ifirst=. >: jch
goto60=. 0
end.
goto_u60.
end.
y=. 0 (< 1 , jch + 1 1)} y
jch=. >: jch
end.
else.
NB. Only test 2 passed - chase the zero to
NB. T[ilast,ilast], then process as in the
NB. case T[ilast,ilast]=0
jch=. j
lisoc=. (>: ilastm) liso4th <: j
lisor=. (2 + j) liso4th ifrstm
while. jch < ilast do.
'y cs'=. rot&.|: rotga y ; (< 1 ; (jch + 0 1) ; (2 }. lisoc)) ; ((< < a: ; 0))
y=. cs&(rot&.|:)&.((< 0 ; (jch + 0 1) ; lisoc)&{) y
dQ1=. dQ1 , (+ cs) , jch + 0 1
'y cs'=. rot rotga y ; (< 0 ; lisor ; (jch - 0 1)) ; _1
y=. cs&rot&.((< 1 ; (_2 }. lisor) ; (jch - 0 1))&{) y
dZ1=. dZ1 , cs , jch - 0 1
lisoc=. }. lisoc
lisor=. lisor , 2 + jch
jch=. >: jch
end.
end.
goto_u50.
elseif. ilazro do.
ifirst=. j
goto60=. 0
goto_u60.
end.
j=. <: j
end.
NB. drop-through is impossible
(((_. ; '')) setdiag"2 y) ; ((,~ a:)) NB. set all eigenvalues to NaN
return.
else.
y=. 0 (< 1 , ,~ ilast)} y
end.
label_u50.
NB. T[ilast,ilast]=0 - clear H[ilast,ilast-1] to
NB. split off a 1x1 block
liso=. (>: ilast) liso4th ifrstm
'y cs'=. rot rotga y ; (< 0 ; liso ; (ilast - 0 1)) ; _1
y=. cs&rot&.((< 1 ; (}: liso) ; (ilast - 0 1))&{) y
dZ1=. dZ1 , cs , ilast - 0 1
else.
y=. 0 (< 0 , ilast - 0 1)} y
end.
end.
label_u60.
if. goto60 do.
NB. H[ilast,ilast-1]=0 - standartize B, set alpha and
NB. beta
'y signbc'=. (ilast , 1) hgeqzxo y NB. process ilast-th eigenvalue (column)
dZ1=. dZ1 , 4 ({."1) signbc ,. ilast
NB. goto next block - exit if finished
ilast=. <: ilast
if. ilast < h do.
NB. normal exit
'y signbc'=. (0 , 0 >. <: h) hgeqzxo y NB. process eigenvalues (columns) 0:h-1
dZ1=. dZ1 , 4 ({."1) signbc ,. i. 0 >. <: h
y ; dQ1 ; dZ1
return.
end.
NB. reset counters
iiter=. 0
eshift=. 0
'ifrstm ilastm'=. (h , ilast) reset (ifrstm , ilastm)
else.
NB. QZ step
NB. This iteration only involves rows/columns
NB. ifirst:ilast. We assume ifirst<ilast, and that the
NB. diagonal of B is non-zero
iiter=. >: iiter
ifrstm=. ifirst step ifrstm
NB. Compute the shift.
NB. At this point, ifirst<ilast, and the diagonal
NB. elements of T[ifirst:ilast,ifirst:ilast] are larger
NB. than btol in magnitude
if. 10 | iiter do.
NB. The Wilkinson shift, i.e., the eigenvalues of the
NB. bottom-right 2x2 block of H*T^_1 which is nearest
NB. to the bottom-right element.
NB. We factor T as U*D, where U is unit upper
NB. triangular, and compute (H*D^_1)*U^_1
'U12 AD11 AD21 AD12 AD22'=. %/ ((5 0 2 1 3 ,: 7 4 4 7 7)) ({,) abscale * ((< a: ; ;~ ilast - 1 0) { y)
ABI22=. AD22 - U12 * AD21
t1=. -: AD11 + ABI22
rtdisc=. %: (t1 , AD12 , -AD11) mp (t1 , AD21 , AD22)
temp=. (+/!.0) */ +. rtdisc , t1 - ABI22
shift=. t1 - temp negneg rtdisc
else.
NB. Exceptional shift. Chosen for no paticularly good
NB. reason
eshift=. eshift + + %/ abscale * (;/ 0 1 ,. ((_1 0 ,: _1 _1)) + ilast) { y
shift=. eshift
end.
NB. now check for two consecutive small subdiagonals
HTd=. (0 _1 (,"0 1) ilast (] , -) ifirst) diag"1 2/ y
ctemp=. (- (shift&*))/ abscale * {. HTd
temp=. (sorim }."1 ctemp) ,: ascale * sorim ((< 1 ; 0 ; <<0)) { HTd
tempr=. (>./) temp
temp=. temp %"1 ((0 , 1 - FP_EPS) I. tempr)} 1 , tempr ,: 1
'istart ctemp'=. (+&ifirst , {&ctemp) (ilast - ifirst) | >: (>:/ temp * atol ,: sorim ((< 1 ; 0 ; <<_1)) { HTd) i: 1
NB. do an implicit-shift QZ sweep
NB. initial Q
cs=. lartg ctemp , ascale * (< 0 , istart + 1 0) { y
NB. sweep
j=. istart
lisoc=. (>: ilastm) liso4th j
lisor=. (j + 2) liso4th ifrstm
while. j < ilast do.
liso=. j + 0 1
NB. is a first iteration?
if. j = istart do.
y=. cs&(rot&.|:)"2&.((< a: ; liso ; lisoc)&{) y
else.
'y cs'=. rot&.|: rotga y ; (< 0 ; liso ; lisoc) ; ((< < a: ; 0))
lisoc=. }. lisoc
y=. cs&(rot&.|:)&.((< 1 ; liso ; lisoc)&{) y
end.
dQ1=. dQ1 , (+ cs) , liso
liso=. j + 1 0
'y cs'=. rot rotga y ; (< 1 ; lisor ; liso) ; _1
NB. isn't a last iteration?
if. j < <: ilast do.
lisor=. lisor , j + 2
end.
y=. cs&rot&.((< 0 ; lisor ; liso)&{) y
dZ1=. dZ1 , cs , liso
j=. >: j
end.
end.
jiter=. >: jiter
end.
NB. drop-through means non-convergence, set incorrect eigenvalues 0:ilast to NaN
((_. ; 0 0 , >: ilast) setdiag"2 y) ; dQ1 ; dZ1
)
NB. ---------------------------------------------------------
NB. hgezqe
NB. hgezqs
NB. hgeqze
NB. hgeqzs
NB.
NB. Description:
NB. Shortcuts, see hgeqzxxxx
NB.
NB. Syntax:
NB. 'HTupd dQ1 dZ1'=. hs hgexxx HT
hgezqe=: hgexxeo`[`(2 1&{@,`[@.((<{:)~{.))`[ hgezq
hgezqs=: hgezqso`]`] `] hgezq
hgeqze=: hgexxeo`[`(2 1&{@,`[@.((<{:)~{.))`[ hgeqz
hgeqzs=: hgeqzso`]`] `] hgeqz
NB. =========================================================
NB. Interface
NB. ---------------------------------------------------------
NB. hgezqenn
NB. hgezqenv
NB. hgezqevn
NB. hgezqevv
NB. hgezqsnn
NB. hgezqsnv
NB. hgezqsvn
NB. hgezqsvv
NB.
NB. Description:
NB. Compute eigenvalues (hgezqxxx) and reduce to
NB. generalized Schur form (hgezqsxx):
NB. dQ1^H * S * dZ1 = H
NB. dQ1^H * P * dZ1 = T
NB. the generalized lower Hessenberg form (H,T) using
NB. single-shift ZQ method. Matrix pairs of this type are
NB. produced by the reduction to generalized lower
NB. Hessenberg form of a matrix pair (A,B):
NB. Q1^H * H * Z1 = A
NB. Q1^H * T * Z1 = B
NB. as computed by gghrdlxx. The unitary (orthogonal)
NB. matrices dQ1 and dZ1 may either be formed explicitly,
NB. or they may be postmultiplied by input matrices Q1 and
NB. Z1, so that:
NB. (dQ1*Q1)^H * S * (dZ1*Z1) = Q1^H * H * Z1
NB. (dQ1*Q1)^H * P * (dZ1*Z1) = Q1^H * T * Z1
NB. To avoid overflow, eigenvalues of the matrix pair (H,T)
NB. (equivalently, of (A,B)) are computed as a pair of
NB. values. Each i-th eigenvector (row) from L and R has a
NB. corresponding eigenvalue represented as a pair of i-th
NB. elements from vectors e1 and e2:
NB. E1=. diagmat e1=. diag S
NB. E2=. diagmat e2=. diag P
NB. If E2 is nonsingular then:
NB. E=. diagmat e1%e2
NB. is a diagonal matrix of eigenvalues, and GNEP can be
NB. expressed as:
NB. L * A = E * L * B
NB. A * R^H = B * R^H * E
NB. and if E1 is nonsingular then:
NB. E=. diagmat e2%e1
NB. is a diagonal matrix of eigenvalues, and GNEP can be
NB. expressed as:
NB. E * L * A = L * B
NB. A * R^H * E = B * R^H * E
NB.
NB. Syntax:
NB. e1e2=. hs hgezqenn H ,: T
NB. 'e1e2 Z2'=. hs hgezqenv H , T ,: Z1
NB. 'e1e2 Q2'=. hs hgezqevn H , T ,: Q1
NB. 'e1e2 Q2Z2'=. hs hgezqevv H , T , Q1 ,: Z1
NB. 'S P'=. hs hgezqsnn H ,: T
NB. 'S P Z2'=. hs hgezqsnv H , T ,: Z1
NB. 'S P Q2'=. hs hgezqsvn H , T ,: Q1
NB. 'S P Q2 Z2'=. hs hgezqsvv H , T , Q1 ,: Z1
NB. where
NB. hs - 2-vector of integers (h,s) 'head' and 'size',
NB. defines submatrices H11 and T11 position in H
NB. and T, respectively, see ggballp and gehrdl
NB. H - n×n-matrix, the lower Hessenberg inside the
NB. submatrix H[h:h+s-1,h:h+s-1], and lower
NB. triangular outside
NB. T - n×n-matrix, the lower triangular
NB. e1e2 - 2×n-matrix of eigenvalues e1 and e2:
NB. e1e2 -: e1 ,: e2
NB. Q1 - n×n-matrix, the unitary (orthogonal)
NB. Q2 - n×n-matrix, the unitary (orthogonal), the left
NB. Schur vectors of (H,T) pair if Q1=I, and of
NB. (A,B) pair otherwise
NB. Z1 - n×n-matrix, the unitary (orthogonal)
NB. Z2 - n×n-matrix, the unitary (orthogonal), the right
NB. Schur vectors of (H,T) pair if Z1=I, and of
NB. (A,B) pair otherwise
NB. Q2Z2 -:Q2 ,: Z2
NB. S - n×n-matrix, the lower triangular
NB. P - n×n-matrix, the lower triangular
NB.
NB. Notes:
NB. - non-converged eigenvalues are set to NaN
NB.
NB. Assertions (with appropriate comparison tolerance):
NB. e1e2 -: diag"2 (S ,: P)
NB. Q2 -: dQ1 mp Q1
NB. Z2 -: dZ1 mp Z1
NB. (H ,: T) -: dQ1 (mp~ ct)~"2 (S ,: P) mp"2 dZ1
NB. (C ,: D) -: Q2 (mp~ ct)~"2 (S ,: P) mp"2 Z2
NB. I -: Q2^H * Q2
NB. I -: Z2^H * Z2
NB. where
NB. C - n×n-matrix, general
NB. D - n×n-matrix, general
NB. n=. # C
NB. hs=. 0 , n
NB. I=. idmat n
NB. 'B Z0'=. (trl@:(}:"1) ,: unglq)@gelqf D
NB. A=. C (mp ct) Z0
NB. 'H T Q1 Z1'=. hs gghrdlvv A , B , I ,: Z0
NB. e1e2=. hs hgezqenn H ,: T
NB. 'S P Q2 Z2'=. hs hgezqsvv H , T , Q1 ,: Z1
NB. 'S P dQ1 dZ1'=. hs hgezqsvv H , T , ,:~ I
NB.
NB. Application:
NB. - detect case of non-convergence (0=converged,
NB. 1=non-converged), any of:
NB. isnan < e1e2
NB. isnan < S,:P NB. too expensive, use the next
NB. isnan < diag"2 S,:P
hgezqenn=: diag"2@(0 {:: hgezqe )
hgezqenv=: (2 { ]) ((diag"2@(0 {:: ])) ; (rotscll 2&{:: )) (hgezqe 2&{.)
hgezqevn=: (2 { ]) ((diag"2@(0 {:: ])) ; (rotscll 1&{:: )) (hgezqe 2&{.)
hgezqevv=: (2 }. ]) ((diag"2@(0 {:: ])) ; (rotscll"2&.:> }.)) (hgezqe 2&{.)
hgezqsnn=: 0 {:: hgezqs
hgezqsnv=: (2 { ]) (( 0 {:: ] ) , (rotscll 2&{:: )) (hgezqs 2&{.)
hgezqsvn=: (2 { ]) (( 0 {:: ] ) , (rotscll 1&{:: )) (hgezqs 2&{.)
hgezqsvv=: (2 }. ]) (( 0 {:: ] ) , (rotscll"2&: > }.)) (hgezqs 2&{.)
NB. ---------------------------------------------------------
NB. hgeqzenn
NB. hgeqzenv
NB. hgeqzevn
NB. hgeqzevv
NB. hgeqzsnn
NB. hgeqzsnv
NB. hgeqzsvn
NB. hgeqzsvv
NB.
NB. Description:
NB. Compute eigenvalues (hgeqzxxx) and reduce to
NB. generalized Schur form (hgeqzsxx):
NB. dQ1 * S * dZ1^H = H
NB. dQ1 * P * dZ1^H = T
NB. the generalized upper Hessenberg form (H,T) using
NB. single-shift QZ method. Matrix pairs of this type are
NB. produced by the reduction to generalized upper
NB. Hessenberg form of a matrix pair (A,B):
NB. Q1 * H * Z1^H = A
NB. Q1 * T * Z1^H = B
NB. as computed by gghrduxx. The unitary (orthogonal)
NB. matrices dQ1 and dZ1 may either be formed explicitly,
NB. or they may be premultiplied by input matrices Q1 and
NB. Z1, so that:
NB. (Q1*dQ1) * S * (Z1*dZ1)^H = Q1 * H * Z1^H
NB. (Q1*dQ1) * P * (Z1*dZ1)^H = Q1 * T * Z1^H
NB. To avoid overflow, eigenvalues of the matrix pair (H,T)
NB. (equivalently, of (A,B)) are computed as a pair of
NB. values. Each i-th eigenvector (column) from L and R has
NB. a corresponding eigenvalue represented as a pair of
NB. i-th elements from vectors e1 and e2:
NB. E1=. diagmat e1=. diag S
NB. E2=. diagmat e2=. diag P
NB. If E2 is nonsingular then:
NB. E=. diagmat e1%e2
NB. is a diagonal matrix of eigenvalues, and GNEP can be
NB. expressed as:
NB. L^H * A = E * L^H * B
NB. A * R = B * R * E
NB. and if E1 is nonsingular then:
NB. E=. diagmat e2%e1
NB. is a diagonal matrix of eigenvalues, and GNEP can be
NB. expressed as:
NB. E * L^H * A = L^H * B
NB. A * R * E = B * R * E
NB.
NB. Syntax:
NB. e1e2=. hs hgeqzenn H ,: T
NB. 'e1e2 Z2'=. hs hgeqzenv H , T ,: Z1
NB. 'e1e2 Q2'=. hs hgeqzevn H , T ,: Q1
NB. 'e1e2 Q2Z2'=. hs hgeqzevv H , T , Q1 ,: Z1
NB. 'S P'=. hs hgeqzsnn H ,: T
NB. 'S P Z2'=. hs hgeqzsnv H , T ,: Z1
NB. 'S P Q2'=. hs hgeqzsvn H , T ,: Q1
NB. 'S P Q2 Z2'=. hs hgeqzsvv H , T , Q1 ,: Z1
NB. where
NB. hs - 2-vector of integers (h,s) 'head' and 'size',
NB. defines submatrices H11 and T11 position in H
NB. and T, respectively, see ggbalup and gehrdu
NB. H - n×n-matrix, the upper Hessenberg inside the
NB. submatrix H[h:h+s-1,h:h+s-1], and upper
NB. triangular outside
NB. T - n×n-matrix, the upper triangular
NB. e1e2 - 2×n-matrix of eigenvalues e1 and e2:
NB. e1e2 -: e1 ,: e2
NB. Q1 - n×n-matrix, the unitary (orthogonal)
NB. Q2 - n×n-matrix, the unitary (orthogonal), the left
NB. Schur vectors of (H,T) pair if Q1=I, and of
NB. (A,B) pair otherwise
NB. Z1 - n×n-matrix, the unitary (orthogonal)
NB. Z2 - n×n-matrix, the unitary (orthogonal), the right
NB. Schur vectors of (H,T) pair if Z1=I, and of
NB. (A,B) pair otherwise
NB. Q2Z2 -:Q2 ,: Z2
NB. S - n×n-matrix, the upper triangular
NB. P - n×n-matrix, the upper triangular
NB.
NB. Notes:
NB. - non-converged eigenvalues are set to NaN
NB. - hgeqzenn models LAPACK's xHGEQZ('E','N','N')
NB. - hgeqzenv models LAPACK's xHGEQZ('E','N','V')
NB. - hgeqzevn models LAPACK's xHGEQZ('E','V','N')
NB. - hgeqzevv models LAPACK's xHGEQZ('E','V','V')
NB. - hgeqzsnn models LAPACK's xHGEQZ('S','N','N')
NB. - hgeqzsnv models LAPACK's xHGEQZ('S','N','V')
NB. - hgeqzsvn models LAPACK's xHGEQZ('S','V','N')
NB. - hgeqzsvv models LAPACK's xHGEQZ('S','V','V')
NB.
NB. Assertions (with appropriate comparison tolerance):
NB. e1e2 -: diag"2 (S ,: P)
NB. Q2 -: Q1 mp dQ1
NB. Z2 -: Z1 mp dZ1
NB. (H ,: T) -: dQ1 mp"2 (S ,: P) (mp ct)"2 dZ1
NB. (C ,: D) -: Q2 mp"2 (S ,: P) (mp ct)"2 Z2
NB. I -: Q2 * Q2^H
NB. I -: Z2 * Z2^H
NB. where
NB. C - n×n-matrix, general
NB. D - n×n-matrix, general
NB. n=. # C
NB. hs=. 0 , n
NB. I=. idmat n
NB. 'Q0 B'=. (ungqr ,: tru@}:)@geqrf D
NB. A=. Q0 (mp~ ct)~ C
NB. 'H T Q1 Z1'=. hs gghrduvv A , B , Q0 ,: I
NB. e1e2=. hs hgeqzenn H ,: T
NB. 'S P Q2 Z2'=. hs hgeqzsvv H , T , Q1 ,: Z1
NB. 'S P dQ1 dZ1'=. hs hgeqzsvv H , T , ,:~ I
NB.
NB. Application:
NB. - models LAPACK's xHGEQZ('E','N','I'):
NB. NB. 'e1e2 dZ1'=. hs hgeqzeni H ,: T
NB. hgeqzeni=: hgeqzenv (, idmat@c)
NB. - models LAPACK's xHGEQZ('E','I','N'):
NB. NB. 'e1e2 dQ1'=. hs hgeqzein H ,: T
NB. hgeqzein=: hgeqzevn (, idmat@c)
NB. - models LAPACK's xHGEQZ('E','I','I'):
NB. NB. 'e1e2 dQ1dZ1'=. hs hgeqzeii H ,: T
NB. hgeqzeii=: hgeqzevv (,~^:2~ idmat@c)
NB. - models LAPACK's xHGEQZ('E','I','V'):
NB. NB. 'e1e2 dQ1Z2'=. hs hgeqzeiv H , T ,: Z1
NB. hgeqzeiv=: hgeqzevv (1&A.@, idmat@c)
NB. - models LAPACK's xHGEQZ('E','V','I'):
NB. NB. 'e1e2 Q2dZ1'=. hs hgeqzevi H , T ,: Q1
NB. hgeqzevi=: hgeqzevv (, idmat@c)
NB. - models LAPACK's xHGEQZ('S','N','I'):
NB. NB. 'S P dZ1'=. hs hgeqzsni H ,: T
NB. hgeqzsni=: hgeqzsnv (, idmat@c)
NB. - models LAPACK's xHGEQZ('S','I','N'):
NB. NB. 'S P dQ1'=. hs hgeqzsin H ,: T
NB. hgeqzsin=: hgeqzsvn (, idmat@c)
NB. - models LAPACK's xHGEQZ('S','I','I'):
NB. NB. 'S P dQ1 dZ1'=. hs hgeqzsii H ,: T
NB. hgeqzsii=: hgeqzsvv (,~^:2~ idmat@c)
NB. - models LAPACK's xHGEQZ('S','I','V'):
NB. NB. 'S P dQ1 Z2'=. hs hgeqzsiv H , T ,: Z1
NB. hgeqzsiv=: hgeqzsvv (1&A.@, idmat@c)
NB. - models LAPACK's xHGEQZ('S','V','I'):
NB. NB. 'S P Q2 dZ1'=. hs hgeqzsvi H , T ,: Q1
NB. hgeqzsvi=: hgeqzsvv (, idmat@c)
NB. - detect case of non-convergence (0=converged,
NB. 1=non-converged), any of:
NB. isnan < e1e2
NB. isnan < S,:P NB. too expensive, use the next
NB. isnan < diag"2 S,:P
NB.
NB. References:
NB. [1] C. B. Moler, G. W. Stewart. An Algorithm for
NB. Generalized Matrix Eigenvalue Problems. SIAM J.
NB. Numer. Anal., 10(1973), pp. 241-256.
hgeqzenn=: diag"2@(0 {:: hgeqze )
hgeqzenv=: (2 { ]) ((diag"2@(0 {:: ])) ; (rotsclu 2&{:: )) (hgeqze 2&{.)
hgeqzevn=: (2 { ]) ((diag"2@(0 {:: ])) ; (rotsclu 1&{:: )) (hgeqze 2&{.)
hgeqzevv=: (2 }. ]) ((diag"2@(0 {:: ])) ; (rotsclu"2&.:> }.)) (hgeqze 2&{.)
hgeqzsnn=: 0 {:: hgeqzs
hgeqzsnv=: (2 { ]) (( 0 {:: ] ) , (rotsclu 2&{:: )) (hgeqzs 2&{.)
hgeqzsvn=: (2 { ]) (( 0 {:: ] ) , (rotsclu 1&{:: )) (hgeqzs 2&{.)
hgeqzsvv=: (2 }. ]) (( 0 {:: ] ) , (rotsclu"2&: > }.)) (hgeqzs 2&{.)
NB. =========================================================
NB. Test suite
NB. ---------------------------------------------------------
NB. testhgeq
NB.
NB. Description:
NB. Test:
NB. - xHGEQZ (math/lapack2 addon)
NB. - hgexxxxx (math/mt addon)
NB. by pair of square matrices
NB.
NB. Syntax:
NB. log=. testhgeq AB
NB. where
NB. AB - 2×n×n-brick
NB. log - 6-vector of boxes, test log
testhgeq=: 3 : 0
load_mttmp_ 'math/mt/external/lapack2/hgeqz'
n=. c y
hs=. 0 , n
I=. idmat n
'Hl Tl'=. HTl=. hs gghrdlnn (((mp ct@unglq) ,: trlpick@(_1 }."1 ])) gelqf)/ y
'Hu Tu'=. HTu=. hs gghrdunn (((mp~ ct@ungqr) ,: trupick@(_1 }. ])) geqrf)/ y
rcondl=. (geconi Hl) <. trlconi Tl
rcondu=. (gecon1 Hu) <. trucon1 Tu
normsl=. ;/ normi"2 HTl
normsu=. ;/ norm1"2 HTu
argslapack=. normsu , ;/ HTu , ,:~ I NB. arguments for xHGEQZ t511u t513u
argsmtl=. normsl , < HTl NB. arguments for hgezqxnn t511l t513l
argsmtvl=. normsl , < HTl , I NB. arguments for hgezqxnv hgezqxvn t511l t513l
argsmtvvl=. normsl , < HTl , ,:~ I NB. arguments for hgezqxvv t511l t513l
argsmtu=. normsu , < HTu NB. arguments for hgeqzxnn t511u t513u
argsmtvu=. normsu , < HTu , I NB. arguments for hgeqzxnv hgeqzxvn t511u t513u
argsmtvvu=. normsu , < HTu , ,:~ I NB. arguments for hgeqzxvv t511u t513u
t511u1=: (t511u"1~ ( 2 0 ,: 3 1)&{ )~ (0 4 5 ,: 1 4 5)&{
t511l2=: (t511l"1~ (((2 ; 0)&{::) ; 0&{::) ,: (2 ; 1)&{:: ; 1 &{::)~ <"2@((0 2 3 ,: 1 2 3)&{)
t511u2=: (t511u"1~ (((2 ; 0)&{::) ; 0&{::) ,: (2 ; 1)&{:: ; 1 &{::)~ <"2@((0 2 3 ,: 1 2 3)&{)
t513u4=: t513u@(4 {:: ])
t513u5=: t513u@(5 {:: ])
t513u45=: (4 {:: ]) >.&t513u 5 {:: ]
t513l1=: t513l@(1 {:: ])
t513u1=: t513u@(1 {:: ])
t513l2=: t513l@(2 { ])
t513u2=: t513u@(2 { ])
t513l01=: ((1;0) {:: ]) >.&t513l (1;1) {:: ]
t513u01=: ((1;0) {:: ]) >.&t513u (1;1) {:: ]
t513l23=: (2 { ]) >.&t513l 3 { ]
t513u23=: (2 { ]) >.&t513u 3 { ]
log=. ('''enn''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`nan )) argslapack
log=. log lcat ('''eni''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u5 ))) argslapack
log=. log lcat ('''env''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u5 ))) argslapack
log=. log lcat ('''ein''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u4 ))) argslapack
log=. log lcat ('''eii''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u45))) argslapack
log=. log lcat ('''eiv''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u45))) argslapack
log=. log lcat ('''evn''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u4 ))) argslapack
log=. log lcat ('''evi''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u45))) argslapack
log=. log lcat ('''evv''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u45))) argslapack
log=. log lcat ('''snn''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`nan )) argslapack
log=. log lcat ('''sni''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u5 ))) argslapack
log=. log lcat ('''snv''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u5 ))) argslapack
log=. log lcat ('''sin''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u4 ))) argslapack
log=. log lcat ('''sii''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`(t511u1 >./@, t513u45))) argslapack
log=. log lcat ('''siv''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`(t511u1 >./@, t513u45))) argslapack
log=. log lcat ('''svn''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u4 ))) argslapack
log=. log lcat ('''svi''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`(t511u1 >./@, t513u45))) argslapack
log=. log lcat ('''svv''&dhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`(t511u1 >./@, t513u45))) argslapack
log=. log lcat ('''enn''&zhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`nan )) argslapack
log=. log lcat ('''eni''&zhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u5 ))) argslapack
log=. log lcat ('''env''&zhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u5 ))) argslapack
log=. log lcat ('''ein''&zhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u4 ))) argslapack
log=. log lcat ('''eii''&zhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u45))) argslapack
log=. log lcat ('''eiv''&zhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u45))) argslapack
log=. log lcat ('''evn''&zhgeqz_mttmp_' tmonad ((0 1}~ 1 ; #@(2&{::)) `]`(rcondu"_)`nan`( t513u4 ))) argslapack