概要
本サンプルはFortran言語によりLAPACKルーチンZHEGVDを利用するサンプルプログラムです。
一般化エルミート固有値問題
のすべての固有値と固有ベクトルを求めます。
及び
の条件数の推定値と計算された固有値と固有ベクトルの誤差限界の近似値も合わせて求めます。ZHEGVの例題プログラムは一般化エルミート固有値問題
の解き方を示します。
入力データ
（本ルーチンの詳細はZHEGVD のマニュアルページを参照）| このデータをダウンロード |
ZHEGVD Example Program Data
4 :Value of N
(-7.36, 0.00) ( 0.77, -0.43) (-0.64, -0.92) ( 3.01, -6.97)
( 3.49, 0.00) ( 2.19, 4.45) ( 1.90, 3.73)
( 0.12, 0.00) ( 2.88, -3.17)
(-2.54, 0.00) :End of matrix A
( 3.23, 0.00) ( 1.51, -1.92) ( 1.90, 0.84) ( 0.42, 2.50)
( 3.58, 0.00) (-0.23, 1.11) (-1.18, 1.37)
( 4.09, 0.00) ( 2.33, -0.14)
( 4.29, 0.00) :End of matrix B
出力結果
（本ルーチンの詳細はZHEGVD のマニュアルページを参照）| この出力例をダウンロード |
ZHEGVD Example Program Results
Eigenvalues
-61.7321 -6.6195 0.0725 43.1883
Eigenvectors
1 2 3 4
1 0.3903 -0.1560 2.2909 -0.1943
0.0000 -0.0404 0.0000 -0.0690
2 -0.1814 -0.1552 -0.5042 0.3884
0.0114 -0.3651 -0.7120 0.0000
3 0.0438 0.5364 -1.2701 0.0657
0.0338 0.0000 -0.4547 -0.2095
4 -0.2221 -0.1298 0.5706 0.2924
-0.2272 -0.1880 1.3132 -0.0675
Estimate of reciprocal condition number for B
2.5E-03
Error estimates (relative to machine precision)
for the eigenvalues:
2.4E+04 2.8E+03 2.3E+02 1.7E+04
for the eigenvectors:
4.7E+02 1.0E+03 1.0E+03 4.9E+02
ソースコード
（本ルーチンの詳細はZHEGVD のマニュアルページを参照）※本サンプルソースコードのご利用手順は「サンプルのコンパイル及び実行方法」をご参照下さい。
| このソースコードをダウンロード |
Program zhegvd_example
! ZHEGVD Example Program Text
! Copyright 2017, Numerical Algorithms Group Ltd. http://www.nag.com
! .. Use Statements ..
Use lapack_example_aux, Only: nagf_file_print_matrix_complex_gen
Use lapack_interfaces, Only: ddisna, zhegvd, zlanhe, ztrcon
Use lapack_precision, Only: dp
! .. Implicit None Statement ..
Implicit None
! .. Parameters ..
Real (Kind=dp), Parameter :: one = 1.0E+0_dp
Integer, Parameter :: nb = 64, nin = 5, nout = 6
! .. Local Scalars ..
Complex (Kind=dp) :: scal
Real (Kind=dp) :: anorm, bnorm, eps, rcond, rcondb, t1, t2
Integer :: i, ifail, info, k, lda, ldb, liwork, lrwork, lwork, n
! .. Local Arrays ..
Complex (Kind=dp), Allocatable :: a(:, :), b(:, :), work(:)
Complex (Kind=dp) :: cdum(1)
Real (Kind=dp), Allocatable :: eerbnd(:), rcondz(:), rwork(:), w(:), &
zerbnd(:)
Real (Kind=dp) :: rdum(1)
Integer :: idum(1)
Integer, Allocatable :: iwork(:)
! .. Intrinsic Procedures ..
Intrinsic :: abs, conjg, epsilon, max, maxloc, nint, real
! .. Executable Statements ..
Write (nout, *) 'ZHEGVD Example Program Results'
Write (nout, *)
! Skip heading in data file
Read (nin, *)
Read (nin, *) n
lda = n
ldb = n
Allocate (a(lda,n), b(ldb,n), eerbnd(n), rcondz(n), w(n), zerbnd(n))
! Use routine workspace query to get optimal workspace.
lwork = -1
liwork = -1
lrwork = -1
Call zhegvd(2, 'Vectors', 'Upper', n, a, lda, b, ldb, w, cdum, lwork, &
rdum, lrwork, idum, liwork, info)
! Make sure that there is enough workspace for block size nb.
lwork = max((nb+2+n)*n, nint(real(cdum(1))))
lrwork = max(1+(5+2*n)*n, nint(rdum(1)))
liwork = max(3+5*n, idum(1))
Allocate (work(lwork), rwork(lrwork), iwork(liwork))
! Read the upper triangular parts of the matrices A and B
Read (nin, *)(a(i,i:n), i=1, n)
Read (nin, *)(b(i,i:n), i=1, n)
! Compute the one-norms of the symmetric matrices A and B
anorm = zlanhe('One norm', 'Upper', n, a, lda, rwork)
bnorm = zlanhe('One norm', 'Upper', n, b, ldb, rwork)
! Solve the generalized Hermitian eigenvalue problem
! A*B*x = lambda*x (itype = 2)
Call zhegvd(2, 'Vectors', 'Upper', n, a, lda, b, ldb, w, work, lwork, &
rwork, lrwork, iwork, liwork, info)
If (info==0) Then
! Print solution
Write (nout, *) 'Eigenvalues'
Write (nout, 100) w(1:n)
Flush (nout)
! Normalize the eigenvectors, largest element real
Do i = 1, n
rwork(1:n) = abs(a(1:n,i))
k = maxloc(rwork(1:n), 1)
scal = conjg(a(k,i))/abs(a(k,i))
a(1:n, i) = a(1:n, i)*scal
End Do
! ifail: behaviour on error exit
! =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft
ifail = 0
Call nagf_file_print_matrix_complex_gen('General', ' ', n, n, a, lda, &
'Eigenvectors', ifail)
! Call ZTRCON to estimate the reciprocal condition
! number of the Cholesky factor of B. Note that:
! cond(B) = 1/rcond**2
Call ztrcon('One norm', 'Upper', 'Non-unit', n, b, ldb, rcond, work, &
rwork, info)
! Print the reciprocal condition number of B
rcondb = rcond**2
Write (nout, *)
Write (nout, *) 'Estimate of reciprocal condition number for B'
Write (nout, 110) rcondb
Flush (nout)
! Get the machine precision, eps, and if rcondb is not less
! than eps**2, compute error estimates for the eigenvalues and
! eigenvectors
eps = epsilon(1.0E0_dp)
If (rcond>=eps) Then
! Call DDISNA to estimate reciprocal condition
! numbers for the eigenvectors of (A*B - lambda*I)
Call ddisna('Eigenvectors', n, n, w, rcondz, info)
! Compute the error estimates for the eigenvalues and
! eigenvectors
t1 = one/rcond
t2 = anorm*bnorm
Do i = 1, n
eerbnd(i) = (t2+abs(w(i))/rcondb)
zerbnd(i) = t1*(t2/rcondz(i)+t1)
End Do
! Print the approximate error bounds for the eigenvalues
! and vectors
Write (nout, *)
Write (nout, *) 'Error estimates (relative to machine precision)'
Write (nout, *) 'for the eigenvalues:'
Write (nout, 110) eerbnd(1:n)
Write (nout, *)
Write (nout, *) 'for the eigenvectors:'
Write (nout, 110) zerbnd(1:n)
Else
Write (nout, *)
Write (nout, *) 'B is very ill-conditioned, error ', &
'estimates have not been computed'
End If
Else If (info>n) Then
i = info - n
Write (nout, 120) 'The leading minor of order ', i, &
' of B is not positive definite'
Else
Write (nout, 130) 'Failure in ZHEGVD. INFO =', info
End If
100 Format (3X, (6F11.4))
110 Format (4X, 1P, 6E11.1)
120 Format (1X, A, I4, A)
130 Format (1X, A, I4)
End Program