概要
本サンプルはFortran言語によりLAPACKルーチンDGGSVDを利用するサンプルプログラムです。
一般化特異値分解を行います。
ここで
の条件数の推定値と計算された一般化特異値の誤差限界も合わせて求めます。この例題プログラムは 
であることを想定していますが、そうでない場合には若干のコード変更が必要です。
入力データ
（本ルーチンの詳細はDGGSVD のマニュアルページを参照）| このデータをダウンロード |
DGGSVD Example Program Data 4 3 2 :Values of M, N and P 1.0 2.0 3.0 3.0 2.0 1.0 4.0 5.0 6.0 7.0 8.0 8.0 :End of matrix A -2.0 -3.0 3.0 4.0 6.0 5.0 :End of matrix B
出力結果
（本ルーチンの詳細はDGGSVD のマニュアルページを参照）| この出力例をダウンロード |
DGGSVD Example Program Results
Number of infinite generalized singular values (K)
1
Number of finite generalized singular values (L)
2
Numerical rank of (A**T B**T)**T (K+L)
3
Finite generalized singular values
1.3151E+00 8.0185E-02
Orthogonal matrix U
1 2 3 4
1 -1.3484E-01 5.2524E-01 -2.0924E-01 8.1373E-01
2 6.7420E-01 -5.2213E-01 -3.8886E-01 3.4874E-01
3 2.6968E-01 5.2757E-01 -6.5782E-01 -4.6499E-01
4 6.7420E-01 4.1615E-01 6.1014E-01 1.5127E-15
Orthogonal matrix V
1 2
1 3.5539E-01 -9.3472E-01
2 9.3472E-01 3.5539E-01
Orthogonal matrix Q
1 2 3
1 -8.3205E-01 -9.4633E-02 -5.4657E-01
2 5.5470E-01 -1.4195E-01 -8.1985E-01
3 0.0000E+00 -9.8534E-01 1.7060E-01
Nonsingular upper triangular matrix R
1 2 3
1 -2.0569E+00 -9.0121E+00 -9.3705E+00
2 -1.0882E+01 -7.2688E+00
3 -6.0405E+00
Estimate of reciprocal condition number for R
4.2E-02
Error estimate for the generalized singular values
5.3E-15
ソースコード
（本ルーチンの詳細はDGGSVD 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 dggsvd_example
! DGGSVD Example Program Text
! Copyright 2017, Numerical Algorithms Group Ltd. http://www.nag.com
! .. Use Statements ..
Use lapack_example_aux, Only: nagf_file_print_matrix_real_gen_comp
Use lapack_interfaces, Only: dggsvd3, dtrcon
Use lapack_precision, Only: dp
! .. Implicit None Statement ..
Implicit None
! .. Parameters ..
Integer, Parameter :: nin = 5, nout = 6
! .. Local Scalars ..
Real (Kind=dp) :: eps, rcond, serrbd
Integer :: i, ifail, info, irank, j, k, l, lda, ldb, ldq, ldu, ldv, &
lwork, m, n, p
! .. Local Arrays ..
Real (Kind=dp), Allocatable :: a(:, :), alpha(:), b(:, :), beta(:), &
q(:, :), u(:, :), v(:, :), work(:)
Integer, Allocatable :: iwork(:)
Character (1) :: clabs(1), rlabs(1)
! .. Intrinsic Procedures ..
Intrinsic :: epsilon
! .. Executable Statements ..
Write (nout, *) 'DGGSVD Example Program Results'
Write (nout, *)
Flush (nout)
! Skip heading in data file
Read (nin, *)
Read (nin, *) m, n, p
lda = m
ldb = p
ldq = n
ldu = m
ldv = p
Allocate (a(lda,n), alpha(n), b(ldb,n), beta(n), q(ldq,n), u(ldu,m), &
v(ldv,p), work(m+3*n), iwork(n))
! Read the m by n matrix A and p by n matrix B from data file
Read (nin, *)(a(i,1:n), i=1, m)
Read (nin, *)(b(i,1:n), i=1, p)
! Compute the generalized singular value decomposition of (A, B)
! (A = U*D1*(0 R)*(Q**T), B = V*D2*(0 R)*(Q**T), m>=n)
lwork = m + 3*n
Call dggsvd3('U', 'V', 'Q', m, n, p, k, l, a, lda, b, ldb, alpha, beta, &
u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)
If (info==0) Then
! Print solution
irank = k + l
Write (nout, *) 'Number of infinite generalized singular values (K)'
Write (nout, 100) k
Write (nout, *) 'Number of finite generalized singular values (L)'
Write (nout, 100) l
Write (nout, *) 'Numerical rank of (A**T B**T)**T (K+L)'
Write (nout, 100) irank
Write (nout, *)
Write (nout, *) 'Finite generalized singular values'
Write (nout, 110)(alpha(j)/beta(j), j=k+1, irank)
Write (nout, *)
Flush (nout)
! ifail: behaviour on error exit
! =0 for hard exit, =1 for quiet-soft, =-1 for noisy-soft
ifail = 0
Call nagf_file_print_matrix_real_gen_comp('General', ' ', m, m, u, &
ldu, '1P,E12.4', 'Orthogonal matrix U', 'Integer', rlabs, 'Integer', &
clabs, 80, 0, ifail)
Write (nout, *)
Flush (nout)
Call nagf_file_print_matrix_real_gen_comp('General', ' ', p, p, v, &
ldv, '1P,E12.4', 'Orthogonal matrix V', 'Integer', rlabs, 'Integer', &
clabs, 80, 0, ifail)
Write (nout, *)
Flush (nout)
Call nagf_file_print_matrix_real_gen_comp('General', ' ', n, n, q, &
ldq, '1P,E12.4', 'Orthogonal matrix Q', 'Integer', rlabs, 'Integer', &
clabs, 80, 0, ifail)
Write (nout, *)
Flush (nout)
Call nagf_file_print_matrix_real_gen_comp('Upper triangular', &
'Non-unit', irank, irank, a(1,n-irank+1), lda, '1P,E12.4', &
'Nonsingular upper triangular matrix R', 'Integer', rlabs, &
'Integer', clabs, 80, 0, ifail)
! Call DTRCON to estimate the reciprocal condition
! number of R
Call dtrcon('Infinity-norm', 'Upper', 'Non-unit', irank, &
a(1,n-irank+1), lda, rcond, work, iwork, info)
Write (nout, *)
Write (nout, *) 'Estimate of reciprocal condition number for R'
Write (nout, 120) rcond
Write (nout, *)
! So long as irank = n, get the machine precision, eps, and
! compute the approximate error bound for the computed
! generalized singular values
If (irank==n) Then
eps = epsilon(1.0E0_dp)
serrbd = eps/rcond
Write (nout, *) 'Error estimate for the generalized singular values'
Write (nout, 120) serrbd
Else
Write (nout, *) '(A**T B**T)**T is not of full rank'
End If
Else
Write (nout, 130) 'Failure in DGGSVD3. INFO =', info
End If
100 Format (1X, I5)
110 Format (3X, 8(1P,E12.4))
120 Format (1X, 1P, E11.1)
130 Format (1X, A, I4)
End Program