走进LAPACK（三）续

五、BLAS和LAPACK的常用函数

  7、特征值和特征向量（Eigenvalues and eigenvectors，A*v(j)= λ(j)*v(j)，u(j)^H*A = λ(j)*u(j)^H）

    lapack函数（dgeev, zgeev for eigenvalues and eigenvectors; dgegv, zgegv for generalized eigenvalues and eigenvectors）

   函数原型：  call dgeev(jobvl, jobvr, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, work, lwork, info)

   输出：wr wi vl vr (具体看程序说明)

program main

    implicit none
    real(8) :: aaa(2,2),work(60),wr(2),wi(2),vl(2,2),vr(2,2)
    integer :: lda,n,ldvl,ldvr,lwork,info
    character :: jobvl,jobvr

    aaa=reshape((/1,3,2,4/),(/2,2/))
    jobvl='N'
    jobvr='V'
    n=2
    lda=2
    ldvl=1
    ldvr=2
    lwork=60
    wr=0.;wi=0.

    call dgeev(jobvl, jobvr, n, aaa, lda, wr, wi, vl, ldvl, vr, ldvr, work, lwork, info)
    
    print *,wr,wi
    print *,vl,vr

end program

输出结果

8、海森伯简化（Hessenberg reduction，A = Q*H*Q^H）

lapack函数（dgehrd, zgehrd; dorghr, zunghr for computing Q）

函数原型：call dgehrd(n, ilo, ihi, a, lda, tau, work, lwork, info)

              call dorghr(n, ilo, ihi, a, lda, tau, work, lwork, info)

program main

    implicit none
    real(8) :: aaa(3,3),hhh(3,3),qqq(3,3),tau(2),work(3)
    integer :: n,ilo,ihi,lda,info,lwork,i,j

    aaa=reshape((/1,4,7,2,5,8,3,6,9/),(/3,3/))
    n=3
    ilo=1
    ihi=3
    lda=3
    lwork=3
    hhh=0.;qqq=0.
    call dgehrd(n, ilo, ihi, aaa, lda, tau, work, lwork, info)
    forall(i=1:3,j=1:3,i<=j+1)
        hhh(i,j)=aaa(i,j)
    end forall
    print *,'hhh='
    print *,hhh

    print *,'---------------------------'
    call dorghr(n, ilo, ihi, aaa, lda, tau, work, lwork, info)
    qqq=aaa
    print *,'qqq='
    print *,qqq

end program

输出结果

9、逆矩阵（Matrix inverse，R=inv(X)）

lapack函数（dgesv, zgesv）

原型同矩阵左除

program main

    implicit none
    real(8) :: aaa(2,2),bbb(2,2),rrr(2,2)
    integer :: n,nrhs,lda,ldb,ipiv(2),info

    aaa=reshape((/1.d0,3.d0,2.d0,4.d0/),(/2,2/))
    !bbb为单位矩阵
    bbb=reshape((/1.d0,0.d0,0.d0,1.d0/),(/2,2/))

    n=2
    nrhs=2
    lda=2
    ldb=2

    call dgesv( n, nrhs, aaa, lda, ipiv, bbb, ldb, info )

    rrr=bbb
    print *,bbb

end program

输出结果

10、LU分解（LU factorization，P*X=L*U）

lapack函数（dgetrf, zgetrf）

同行列式

program main

    implicit none
    real(8) :: aaa(3,3),ppp(3,3),lll(3,3),uuu(3,3),temp
    integer :: m,n,lda,ipiv(3),info,i,j

    aaa=reshape((/1,4,7,2,5,8,3,6,8/),(/3,3/))
    ppp=0.;lll=0.;uuu=0.
    m=3
    n=3
    lda=3

    call dgetrf( m, n, aaa, lda, ipiv, info )

    do i=1,3
        do j=1,3
            if(i>j) then
                lll(i,j)=aaa(i,j)
            elseif(i==j) then
                lll(i,j)=1.
                uuu(i,j)=aaa(i,j)
            else
                uuu(i,j)=aaa(i,j)
            endif
        enddo
    enddo

    forall(i=1:3,j=1:3,i==j)
        ppp(i,j)=1.
    end forall

    do i=1,3
        do j=1,3
            temp=ppp(i,j)
            ppp(i,j)=ppp(ipiv(i),j)
            ppp(ipiv(i),j)=temp
        enddo
    enddo
    
    print *,'P='
    print *,ppp
    print *,'L='
    print *,lll
    print *,'U='
    print *,uuu
end program

输出结果