Numpy scipy.linalg.eigvals实际上是如何计算特征值的？
我找不到任何关于这个东西是如何计算特征值的文档，文档只是说它使用了“geev-LAPACK例程”，但我已经搜索了很多关于它的文档，也找不到。获取到英特尔网站的奇怪链接，这进一步使我的搜索徒劳无功。谢谢您的帮助。这取决于您的具体情况。因此，有多个候选者，英特尔的MKL就是其中之一 原始的LAPACK（可能没有太多的使用；但我只是在这里猜测）作为开源（Fortran）和良好的文档记录非常好 这是双版本（请参见Numpy scipy.linalg.eigvals实际上是如何计算特征值的？,numpy,scipy,eigenvalue,Numpy,Scipy,Eigenvalue,我找不到任何关于这个东西是如何计算特征值的文档，文档只是说它使用了“geev-LAPACK例程”，但我已经搜索了很多关于它的文档，也找不到。获取到英特尔网站的奇怪链接，这进一步使我的搜索徒劳无功。谢谢您的帮助。这取决于您的具体情况。因此，有多个候选者，英特尔的MKL就是其中之一 原始的LAPACK（可能没有太多的使用；但我只是在这里猜测）作为开源（Fortran）和良好的文档记录非常好 这是双版本（请参见D-前缀），来自： 子程序DGEV（JOBVL、JOBVR、N、A、LDA、WR、WI、VL
D子程序DGEV（JOBVL、JOBVR、N、A、LDA、WR、WI、VL、LDVL、VR、，
$LDVR，工作，工作，信息）
*
*--LAPACK驱动程序例程（版本3.1）--
*田纳西大学、加利福尼亚大学伯克利分校和纳格有限公司。。
*2006年11月
*
* .. 标量参数。。
字符JOBVL，JOBVR
整数信息，LDA，LDVL，LDVR，LWORK，N
* ..
* .. 数组参数。。
双精度A（LDA，*）、VL（LDVL，*）、VR（LDVR，*）、，
$WI（*）、工时（*）、工时（*）
* ..
*
*目的
* =======
*
*DGEEV计算一个N×N实非对称矩阵A，其中
*特征值和（可选）左和/或右特征向量。
*
*A的右特征向量v（j）满足
*A*v（j）=λ（j）*v（j）
*其中λ（j）是其特征值。
*A的左特征向量u（j）满足
*u（j）**H*A=λ（j）*u（j）**H
*其中u（j）**H表示u（j）的共轭转置。
*
*将计算出的特征向量归一化为欧几里德范数
*等于1和最大分量实数。
*
*论据
* =========
*
*JOBVL（输入）字符*1
*='N'：不计算A的左特征向量；
*='V'：计算A的左特征向量。
*
*JOBVR（输入）字符*1
*='N'：不计算A的右特征向量；
*='V'：计算A的右特征向量。
*
*N（输入）整数
*矩阵的阶数A.N>=0。
*
*A（输入/输出）双精度阵列，尺寸（LDA，N）
*输入时，N×N矩阵A。
*退出时，A已被覆盖。
*
*LDA（输入）整数
*数组的前导维度A.LDA>=max（1，N）。
*
*WR（输出）双精度阵列，尺寸（N）
*WI（输出）双精度阵列，尺寸（N）
*WR和WI包含实部和虚部，
*分别计算特征值的。复杂的
*特征值的共轭对连续出现
*特征值具有正虚部
*首先。
*
*VL（输出）双精度阵列，尺寸（LDVL，N）
*如果JOBVL='V'，则左特征向量u（j）存储一个
*在VL的列中，以相同的顺序依次显示
*作为它们的特征值。
*如果JOBVL='N'，则不引用VL。
*如果第j个特征值为实数，则u（j）=VL（：，j），
*VL的第j列。
*如果j-th和（j+1）-st特征值形成复数
*共轭对，然后u（j）=VL（：，j）+i*VL（：，j+1）和
*u（j+1）=VL（：，j）-i*VL（：，j+1）。
*
*LDVL（输入）整数
*阵列VL的前导尺寸。LDVL>=1；如果
*JOBVL='V'，LDVL>=N。
*
*VR（输出）双精度阵列，尺寸（LDVR，N）
*如果JOBVR='V'，则右特征向量V（j）存储一个
*在VR列中，按相同顺序依次显示
*作为它们的特征值。
*如果JOBVR='N'，则不引用VR。
*如果第j个特征值为实数，则v（j）=VR（：，j），
*VR的第j列。
*如果j-th和（j+1）-st特征值形成复数
*共轭对，然后v（j）=VR（：，j）+i*VR（：，j+1）和
*v（j+1）=VR（：，j）-i*VR（：，j+1）。
*
*LDVR（输入）整数
*阵列VR的前导维度。LDVR>=1；如果
*JOBVR='V'，LDVR>=N。
*
*工时（工作空间/输出）双精度阵列，尺寸（最大（1，LWORK））
*退出时，如果INFO=0，则WORK（1）返回最佳LWORK。
*
*LWORK（输入）整数
*数组的维度将起作用。LWORK>=最大值（1,3*N），以及
*如果JOBVL='V'或JOBVR='V'，则LWORK>=4*N.表示永久
*性能方面，LWORK通常必须更大。
*
*如果LWORK=-1，则假定为工作区查询；例行公事
*仅计算工作数组的最佳大小，返回
*此值作为工作数组的第一个条目，没有错误
*与LWORK相关的信息由XERBLA发布。
*
*信息（输出）整数
*=0：成功退出
*<0:如果INFO=-i，则第i个参数的值非法。
*>0：如果INFO=i，则QR算法无法计算所有
*特征值，且未计算特征向量；
*WR和WI的元素i+1:N包含以下特征值：
*已经趋同。
*
* =====================================================================
*
* .. 参数。。
双精度零，一
参数（零=0.0D0，一=1.0D0）
* ..
* .. 局部标量。。
逻辑LQUERY、SCALEA、WANTVL、WANTVR
角色方面
整数HSWORK、I、IBAL、IERR、IHI、ILO、ITAU、IWRK、K、，
$MAXWRK，MINWRK，NOUT
双精度ANRM、BIGNUM、CS、CSCALE、EPS、R、SCL、SMLNUM、，
$SN
* ..
* .. 本地阵列。。
逻辑选择（1）
双精度DUM（1）
* ..
* .. 外部子程序。。
外部DGEBAK、DGEBAL、DGEHRD、DHSEQR、DLABAD、DLACPY、，
$DLARTG、DLASCL、DORGHR、DROT、DSCAL、DTREVC、，
$XE
SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
$ LDVR, WORK, LWORK, INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WI( * ), WORK( * ), WR( * )
* ..
*
* Purpose
* =======
*
* DGEEV computes for an N-by-N real nonsymmetric matrix A, the
* eigenvalues and, optionally, the left and/or right eigenvectors.
*
* The right eigenvector v(j) of A satisfies
* A * v(j) = lambda(j) * v(j)
* where lambda(j) is its eigenvalue.
* The left eigenvector u(j) of A satisfies
* u(j)**H * A = lambda(j) * u(j)**H
* where u(j)**H denotes the conjugate transpose of u(j).
*
* The computed eigenvectors are normalized to have Euclidean norm
* equal to 1 and largest component real.
*
* Arguments
* =========
*
* JOBVL (input) CHARACTER*1
* = 'N': left eigenvectors of A are not computed;
* = 'V': left eigenvectors of A are computed.
*
* JOBVR (input) CHARACTER*1
* = 'N': right eigenvectors of A are not computed;
* = 'V': right eigenvectors of A are computed.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the N-by-N matrix A.
* On exit, A has been overwritten.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* WR (output) DOUBLE PRECISION array, dimension (N)
* WI (output) DOUBLE PRECISION array, dimension (N)
* WR and WI contain the real and imaginary parts,
* respectively, of the computed eigenvalues. Complex
* conjugate pairs of eigenvalues appear consecutively
* with the eigenvalue having the positive imaginary part
* first.
*
* VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
* If JOBVL = 'V', the left eigenvectors u(j) are stored one
* after another in the columns of VL, in the same order
* as their eigenvalues.
* If JOBVL = 'N', VL is not referenced.
* If the j-th eigenvalue is real, then u(j) = VL(:,j),
* the j-th column of VL.
* If the j-th and (j+1)-st eigenvalues form a complex
* conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
* u(j+1) = VL(:,j) - i*VL(:,j+1).
*
* LDVL (input) INTEGER
* The leading dimension of the array VL. LDVL >= 1; if
* JOBVL = 'V', LDVL >= N.
*
* VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
* If JOBVR = 'V', the right eigenvectors v(j) are stored one
* after another in the columns of VR, in the same order
* as their eigenvalues.
* If JOBVR = 'N', VR is not referenced.
* If the j-th eigenvalue is real, then v(j) = VR(:,j),
* the j-th column of VR.
* If the j-th and (j+1)-st eigenvalues form a complex
* conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
* v(j+1) = VR(:,j) - i*VR(:,j+1).
*
* LDVR (input) INTEGER
* The leading dimension of the array VR. LDVR >= 1; if
* JOBVR = 'V', LDVR >= N.
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,3*N), and
* if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good
* performance, LWORK must generally be larger.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if INFO = i, the QR algorithm failed to compute all the
* eigenvalues, and no eigenvectors have been computed;
* elements i+1:N of WR and WI contain eigenvalues which
* have converged.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, SCALEA, WANTVL, WANTVR
CHARACTER SIDE
INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K,
$ MAXWRK, MINWRK, NOUT
DOUBLE PRECISION ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
$ SN
* ..
* .. Local Arrays ..
LOGICAL SELECT( 1 )
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,
$ DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX, ILAENV
DOUBLE PRECISION DLAMCH, DLANGE, DLAPY2, DNRM2
EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2,
$ DNRM2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
WANTVL = LSAME( JOBVL, 'V' )
WANTVR = LSAME( JOBVR, 'V' )
IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
INFO = -9
ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
INFO = -11
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "Workspace:" describe the
* minimal amount of workspace needed at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.
* HSWORK refers to the workspace preferred by DHSEQR, as
* calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
* the worst case.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
ELSE
MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
IF( WANTVL ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
$ WORK, -1, INFO )
HSWORK = WORK( 1 )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE IF( WANTVR ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
$ WORK, -1, INFO )
HSWORK = WORK( 1 )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE
MINWRK = 3*N
CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR,
$ WORK, -1, INFO )
HSWORK = WORK( 1 )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEEV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
SCALEA = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
SCALEA = .TRUE.
CSCALE = SMLNUM
ELSE IF( ANRM.GT.BIGNUM ) THEN
SCALEA = .TRUE.
CSCALE = BIGNUM
END IF
IF( SCALEA )
$ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
* Balance the matrix
* (Workspace: need N)
*
IBAL = 1
CALL DGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
* Reduce to upper Hessenberg form
* (Workspace: need 3*N, prefer 2*N+N*NB)
*
ITAU = IBAL + N
IWRK = ITAU + N
CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
IF( WANTVL ) THEN
*
* Want left eigenvectors
* Copy Householder vectors to VL
*
SIDE = 'L'
CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
* Generate orthogonal matrix in VL
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VL
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
IF( WANTVR ) THEN
*
* Want left and right eigenvectors
* Copy Schur vectors to VR
*
SIDE = 'B'
CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
END IF
*
ELSE IF( WANTVR ) THEN
*
* Want right eigenvectors
* Copy Householder vectors to VR
*
SIDE = 'R'
CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
* Generate orthogonal matrix in VR
* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
* Perform QR iteration, accumulating Schur vectors in VR
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
*
ELSE
*
* Compute eigenvalues only
* (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
$ WORK( IWRK ), LWORK-IWRK+1, INFO )
END IF
*
* If INFO > 0 from DHSEQR, then quit
*
IF( INFO.GT.0 )
$ GO TO 50
*
IF( WANTVL .OR. WANTVR ) THEN
*
* Compute left and/or right eigenvectors
* (Workspace: need 4*N)
*
CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
$ N, NOUT, WORK( IWRK ), IERR )
END IF
*
IF( WANTVL ) THEN
*
* Undo balancing of left eigenvectors
* (Workspace: need N)
*
CALL DGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL, LDVL,
$ IERR )
*
* Normalize left eigenvectors and make largest component real
*
DO 20 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
$ DNRM2( N, VL( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
DO 10 K = 1, N
WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2
10 CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
VL( K, I+1 ) = ZERO
END IF
20 CONTINUE
END IF
*
IF( WANTVR ) THEN
*
* Undo balancing of right eigenvectors
* (Workspace: need N)
*
CALL DGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR, LDVR,
$ IERR )
*
* Normalize right eigenvectors and make largest component real
*
DO 40 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
$ DNRM2( N, VR( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
DO 30 K = 1, N
WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2
30 CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
VR( K, I+1 ) = ZERO
END IF
40 CONTINUE
END IF
*
* Undo scaling if necessary
*
50 CONTINUE
IF( SCALEA ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
$ MAX( N-INFO, 1 ), IERR )
IF( INFO.GT.0 ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
$ IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
$ IERR )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
* End of DGEEV
*
END