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auxiliary Computational routines(3) | LAPACK | auxiliary Computational routines(3) |
NAME¶
auxiliary Computational routines -Functions¶
character *1 function chla_transtype (TRANS)
Detailed Description¶
This is the group of auxiliary Computational routinesFunction Documentation¶
character*1 function chla_transtype (integerTRANS)¶
CHLA_TRANSTYPE Purpose:This subroutine translates from a BLAST-specified integer constant to the character string specifying a transposition operation. CHLA_TRANSTYPE returns an CHARACTER*1. If CHLA_TRANSTYPE is 'X', then input is not an integer indicating a transposition operator. Otherwise CHLA_TRANSTYPE returns the constant value corresponding to TRANS.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine dbdsdc (characterUPLO, characterCOMPQ, integerN, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( ldu, * )U, integerLDU, double precision, dimension( ldvt, * )VT, integerLDVT, double precision, dimension( * )Q, integer, dimension( * )IQ, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
DBDSDC Purpose:DBDSDC computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT, using a divide and conquer method, where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively. DBDSDC can be used to compute all singular values, and optionally, singular vectors or singular vectors in compact form. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. See DLASD3 for details. The code currently calls DLASDQ if singular values only are desired. However, it can be slightly modified to compute singular values using the divide and conquer method.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': B is upper bidiagonal. = 'L': B is lower bidiagonal.COMPQ
COMPQ is CHARACTER*1 Specifies whether singular vectors are to be computed as follows: = 'N': Compute singular values only; = 'P': Compute singular values and compute singular vectors in compact form; = 'I': Compute singular values and singular vectors.N
N is INTEGER The order of the matrix B. N >= 0.D
D is DOUBLE PRECISION array, dimension (N) On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B.E
E is DOUBLE PRECISION array, dimension (N-1) On entry, the elements of E contain the offdiagonal elements of the bidiagonal matrix whose SVD is desired. On exit, E has been destroyed.U
U is DOUBLE PRECISION array, dimension (LDU,N) If COMPQ = 'I', then: On exit, if INFO = 0, U contains the left singular vectors of the bidiagonal matrix. For other values of COMPQ, U is not referenced.LDU
LDU is INTEGER The leading dimension of the array U. LDU >= 1. If singular vectors are desired, then LDU >= max( 1, N ).VT
VT is DOUBLE PRECISION array, dimension (LDVT,N) If COMPQ = 'I', then: On exit, if INFO = 0, VT**T contains the right singular vectors of the bidiagonal matrix. For other values of COMPQ, VT is not referenced.LDVT
LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1. If singular vectors are desired, then LDVT >= max( 1, N ).Q
Q is DOUBLE PRECISION array, dimension (LDQ) If COMPQ = 'P', then: On exit, if INFO = 0, Q and IQ contain the left and right singular vectors in a compact form, requiring O(N log N) space instead of 2*N**2. In particular, Q contains all the DOUBLE PRECISION data in LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1)))) words of memory, where SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25). For other values of COMPQ, Q is not referenced.IQ
IQ is INTEGER array, dimension (LDIQ) If COMPQ = 'P', then: On exit, if INFO = 0, Q and IQ contain the left and right singular vectors in a compact form, requiring O(N log N) space instead of 2*N**2. In particular, IQ contains all INTEGER data in LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1)))) words of memory, where SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25). For other values of COMPQ, IQ is not referenced.WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) If COMPQ = 'N' then LWORK >= (4 * N). If COMPQ = 'P' then LWORK >= (6 * N). If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).IWORK
IWORK is INTEGER array, dimension (8*N)INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute a singular value. The update process of divide and conquer failed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
Contributors:
Ming Gu and Huan Ren, Computer Science Division,
University of California at Berkeley, USA
subroutine dbdsqr (characterUPLO, integerN, integerNCVT, integerNRU, integerNCC, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( ldvt, * )VT, integerLDVT, double precision, dimension( ldu, * )U, integerLDU, double precision, dimension( ldc, * )C, integerLDC, double precision, dimension( * )WORK, integerINFO)¶
DBDSQR Purpose:DBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm. The SVD of B has the form B = Q * S * P**T where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns P**T*VT instead of P**T, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by DGEBRD, then A = (U*Q) * S * (P**T*VT) is the SVD of A. Optionally, the subroutine may also compute Q**T*C for a given real input matrix C. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873-912, Sept 1990) and "Accurate singular values and differential qd algorithms," by B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992 for a detailed description of the algorithm.
UPLO
Internal Parameters:
UPLO is CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal.N
N is INTEGER The order of the matrix B. N >= 0.NCVT
NCVT is INTEGER The number of columns of the matrix VT. NCVT >= 0.NRU
NRU is INTEGER The number of rows of the matrix U. NRU >= 0.NCC
NCC is INTEGER The number of columns of the matrix C. NCC >= 0.D
D is DOUBLE PRECISION array, dimension (N) On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B in decreasing order.E
E is DOUBLE PRECISION array, dimension (N-1) On entry, the N-1 offdiagonal elements of the bidiagonal matrix B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input.VT
VT is DOUBLE PRECISION array, dimension (LDVT, NCVT) On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by P**T * VT. Not referenced if NCVT = 0.LDVT
LDVT is INTEGER The leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.U
U is DOUBLE PRECISION array, dimension (LDU, N) On entry, an NRU-by-N matrix U. On exit, U is overwritten by U * Q. Not referenced if NRU = 0.LDU
LDU is INTEGER The leading dimension of the array U. LDU >= max(1,NRU).C
C is DOUBLE PRECISION array, dimension (LDC, NCC) On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q**T * C. Not referenced if NCC = 0.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.WORK
WORK is DOUBLE PRECISION array, dimension (4*N)INFO
INFO is INTEGER = 0: successful exit < 0: If INFO = -i, the i-th argument had an illegal value > 0: if NCVT = NRU = NCC = 0, = 1, a split was marked by a positive value in E = 2, current block of Z not diagonalized after 30*N iterations (in inner while loop) = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks) else NCVT = NRU = NCC = 0, the algorithm did not converge; D and E contain the elements of a bidiagonal matrix which is orthogonally similar to the input matrix B; if INFO = i, i elements of E have not converged to zero.
TOLMUL DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8))) TOLMUL controls the convergence criterion of the QR loop. If it is positive, TOLMUL*EPS is the desired relative precision in the computed singular values. If it is negative, abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the computed singular values (corresponds to relative accuracy abs(TOLMUL*EPS) in the largest singular value. abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for fast convergence) and .1/EPS (for there to be some accuracy in the results). Default is to lose at either one eighth or 2 of the available decimal digits in each computed singular value (whichever is smaller). MAXITR INTEGER, default = 6 MAXITR controls the maximum number of passes of the algorithm through its inner loop. The algorithms stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine ddisna (characterJOB, integerM, integerN, double precision, dimension( * )D, double precision, dimension( * )SEP, integerINFO)¶
DDISNA Purpose:DDISNA computes the reciprocal condition numbers for the eigenvectors of a real symmetric or complex Hermitian matrix or for the left or right singular vectors of a general m-by-n matrix. The reciprocal condition number is the 'gap' between the corresponding eigenvalue or singular value and the nearest other one. The bound on the error, measured by angle in radians, in the I-th computed vector is given by DLAMCH( 'E' ) * ( ANORM / SEP( I ) ) where ANORM = 2-norm(A) = max( abs( D(j) ) ). SEP(I) is not allowed to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of the error bound. DDISNA may also be used to compute error bounds for eigenvectors of the generalized symmetric definite eigenproblem.
JOB
Author:
JOB is CHARACTER*1 Specifies for which problem the reciprocal condition numbers should be computed: = 'E': the eigenvectors of a symmetric/Hermitian matrix; = 'L': the left singular vectors of a general matrix; = 'R': the right singular vectors of a general matrix.M
M is INTEGER The number of rows of the matrix. M >= 0.N
N is INTEGER If JOB = 'L' or 'R', the number of columns of the matrix, in which case N >= 0. Ignored if JOB = 'E'.D
D is DOUBLE PRECISION array, dimension (M) if JOB = 'E' dimension (min(M,N)) if JOB = 'L' or 'R' The eigenvalues (if JOB = 'E') or singular values (if JOB = 'L' or 'R') of the matrix, in either increasing or decreasing order. If singular values, they must be non-negative.SEP
SEP is DOUBLE PRECISION array, dimension (M) if JOB = 'E' dimension (min(M,N)) if JOB = 'L' or 'R' The reciprocal condition numbers of the vectors.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine dlaed0 (integerICOMPQ, integerQSIZ, integerN, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( ldq, * )Q, integerLDQ, double precision, dimension( ldqs, * )QSTORE, integerLDQS, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
DLAED0 used by sstedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. Purpose:DLAED0 computes all eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method.
ICOMPQ
Author:
ICOMPQ is INTEGER = 0: Compute eigenvalues only. = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form. = 2: Compute eigenvalues and eigenvectors of tridiagonal matrix.QSIZ
QSIZ is INTEGER The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.D
D is DOUBLE PRECISION array, dimension (N) On entry, the main diagonal of the tridiagonal matrix. On exit, its eigenvalues.E
E is DOUBLE PRECISION array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix. On exit, E has been destroyed.Q
Q is DOUBLE PRECISION array, dimension (LDQ, N) On entry, Q must contain an N-by-N orthogonal matrix. If ICOMPQ = 0 Q is not referenced. If ICOMPQ = 1 On entry, Q is a subset of the columns of the orthogonal matrix used to reduce the full matrix to tridiagonal form corresponding to the subset of the full matrix which is being decomposed at this time. If ICOMPQ = 2 On entry, Q will be the identity matrix. On exit, Q contains the eigenvectors of the tridiagonal matrix.LDQ
LDQ is INTEGER The leading dimension of the array Q. If eigenvectors are desired, then LDQ >= max(1,N). In any case, LDQ >= 1.QSTORE
QSTORE is DOUBLE PRECISION array, dimension (LDQS, N) Referenced only when ICOMPQ = 1. Used to store parts of the eigenvector matrix when the updating matrix multiplies take place.LDQS
LDQS is INTEGER The leading dimension of the array QSTORE. If ICOMPQ = 1, then LDQS >= max(1,N). In any case, LDQS >= 1.WORK
WORK is DOUBLE PRECISION array, If ICOMPQ = 0 or 1, the dimension of WORK must be at least 1 + 3*N + 2*N*lg N + 3*N**2 ( lg( N ) = smallest integer k such that 2^k >= N ) If ICOMPQ = 2, the dimension of WORK must be at least 4*N + N**2.IWORK
IWORK is INTEGER array, If ICOMPQ = 0 or 1, the dimension of IWORK must be at least 6 + 6*N + 5*N*lg N. ( lg( N ) = smallest integer k such that 2^k >= N ) If ICOMPQ = 2, the dimension of IWORK must be at least 3 + 5*N.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1).
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine dlaed1 (integerN, double precision, dimension( * )D, double precision, dimension( ldq, * )Q, integerLDQ, integer, dimension( * )INDXQ, double precisionRHO, integerCUTPNT, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
DLAED1 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal. Purpose:DLAED1 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and eigenvectors of a tridiagonal matrix. DLAED7 handles the case in which eigenvalues only or eigenvalues and eigenvectors of a full symmetric matrix (which was reduced to tridiagonal form) are desired. T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) where Z = Q**T*u, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED2. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by DLAED3). This routine also calculates the eigenvectors of the current problem. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.
N
Author:
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.D
D is DOUBLE PRECISION array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.Q
Q is DOUBLE PRECISION array, dimension (LDQ,N) On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).INDXQ
INDXQ is INTEGER array, dimension (N) On entry, the permutation which separately sorts the two subproblems in D into ascending order. On exit, the permutation which will reintegrate the subproblems back into sorted order, i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.RHO
RHO is DOUBLE PRECISION The subdiagonal entry used to create the rank-1 modification.CUTPNT
CUTPNT is INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2.WORK
WORK is DOUBLE PRECISION array, dimension (4*N + N**2)IWORK
IWORK is INTEGER array, dimension (4*N)INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified by Francoise Tisseur, University of Tennessee
subroutine dlaed2 (integerK, integerN, integerN1, double precision, dimension( * )D, double precision, dimension( ldq, * )Q, integerLDQ, integer, dimension( * )INDXQ, double precisionRHO, double precision, dimension( * )Z, double precision, dimension( * )DLAMDA, double precision, dimension( * )W, double precision, dimension( * )Q2, integer, dimension( * )INDX, integer, dimension( * )INDXC, integer, dimension( * )INDXP, integer, dimension( * )COLTYP, integerINFO)¶
DLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal. Purpose:DLAED2 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.
K
Author:
K is INTEGER The number of non-deflated eigenvalues, and the order of the related secular equation. 0 <= K <=N.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.N1
N1 is INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= N1 <= N/2.D
D is DOUBLE PRECISION array, dimension (N) On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.Q
Q is DOUBLE PRECISION array, dimension (LDQ, N) On entry, Q contains the eigenvectors of two submatrices in the two square blocks with corners at (1,1), (N1,N1) and (N1+1, N1+1), (N,N). On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).INDXQ
INDXQ is INTEGER array, dimension (N) The permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have N1 added to their values. Destroyed on exit.RHO
RHO is DOUBLE PRECISION On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by DLAED3.Z
Z is DOUBLE PRECISION array, dimension (N) On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z have been destroyed by the updating process.DLAMDA
DLAMDA is DOUBLE PRECISION array, dimension (N) A copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation.W
W is DOUBLE PRECISION array, dimension (N) The first k values of the final deflation-altered z-vector which will be passed to DLAED3.Q2
Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2) A copy of the first K eigenvectors which will be used by DLAED3 in a matrix multiply (DGEMM) to solve for the new eigenvectors.INDX
INDX is INTEGER array, dimension (N) The permutation used to sort the contents of DLAMDA into ascending order.INDXC
INDXC is INTEGER array, dimension (N) The permutation used to arrange the columns of the deflated Q matrix into three groups: the first group contains non-zero elements only at and above N1, the second contains non-zero elements only below N1, and the third is dense.INDXP
INDXP is INTEGER array, dimension (N) The permutation used to place deflated values of D at the end of the array. INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues.COLTYP
COLTYP is INTEGER array, dimension (N) During execution, a label which will indicate which of the following types a column in the Q2 matrix is: 1 : non-zero in the upper half only; 2 : dense; 3 : non-zero in the lower half only; 4 : deflated. On exit, COLTYP(i) is the number of columns of type i, for i=1 to 4 only.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified by Francoise Tisseur, University of Tennessee
subroutine dlaed3 (integerK, integerN, integerN1, double precision, dimension( * )D, double precision, dimension( ldq, * )Q, integerLDQ, double precisionRHO, double precision, dimension( * )DLAMDA, double precision, dimension( * )Q2, integer, dimension( * )INDX, integer, dimension( * )CTOT, double precision, dimension( * )W, double precision, dimension( * )S, integerINFO)¶
DLAED3 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal. Purpose:DLAED3 finds the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K. It makes the appropriate calls to DLAED4 and then updates the eigenvectors by multiplying the matrix of eigenvectors of the pair of eigensystems being combined by the matrix of eigenvectors of the K-by-K system which is solved here. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
K
Author:
K is INTEGER The number of terms in the rational function to be solved by DLAED4. K >= 0.N
N is INTEGER The number of rows and columns in the Q matrix. N >= K (deflation may result in N>K).N1
N1 is INTEGER The location of the last eigenvalue in the leading submatrix. min(1,N) <= N1 <= N/2.D
D is DOUBLE PRECISION array, dimension (N) D(I) contains the updated eigenvalues for 1 <= I <= K.Q
Q is DOUBLE PRECISION array, dimension (LDQ,N) Initially the first K columns are used as workspace. On output the columns 1 to K contain the updated eigenvectors.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).RHO
RHO is DOUBLE PRECISION The value of the parameter in the rank one update equation. RHO >= 0 required.DLAMDA
DLAMDA is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation. May be changed on output by having lowest order bit set to zero on Cray X-MP, Cray Y-MP, Cray-2, or Cray C-90, as described above.Q2
Q2 is DOUBLE PRECISION array, dimension (LDQ2, N) The first K columns of this matrix contain the non-deflated eigenvectors for the split problem.INDX
INDX is INTEGER array, dimension (N) The permutation used to arrange the columns of the deflated Q matrix into three groups (see DLAED2). The rows of the eigenvectors found by DLAED4 must be likewise permuted before the matrix multiply can take place.CTOT
CTOT is INTEGER array, dimension (4) A count of the total number of the various types of columns in Q, as described in INDX. The fourth column type is any column which has been deflated.W
W is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating vector. Destroyed on output.S
S is DOUBLE PRECISION array, dimension (N1 + 1)*K Will contain the eigenvectors of the repaired matrix which will be multiplied by the previously accumulated eigenvectors to update the system.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified by Francoise Tisseur, University of Tennessee
subroutine dlaed4 (integerN, integerI, double precision, dimension( * )D, double precision, dimension( * )Z, double precision, dimension( * )DELTA, double precisionRHO, double precisionDLAM, integerINFO)¶
DLAED4 used by sstedc. Finds a single root of the secular equation. Purpose:This subroutine computes the I-th updated eigenvalue of a symmetric rank-one modification to a diagonal matrix whose elements are given in the array d, and that D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rank-one modified system is thus diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.
N
Internal Parameters:
N is INTEGER The length of all arrays.I
I is INTEGER The index of the eigenvalue to be computed. 1 <= I <= N.D
D is DOUBLE PRECISION array, dimension (N) The original eigenvalues. It is assumed that they are in order, D(I) < D(J) for I < J.Z
Z is DOUBLE PRECISION array, dimension (N) The components of the updating vector.DELTA
DELTA is DOUBLE PRECISION array, dimension (N) If N .GT. 2, DELTA contains (D(j) - lambda_I) in its j-th component. If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5 for detail. The vector DELTA contains the information necessary to construct the eigenvectors by DLAED3 and DLAED9.RHO
RHO is DOUBLE PRECISION The scalar in the symmetric updating formula.DLAM
DLAM is DOUBLE PRECISION The computed lambda_I, the I-th updated eigenvalue.INFO
INFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed.
Logical variable ORGATI (origin-at-i?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin. ORGATI = .true. origin at i ORGATI = .false. origin at i+1 Logical variable SWTCH3 (switch-for-3-poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
subroutine dlaed5 (integerI, double precision, dimension( 2 )D, double precision, dimension( 2 )Z, double precision, dimension( 2 )DELTA, double precisionRHO, double precisionDLAM)¶
DLAED5 used by sstedc. Solves the 2-by-2 secular equation. Purpose:This subroutine computes the I-th eigenvalue of a symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) + RHO * Z * transpose(Z) . The diagonal elements in the array D are assumed to satisfy D(i) < D(j) for i < j . We also assume RHO > 0 and that the Euclidean norm of the vector Z is one.
I
Author:
I is INTEGER The index of the eigenvalue to be computed. I = 1 or I = 2.D
D is DOUBLE PRECISION array, dimension (2) The original eigenvalues. We assume D(1) < D(2).Z
Z is DOUBLE PRECISION array, dimension (2) The components of the updating vector.DELTA
DELTA is DOUBLE PRECISION array, dimension (2) The vector DELTA contains the information necessary to construct the eigenvectors.RHO
RHO is DOUBLE PRECISION The scalar in the symmetric updating formula.DLAM
DLAM is DOUBLE PRECISION The computed lambda_I, the I-th updated eigenvalue.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
subroutine dlaed6 (integerKNITER, logicalORGATI, double precisionRHO, double precision, dimension( 3 )D, double precision, dimension( 3 )Z, double precisionFINIT, double precisionTAU, integerINFO)¶
DLAED6 used by sstedc. Computes one Newton step in solution of the secular equation. Purpose:DLAED6 computes the positive or negative root (closest to the origin) of z(1) z(2) z(3) f(x) = rho + --------- + ---------- + --------- d(1)-x d(2)-x d(3)-x It is assumed that if ORGATI = .true. the root is between d(2) and d(3); otherwise it is between d(1) and d(2) This routine will be called by DLAED4 when necessary. In most cases, the root sought is the smallest in magnitude, though it might not be in some extremely rare situations.
KNITER
Author:
KNITER is INTEGER Refer to DLAED4 for its significance.ORGATI
ORGATI is LOGICAL If ORGATI is true, the needed root is between d(2) and d(3); otherwise it is between d(1) and d(2). See DLAED4 for further details.RHO
RHO is DOUBLE PRECISION Refer to the equation f(x) above.D
D is DOUBLE PRECISION array, dimension (3) D satisfies d(1) < d(2) < d(3).Z
Z is DOUBLE PRECISION array, dimension (3) Each of the elements in z must be positive.FINIT
FINIT is DOUBLE PRECISION The value of f at 0. It is more accurate than the one evaluated inside this routine (if someone wants to do so).TAU
TAU is DOUBLE PRECISION The root of the equation f(x).INFO
INFO is INTEGER = 0: successful exit > 0: if INFO = 1, failure to converge
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NAG Ltd.
Date:
December 2016
Further Details:
10/02/03: This version has a few statements commented out for thread safety (machine parameters are computed on each entry). SJH. 05/10/06: Modified from a new version of Ren-Cang Li, use Gragg-Thornton-Warner cubic convergent scheme for better stability.
Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
subroutine dlaed7 (integerICOMPQ, integerN, integerQSIZ, integerTLVLS, integerCURLVL, integerCURPBM, double precision, dimension( * )D, double precision, dimension( ldq, * )Q, integerLDQ, integer, dimension( * )INDXQ, double precisionRHO, integerCUTPNT, double precision, dimension( * )QSTORE, integer, dimension( * )QPTR, integer, dimension( * )PRMPTR, integer, dimension( * )PERM, integer, dimension( * )GIVPTR, integer, dimension( 2, * )GIVCOL, double precision, dimension( 2, * )GIVNUM, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
DLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense. Purpose:DLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense symmetric matrix that has been reduced to tridiagonal form. DLAED1 handles the case in which all eigenvalues and eigenvectors of a symmetric tridiagonal matrix are desired. T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) where Z = Q**Tu, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine DLAED8. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine DLAED4 (as called by DLAED9). This routine also calculates the eigenvectors of the current problem. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.
ICOMPQ
Author:
ICOMPQ is INTEGER = 0: Compute eigenvalues only. = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.QSIZ
QSIZ is INTEGER The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.TLVLS
TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree.CURLVL
CURLVL is INTEGER The current level in the overall merge routine, 0 <= CURLVL <= TLVLS.CURPBM
CURPBM is INTEGER The current problem in the current level in the overall merge routine (counting from upper left to lower right).D
D is DOUBLE PRECISION array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.Q
Q is DOUBLE PRECISION array, dimension (LDQ, N) On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).INDXQ
INDXQ is INTEGER array, dimension (N) The permutation which will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will be in ascending order.RHO
RHO is DOUBLE PRECISION The subdiagonal element used to create the rank-1 modification.CUTPNT
CUTPNT is INTEGER Contains the location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.QSTORE
QSTORE is DOUBLE PRECISION array, dimension (N**2+1) Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices.QPTR
QPTR is INTEGER array, dimension (N+2) List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top.PRMPTR
PRMPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and also the size of the full, non-deflated problem.PERM
PERM is INTEGER array, dimension (N lg N) Contains the permutations (from deflation and sorting) to be applied to each eigenblock.GIVPTR
GIVPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.GIVCOL
GIVCOL is INTEGER array, dimension (2, N lg N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation.GIVNUM
GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N) Each number indicates the S value to be used in the corresponding Givens rotation.WORK
WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)IWORK
IWORK is INTEGER array, dimension (4*N)INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine dlaed8 (integerICOMPQ, integerK, integerN, integerQSIZ, double precision, dimension( * )D, double precision, dimension( ldq, * )Q, integerLDQ, integer, dimension( * )INDXQ, double precisionRHO, integerCUTPNT, double precision, dimension( * )Z, double precision, dimension( * )DLAMDA, double precision, dimension( ldq2, * )Q2, integerLDQ2, double precision, dimension( * )W, integer, dimension( * )PERM, integerGIVPTR, integer, dimension( 2, * )GIVCOL, double precision, dimension( 2, * )GIVNUM, integer, dimension( * )INDXP, integer, dimension( * )INDX, integerINFO)¶
DLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense. Purpose:DLAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny element in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.
ICOMPQ
Author:
ICOMPQ is INTEGER = 0: Compute eigenvalues only. = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.K
K is INTEGER The number of non-deflated eigenvalues, and the order of the related secular equation.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.QSIZ
QSIZ is INTEGER The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.D
D is DOUBLE PRECISION array, dimension (N) On entry, the eigenvalues of the two submatrices to be combined. On exit, the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.Q
Q is DOUBLE PRECISION array, dimension (LDQ,N) If ICOMPQ = 0, Q is not referenced. Otherwise, on entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).INDXQ
INDXQ is INTEGER array, dimension (N) The permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have CUTPNT added to their values in order to be accurate.RHO
RHO is DOUBLE PRECISION On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by DLAED3.CUTPNT
CUTPNT is INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.Z
Z is DOUBLE PRECISION array, dimension (N) On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z are destroyed by the updating process.DLAMDA
DLAMDA is DOUBLE PRECISION array, dimension (N) A copy of the first K eigenvalues which will be used by DLAED3 to form the secular equation.Q2
Q2 is DOUBLE PRECISION array, dimension (LDQ2,N) If ICOMPQ = 0, Q2 is not referenced. Otherwise, a copy of the first K eigenvectors which will be used by DLAED7 in a matrix multiply (DGEMM) to update the new eigenvectors.LDQ2
LDQ2 is INTEGER The leading dimension of the array Q2. LDQ2 >= max(1,N).W
W is DOUBLE PRECISION array, dimension (N) The first k values of the final deflation-altered z-vector and will be passed to DLAED3.PERM
PERM is INTEGER array, dimension (N) The permutations (from deflation and sorting) to be applied to each eigenblock.GIVPTR
GIVPTR is INTEGER The number of Givens rotations which took place in this subproblem.GIVCOL
GIVCOL is INTEGER array, dimension (2, N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation.GIVNUM
GIVNUM is DOUBLE PRECISION array, dimension (2, N) Each number indicates the S value to be used in the corresponding Givens rotation.INDXP
INDXP is INTEGER array, dimension (N) The permutation used to place deflated values of D at the end of the array. INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues.INDX
INDX is INTEGER array, dimension (N) The permutation used to sort the contents of D into ascending order.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine dlaed9 (integerK, integerKSTART, integerKSTOP, integerN, double precision, dimension( * )D, double precision, dimension( ldq, * )Q, integerLDQ, double precisionRHO, double precision, dimension( * )DLAMDA, double precision, dimension( * )W, double precision, dimension( lds, * )S, integerLDS, integerINFO)¶
DLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense. Purpose:DLAED9 finds the roots of the secular equation, as defined by the values in D, Z, and RHO, between KSTART and KSTOP. It makes the appropriate calls to DLAED4 and then stores the new matrix of eigenvectors for use in calculating the next level of Z vectors.
K
Author:
K is INTEGER The number of terms in the rational function to be solved by DLAED4. K >= 0.KSTART
KSTART is INTEGERKSTOP
KSTOP is INTEGER The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP are to be computed. 1 <= KSTART <= KSTOP <= K.N
N is INTEGER The number of rows and columns in the Q matrix. N >= K (delation may result in N > K).D
D is DOUBLE PRECISION array, dimension (N) D(I) contains the updated eigenvalues for KSTART <= I <= KSTOP.Q
Q is DOUBLE PRECISION array, dimension (LDQ,N)LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max( 1, N ).RHO
RHO is DOUBLE PRECISION The value of the parameter in the rank one update equation. RHO >= 0 required.DLAMDA
DLAMDA is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.W
W is DOUBLE PRECISION array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating vector.S
S is DOUBLE PRECISION array, dimension (LDS, K) Will contain the eigenvectors of the repaired matrix which will be stored for subsequent Z vector calculation and multiplied by the previously accumulated eigenvectors to update the system.LDS
LDS is INTEGER The leading dimension of S. LDS >= max( 1, K ).INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine dlaeda (integerN, integerTLVLS, integerCURLVL, integerCURPBM, integer, dimension( * )PRMPTR, integer, dimension( * )PERM, integer, dimension( * )GIVPTR, integer, dimension( 2, * )GIVCOL, double precision, dimension( 2, * )GIVNUM, double precision, dimension( * )Q, integer, dimension( * )QPTR, double precision, dimension( * )Z, double precision, dimension( * )ZTEMP, integerINFO)¶
DLAEDA used by sstedc. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense. Purpose:DLAEDA computes the Z vector corresponding to the merge step in the CURLVLth step of the merge process with TLVLS steps for the CURPBMth problem.
N
Author:
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.TLVLS
TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree.CURLVL
CURLVL is INTEGER The current level in the overall merge routine, 0 <= curlvl <= tlvls.CURPBM
CURPBM is INTEGER The current problem in the current level in the overall merge routine (counting from upper left to lower right).PRMPTR
PRMPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and incidentally the size of the full, non-deflated problem.PERM
PERM is INTEGER array, dimension (N lg N) Contains the permutations (from deflation and sorting) to be applied to each eigenblock.GIVPTR
GIVPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.GIVCOL
GIVCOL is INTEGER array, dimension (2, N lg N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation.GIVNUM
GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N) Each number indicates the S value to be used in the corresponding Givens rotation.Q
Q is DOUBLE PRECISION array, dimension (N**2) Contains the square eigenblocks from previous levels, the starting positions for blocks are given by QPTR.QPTR
QPTR is INTEGER array, dimension (N+2) Contains a list of pointers which indicate where in Q an eigenblock is stored. SQRT( QPTR(i+1) - QPTR(i) ) indicates the size of the block.Z
Z is DOUBLE PRECISION array, dimension (N) On output this vector contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix).ZTEMP
ZTEMP is DOUBLE PRECISION array, dimension (N)INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine dlagtf (integerN, double precision, dimension( * )A, double precisionLAMBDA, double precision, dimension( * )B, double precision, dimension( * )C, double precisionTOL, double precision, dimension( * )D, integer, dimension( * )IN, integerINFO)¶
DLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges. Purpose:DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n tridiagonal matrix and lambda is a scalar, as T - lambda*I = PLU, where P is a permutation matrix, L is a unit lower tridiagonal matrix with at most one non-zero sub-diagonal elements per column and U is an upper triangular matrix with at most two non-zero super-diagonal elements per column. The factorization is obtained by Gaussian elimination with partial pivoting and implicit row scaling. The parameter LAMBDA is included in the routine so that DLAGTF may be used, in conjunction with DLAGTS, to obtain eigenvectors of T by inverse iteration.
N
Author:
N is INTEGER The order of the matrix T.A
A is DOUBLE PRECISION array, dimension (N) On entry, A must contain the diagonal elements of T. On exit, A is overwritten by the n diagonal elements of the upper triangular matrix U of the factorization of T.LAMBDA
LAMBDA is DOUBLE PRECISION On entry, the scalar lambda.B
B is DOUBLE PRECISION array, dimension (N-1) On entry, B must contain the (n-1) super-diagonal elements of T. On exit, B is overwritten by the (n-1) super-diagonal elements of the matrix U of the factorization of T.C
C is DOUBLE PRECISION array, dimension (N-1) On entry, C must contain the (n-1) sub-diagonal elements of T. On exit, C is overwritten by the (n-1) sub-diagonal elements of the matrix L of the factorization of T.TOL
TOL is DOUBLE PRECISION On entry, a relative tolerance used to indicate whether or not the matrix (T - lambda*I) is nearly singular. TOL should normally be chose as approximately the largest relative error in the elements of T. For example, if the elements of T are correct to about 4 significant figures, then TOL should be set to about 5*10**(-4). If TOL is supplied as less than eps, where eps is the relative machine precision, then the value eps is used in place of TOL.D
D is DOUBLE PRECISION array, dimension (N-2) On exit, D is overwritten by the (n-2) second super-diagonal elements of the matrix U of the factorization of T.IN
IN is INTEGER array, dimension (N) On exit, IN contains details of the permutation matrix P. If an interchange occurred at the kth step of the elimination, then IN(k) = 1, otherwise IN(k) = 0. The element IN(n) returns the smallest positive integer j such that abs( u(j,j) ).le. norm( (T - lambda*I)(j) )*TOL, where norm( A(j) ) denotes the sum of the absolute values of the jth row of the matrix A. If no such j exists then IN(n) is returned as zero. If IN(n) is returned as positive, then a diagonal element of U is small, indicating that (T - lambda*I) is singular or nearly singular,INFO
INFO is INTEGER = 0 : successful exit .lt. 0: if INFO = -k, the kth argument had an illegal value
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Date:
December 2016
subroutine dlamrg (integerN1, integerN2, double precision, dimension( * )A, integerDTRD1, integerDTRD2, integer, dimension( * )INDEX)¶
DLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order. Purpose:DLAMRG will create a permutation list which will merge the elements of A (which is composed of two independently sorted sets) into a single set which is sorted in ascending order.
N1
Author:
N1 is INTEGERN2
N2 is INTEGER These arguments contain the respective lengths of the two sorted lists to be merged.A
A is DOUBLE PRECISION array, dimension (N1+N2) The first N1 elements of A contain a list of numbers which are sorted in either ascending or descending order. Likewise for the final N2 elements.DTRD1
DTRD1 is INTEGERDTRD2
DTRD2 is INTEGER These are the strides to be taken through the array A. Allowable strides are 1 and -1. They indicate whether a subset of A is sorted in ascending (DTRDx = 1) or descending (DTRDx = -1) order.INDEX
INDEX is INTEGER array, dimension (N1+N2) On exit this array will contain a permutation such that if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be sorted in ascending order.
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NAG Ltd.
Date:
June 2016
subroutine dlartgs (double precisionX, double precisionY, double precisionSIGMA, double precisionCS, double precisionSN)¶
DLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem. Purpose:DLARTGS generates a plane rotation designed to introduce a bulge in Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD problem. X and Y are the top-row entries, and SIGMA is the shift. The computed CS and SN define a plane rotation satisfying [ CS SN ] . [ X^2 - SIGMA ] = [ R ], [ -SN CS ] [ X * Y ] [ 0 ] with R nonnegative. If X^2 - SIGMA and X * Y are 0, then the rotation is by PI/2.
X
Author:
X is DOUBLE PRECISION The (1,1) entry of an upper bidiagonal matrix.Y
Y is DOUBLE PRECISION The (1,2) entry of an upper bidiagonal matrix.SIGMA
SIGMA is DOUBLE PRECISION The shift.CS
CS is DOUBLE PRECISION The cosine of the rotation.SN
SN is DOUBLE PRECISION The sine of the rotation.
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NAG Ltd.
Date:
December 2016
subroutine dlasq1 (integerN, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( * )WORK, integerINFO)¶
DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr. Purpose:DLASQ1 computes the singular values of a real N-by-N bidiagonal matrix with diagonal D and off-diagonal E. The singular values are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. The algorithm was first presented in "Accurate singular values and differential qd algorithms" by K. V. Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, 1994, and the present implementation is described in "An implementation of the dqds Algorithm (Positive Case)", LAPACK Working Note.
N
Author:
N is INTEGER The number of rows and columns in the matrix. N >= 0.D
D is DOUBLE PRECISION array, dimension (N) On entry, D contains the diagonal elements of the bidiagonal matrix whose SVD is desired. On normal exit, D contains the singular values in decreasing order.E
E is DOUBLE PRECISION array, dimension (N) On entry, elements E(1:N-1) contain the off-diagonal elements of the bidiagonal matrix whose SVD is desired. On exit, E is overwritten.WORK
WORK is DOUBLE PRECISION array, dimension (4*N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm failed = 1, a split was marked by a positive value in E = 2, current block of Z not diagonalized after 100*N iterations (in inner while loop) On exit D and E represent a matrix with the same singular values which the calling subroutine could use to finish the computation, or even feed back into DLASQ1 = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)
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Date:
December 2016
subroutine dlasq2 (integerN, double precision, dimension( * )Z, integerINFO)¶
DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr. Purpose:DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd array Z to high relative accuracy are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. To see the relation of Z to the tridiagonal matrix, let L be a unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and let U be an upper bidiagonal matrix with 1's above and diagonal Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the symmetric tridiagonal to which it is similar. Note : DLASQ2 defines a logical variable, IEEE, which is true on machines which follow ieee-754 floating-point standard in their handling of infinities and NaNs, and false otherwise. This variable is passed to DLASQ3.
N
Author:
N is INTEGER The number of rows and columns in the matrix. N >= 0.Z
Z is DOUBLE PRECISION array, dimension ( 4*N ) On entry Z holds the qd array. On exit, entries 1 to N hold the eigenvalues in decreasing order, Z( 2*N+1 ) holds the trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 ) holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of shifts that failed.INFO
INFO is INTEGER = 0: successful exit < 0: if the i-th argument is a scalar and had an illegal value, then INFO = -i, if the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j) > 0: the algorithm failed = 1, a split was marked by a positive value in E = 2, current block of Z not diagonalized after 100*N iterations (in inner while loop). On exit Z holds a qd array with the same eigenvalues as the given Z. = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)
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Date:
December 2016
Further Details:
Local Variables: I0:N0 defines a current unreduced segment of Z. The shifts are accumulated in SIGMA. Iteration count is in ITER. Ping-pong is controlled by PP (alternates between 0 and 1).
subroutine dlasq3 (integerI0, integerN0, double precision, dimension( * )Z, integerPP, double precisionDMIN, double precisionSIGMA, double precisionDESIG, double precisionQMAX, integerNFAIL, integerITER, integerNDIV, logicalIEEE, integerTTYPE, double precisionDMIN1, double precisionDMIN2, double precisionDN, double precisionDN1, double precisionDN2, double precisionG, double precisionTAU)¶
DLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr. Purpose:DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds. In case of failure it changes shifts, and tries again until output is positive.
I0
Author:
I0 is INTEGER First index.N0
N0 is INTEGER Last index.Z
Z is DOUBLE PRECISION array, dimension ( 4*N0 ) Z holds the qd array.PP
PP is INTEGER PP=0 for ping, PP=1 for pong. PP=2 indicates that flipping was applied to the Z array and that the initial tests for deflation should not be performed.DMIN
DMIN is DOUBLE PRECISION Minimum value of d.SIGMA
SIGMA is DOUBLE PRECISION Sum of shifts used in current segment.DESIG
DESIG is DOUBLE PRECISION Lower order part of SIGMAQMAX
QMAX is DOUBLE PRECISION Maximum value of q.NFAIL
NFAIL is INTEGER Increment NFAIL by 1 each time the shift was too big.ITER
ITER is INTEGER Increment ITER by 1 for each iteration.NDIV
NDIV is INTEGER Increment NDIV by 1 for each division.IEEE
IEEE is LOGICAL Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).TTYPE
TTYPE is INTEGER Shift type.DMIN1
DMIN1 is DOUBLE PRECISIONDMIN2
DMIN2 is DOUBLE PRECISIONDN
DN is DOUBLE PRECISIONDN1
DN1 is DOUBLE PRECISIONDN2
DN2 is DOUBLE PRECISIONG
G is DOUBLE PRECISIONTAU
TAU is DOUBLE PRECISION These are passed as arguments in order to save their values between calls to DLASQ3.
Univ. of Tennessee
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Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
subroutine dlasq4 (integerI0, integerN0, double precision, dimension( * )Z, integerPP, integerN0IN, double precisionDMIN, double precisionDMIN1, double precisionDMIN2, double precisionDN, double precisionDN1, double precisionDN2, double precisionTAU, integerTTYPE, double precisionG)¶
DLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr. Purpose:DLASQ4 computes an approximation TAU to the smallest eigenvalue using values of d from the previous transform.
I0
Author:
I0 is INTEGER First index.N0
N0 is INTEGER Last index.Z
Z is DOUBLE PRECISION array, dimension ( 4*N0 ) Z holds the qd array.PP
PP is INTEGER PP=0 for ping, PP=1 for pong.N0IN
N0IN is INTEGER The value of N0 at start of EIGTEST.DMIN
DMIN is DOUBLE PRECISION Minimum value of d.DMIN1
DMIN1 is DOUBLE PRECISION Minimum value of d, excluding D( N0 ).DMIN2
DMIN2 is DOUBLE PRECISION Minimum value of d, excluding D( N0 ) and D( N0-1 ).DN
DN is DOUBLE PRECISION d(N)DN1
DN1 is DOUBLE PRECISION d(N-1)DN2
DN2 is DOUBLE PRECISION d(N-2)TAU
TAU is DOUBLE PRECISION This is the shift.TTYPE
TTYPE is INTEGER Shift type.G
G is DOUBLE PRECISION G is passed as an argument in order to save its value between calls to DLASQ4.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
Further Details:
CNST1 = 9/16
subroutine dlasq5 (integerI0, integerN0, double precision, dimension( * )Z, integerPP, double precisionTAU, double precisionSIGMA, double precisionDMIN, double precisionDMIN1, double precisionDMIN2, double precisionDN, double precisionDNM1, double precisionDNM2, logicalIEEE, double precisionEPS)¶
DLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr. Purpose:DLASQ5 computes one dqds transform in ping-pong form, one version for IEEE machines another for non IEEE machines.
I0
Author:
I0 is INTEGER First index.N0
N0 is INTEGER Last index.Z
Z is DOUBLE PRECISION array, dimension ( 4*N ) Z holds the qd array. EMIN is stored in Z(4*N0) to avoid an extra argument.PP
PP is INTEGER PP=0 for ping, PP=1 for pong.TAU
TAU is DOUBLE PRECISION This is the shift.SIGMA
SIGMA is DOUBLE PRECISION This is the accumulated shift up to this step.DMIN
DMIN is DOUBLE PRECISION Minimum value of d.DMIN1
DMIN1 is DOUBLE PRECISION Minimum value of d, excluding D( N0 ).DMIN2
DMIN2 is DOUBLE PRECISION Minimum value of d, excluding D( N0 ) and D( N0-1 ).DN
DN is DOUBLE PRECISION d(N0), the last value of d.DNM1
DNM1 is DOUBLE PRECISION d(N0-1).DNM2
DNM2 is DOUBLE PRECISION d(N0-2).IEEE
IEEE is LOGICAL Flag for IEEE or non IEEE arithmetic.EPS
EPS is DOUBLE PRECISION This is the value of epsilon used.
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Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine dlasq6 (integerI0, integerN0, double precision, dimension( * )Z, integerPP, double precisionDMIN, double precisionDMIN1, double precisionDMIN2, double precisionDN, double precisionDNM1, double precisionDNM2)¶
DLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr. Purpose:DLASQ6 computes one dqd (shift equal to zero) transform in ping-pong form, with protection against underflow and overflow.
I0
Author:
I0 is INTEGER First index.N0
N0 is INTEGER Last index.Z
Z is DOUBLE PRECISION array, dimension ( 4*N ) Z holds the qd array. EMIN is stored in Z(4*N0) to avoid an extra argument.PP
PP is INTEGER PP=0 for ping, PP=1 for pong.DMIN
DMIN is DOUBLE PRECISION Minimum value of d.DMIN1
DMIN1 is DOUBLE PRECISION Minimum value of d, excluding D( N0 ).DMIN2
DMIN2 is DOUBLE PRECISION Minimum value of d, excluding D( N0 ) and D( N0-1 ).DN
DN is DOUBLE PRECISION d(N0), the last value of d.DNM1
DNM1 is DOUBLE PRECISION d(N0-1).DNM2
DNM2 is DOUBLE PRECISION d(N0-2).
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NAG Ltd.
Date:
December 2016
subroutine dlasrt (characterID, integerN, double precision, dimension( * )D, integerINFO)¶
DLASRT sorts numbers in increasing or decreasing order. Purpose:Sort the numbers in D in increasing order (if ID = 'I') or in decreasing order (if ID = 'D' ). Use Quick Sort, reverting to Insertion sort on arrays of size <= 20. Dimension of STACK limits N to about 2**32.
ID
Author:
ID is CHARACTER*1 = 'I': sort D in increasing order; = 'D': sort D in decreasing order.N
N is INTEGER The length of the array D.D
D is DOUBLE PRECISION array, dimension (N) On entry, the array to be sorted. On exit, D has been sorted into increasing order (D(1) <= ... <= D(N) ) or into decreasing order (D(1) >= ... >= D(N) ), depending on ID.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
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NAG Ltd.
Date:
June 2016
subroutine dstebz (characterRANGE, characterORDER, integerN, double precisionVL, double precisionVU, integerIL, integerIU, double precisionABSTOL, double precision, dimension( * )D, double precision, dimension( * )E, integerM, integerNSPLIT, double precision, dimension( * )W, integer, dimension( * )IBLOCK, integer, dimension( * )ISPLIT, double precision, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
DSTEBZ Purpose:DSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
RANGE
Internal Parameters:
RANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the half-open interval (VL, VU] will be found. = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found.ORDER
ORDER is CHARACTER*1 = 'B': ("By Block") the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest.N
N is INTEGER The order of the tridiagonal matrix T. N >= 0.VL
VL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.VU
VU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.IL
IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.IU
IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.ABSTOL
ABSTOL is DOUBLE PRECISION The absolute tolerance for the eigenvalues. An eigenvalue (or cluster) is considered to be located if it has been determined to lie in an interval whose width is ABSTOL or less. If ABSTOL is less than or equal to zero, then ULP*|T| will be used, where |T| means the 1-norm of T. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero.D
D is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the tridiagonal matrix T.E
E is DOUBLE PRECISION array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T.M
M is INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.)NSPLIT
NSPLIT is INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.W
W is DOUBLE PRECISION array, dimension (N) On exit, the first M elements of W will contain the eigenvalues. (DSTEBZ may use the remaining N-M elements as workspace.)IBLOCK
IBLOCK is INTEGER array, dimension (N) At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the number of blocks) the eigenvalue W(i) belongs. (DSTEBZ may use the remaining N-M elements as workspace.)ISPLIT
ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.)WORK
WORK is DOUBLE PRECISION array, dimension (4*N)IWORK
IWORK is INTEGER array, dimension (3*N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: some or all of the eigenvalues failed to converge or were not computed: =1 or 3: Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only: Not all of the eigenvalues IL:IU were found. Effect: M < IU+1-IL Cause: non-monotonic arithmetic, causing the Sturm sequence to be non-monotonic. Cure: recalculate, using RANGE='A', and pick out eigenvalues IL:IU. In some cases, increasing the PARAMETER "FUDGE" may make things work. = 4: RANGE='I', and the Gershgorin interval initially used was too small. No eigenvalues were computed. Probable cause: your machine has sloppy floating-point arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again.
RELFAC DOUBLE PRECISION, default = 2.0e0 The relative tolerance. An interval (a,b] lies within "relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), where "ulp" is the machine precision (distance from 1 to the next larger floating point number.) FUDGE DOUBLE PRECISION, default = 2 A "fudge factor" to widen the Gershgorin intervals. Ideally, a value of 1 should work, but on machines with sloppy arithmetic, this needs to be larger. The default for publicly released versions should be large enough to handle the worst machine around. Note that this has no effect on accuracy of the solution.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
subroutine dstedc (characterCOMPZ, integerN, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( ldz, * )Z, integerLDZ, double precision, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)¶
DSTEDC Purpose:DSTEDC computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method. The eigenvectors of a full or band real symmetric matrix can also be found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to tridiagonal form. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. See DLAED3 for details.
COMPZ
Author:
COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'I': Compute eigenvectors of tridiagonal matrix also. = 'V': Compute eigenvectors of original dense symmetric matrix also. On entry, Z contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.D
D is DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order.E
E is DOUBLE PRECISION array, dimension (N-1) On entry, the subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.Z
Z is DOUBLE PRECISION array, dimension (LDZ,N) On entry, if COMPZ = 'V', then Z contains the orthogonal matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original symmetric matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If eigenvectors are desired, then LDZ >= max(1,N).WORK
WORK is DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK
LWORK is INTEGER The dimension of the array WORK. If COMPZ = 'N' or N <= 1 then LWORK must be at least 1. If COMPZ = 'V' and N > 1 then LWORK must be at least ( 1 + 3*N + 2*N*lg N + 4*N**2 ), where lg( N ) = smallest integer k such that 2**k >= N. If COMPZ = 'I' and N > 1 then LWORK must be at least ( 1 + 4*N + N**2 ). Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LWORK need only be max(1,2*(N-1)). 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.IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.LIWORK
LIWORK is INTEGER The dimension of the array IWORK. If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1. If COMPZ = 'V' and N > 1 then LIWORK must be at least ( 6 + 6*N + 5*N*lg N ). If COMPZ = 'I' and N > 1 then LIWORK must be at least ( 3 + 5*N ). Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LIWORK need only be 1. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1).
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified by Francoise Tisseur, University of Tennessee
subroutine dsteqr (characterCOMPZ, integerN, double precision, dimension( * )D, double precision, dimension( * )E, double precision, dimension( ldz, * )Z, integerLDZ, double precision, dimension( * )WORK, integerINFO)¶
DSTEQR Purpose:DSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band symmetric matrix can also be found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to tridiagonal form.
COMPZ
Author:
COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvalues and eigenvectors of the original symmetric matrix. On entry, Z must contain the orthogonal matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvalues and eigenvectors of the tridiagonal matrix. Z is initialized to the identity matrix.N
N is INTEGER The order of the matrix. N >= 0.D
D is DOUBLE PRECISION array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order.E
E is DOUBLE PRECISION array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.Z
Z is DOUBLE PRECISION array, dimension (LDZ, N) On entry, if COMPZ = 'V', then Z contains the orthogonal matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original symmetric matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if eigenvectors are desired, then LDZ >= max(1,N).WORK
WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2)) If COMPZ = 'N', then WORK is not referenced.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm has failed to find all the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero; on exit, D and E contain the elements of a symmetric tridiagonal matrix which is orthogonally similar to the original matrix.
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Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine dsterf (integerN, double precision, dimension( * )D, double precision, dimension( * )E, integerINFO)¶
DSTERF Purpose:DSTERF computes all eigenvalues of a symmetric tridiagonal matrix using the Pal-Walker-Kahan variant of the QL or QR algorithm.
N
Author:
N is INTEGER The order of the matrix. N >= 0.D
D is DOUBLE PRECISION array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order.E
E is DOUBLE PRECISION array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm failed to find all of the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero.
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NAG Ltd.
Date:
December 2016
integer function iladiag (characterDIAG)¶
ILADIAG Purpose:This subroutine translated from a character string specifying if a matrix has unit diagonal or not to the relevant BLAST-specified integer constant. ILADIAG returns an INTEGER. If ILADIAG < 0, then the input is not a character indicating a unit or non-unit diagonal. Otherwise ILADIAG returns the constant value corresponding to DIAG.
Univ. of Tennessee
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Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
integer function ilaprec (characterPREC)¶
ILAPREC Purpose:This subroutine translated from a character string specifying an intermediate precision to the relevant BLAST-specified integer constant. ILAPREC returns an INTEGER. If ILAPREC < 0, then the input is not a character indicating a supported intermediate precision. Otherwise ILAPREC returns the constant value corresponding to PREC.
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NAG Ltd.
Date:
December 2016
integer function ilatrans (characterTRANS)¶
ILATRANS Purpose:This subroutine translates from a character string specifying a transposition operation to the relevant BLAST-specified integer constant. ILATRANS returns an INTEGER. If ILATRANS < 0, then the input is not a character indicating a transposition operator. Otherwise ILATRANS returns the constant value corresponding to TRANS.
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NAG Ltd.
Date:
December 2016
integer function ilauplo (characterUPLO)¶
ILAUPLO Purpose:This subroutine translated from a character string specifying a upper- or lower-triangular matrix to the relevant BLAST-specified integer constant. ILAUPLO returns an INTEGER. If ILAUPLO < 0, then the input is not a character indicating an upper- or lower-triangular matrix. Otherwise ILAUPLO returns the constant value corresponding to UPLO.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine sbdsdc (characterUPLO, characterCOMPQ, integerN, real, dimension( * )D, real, dimension( * )E, real, dimension( ldu, * )U, integerLDU, real, dimension( ldvt, * )VT, integerLDVT, real, dimension( * )Q, integer, dimension( * )IQ, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
SBDSDC Purpose:SBDSDC computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT, using a divide and conquer method, where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively. SBDSDC can be used to compute all singular values, and optionally, singular vectors or singular vectors in compact form. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. See SLASD3 for details. The code currently calls SLASDQ if singular values only are desired. However, it can be slightly modified to compute singular values using the divide and conquer method.
UPLO
Author:
UPLO is CHARACTER*1 = 'U': B is upper bidiagonal. = 'L': B is lower bidiagonal.COMPQ
COMPQ is CHARACTER*1 Specifies whether singular vectors are to be computed as follows: = 'N': Compute singular values only; = 'P': Compute singular values and compute singular vectors in compact form; = 'I': Compute singular values and singular vectors.N
N is INTEGER The order of the matrix B. N >= 0.D
D is REAL array, dimension (N) On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B.E
E is REAL array, dimension (N-1) On entry, the elements of E contain the offdiagonal elements of the bidiagonal matrix whose SVD is desired. On exit, E has been destroyed.U
U is REAL array, dimension (LDU,N) If COMPQ = 'I', then: On exit, if INFO = 0, U contains the left singular vectors of the bidiagonal matrix. For other values of COMPQ, U is not referenced.LDU
LDU is INTEGER The leading dimension of the array U. LDU >= 1. If singular vectors are desired, then LDU >= max( 1, N ).VT
VT is REAL array, dimension (LDVT,N) If COMPQ = 'I', then: On exit, if INFO = 0, VT**T contains the right singular vectors of the bidiagonal matrix. For other values of COMPQ, VT is not referenced.LDVT
LDVT is INTEGER The leading dimension of the array VT. LDVT >= 1. If singular vectors are desired, then LDVT >= max( 1, N ).Q
Q is REAL array, dimension (LDQ) If COMPQ = 'P', then: On exit, if INFO = 0, Q and IQ contain the left and right singular vectors in a compact form, requiring O(N log N) space instead of 2*N**2. In particular, Q contains all the REAL data in LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1)))) words of memory, where SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25). For other values of COMPQ, Q is not referenced.IQ
IQ is INTEGER array, dimension (LDIQ) If COMPQ = 'P', then: On exit, if INFO = 0, Q and IQ contain the left and right singular vectors in a compact form, requiring O(N log N) space instead of 2*N**2. In particular, IQ contains all INTEGER data in LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1)))) words of memory, where SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25). For other values of COMPQ, IQ is not referenced.WORK
WORK is REAL array, dimension (MAX(1,LWORK)) If COMPQ = 'N' then LWORK >= (4 * N). If COMPQ = 'P' then LWORK >= (6 * N). If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).IWORK
IWORK is INTEGER array, dimension (8*N)INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute a singular value. The update process of divide and conquer failed.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
Contributors:
Ming Gu and Huan Ren, Computer Science Division,
University of California at Berkeley, USA
subroutine sbdsqr (characterUPLO, integerN, integerNCVT, integerNRU, integerNCC, real, dimension( * )D, real, dimension( * )E, real, dimension( ldvt, * )VT, integerLDVT, real, dimension( ldu, * )U, integerLDU, real, dimension( ldc, * )C, integerLDC, real, dimension( * )WORK, integerINFO)¶
SBDSQR Purpose:SBDSQR computes the singular values and, optionally, the right and/or left singular vectors from the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B using the implicit zero-shift QR algorithm. The SVD of B has the form B = Q * S * P**T where S is the diagonal matrix of singular values, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular vectors. If left singular vectors are requested, this subroutine actually returns U*Q instead of Q, and, if right singular vectors are requested, this subroutine returns P**T*VT instead of P**T, for given real input matrices U and VT. When U and VT are the orthogonal matrices that reduce a general matrix A to bidiagonal form: A = U*B*VT, as computed by SGEBRD, then A = (U*Q) * S * (P**T*VT) is the SVD of A. Optionally, the subroutine may also compute Q**T*C for a given real input matrix C. See "Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873-912, Sept 1990) and "Accurate singular values and differential qd algorithms," by B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics Department, University of California at Berkeley, July 1992 for a detailed description of the algorithm.
UPLO
Internal Parameters:
UPLO is CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal.N
N is INTEGER The order of the matrix B. N >= 0.NCVT
NCVT is INTEGER The number of columns of the matrix VT. NCVT >= 0.NRU
NRU is INTEGER The number of rows of the matrix U. NRU >= 0.NCC
NCC is INTEGER The number of columns of the matrix C. NCC >= 0.D
D is REAL array, dimension (N) On entry, the n diagonal elements of the bidiagonal matrix B. On exit, if INFO=0, the singular values of B in decreasing order.E
E is REAL array, dimension (N-1) On entry, the N-1 offdiagonal elements of the bidiagonal matrix B. On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E will contain the diagonal and superdiagonal elements of a bidiagonal matrix orthogonally equivalent to the one given as input.VT
VT is REAL array, dimension (LDVT, NCVT) On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by P**T * VT. Not referenced if NCVT = 0.LDVT
LDVT is INTEGER The leading dimension of the array VT. LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.U
U is REAL array, dimension (LDU, N) On entry, an NRU-by-N matrix U. On exit, U is overwritten by U * Q. Not referenced if NRU = 0.LDU
LDU is INTEGER The leading dimension of the array U. LDU >= max(1,NRU).C
C is REAL array, dimension (LDC, NCC) On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q**T * C. Not referenced if NCC = 0.LDC
LDC is INTEGER The leading dimension of the array C. LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.WORK
WORK is REAL array, dimension (4*N)INFO
INFO is INTEGER = 0: successful exit < 0: If INFO = -i, the i-th argument had an illegal value > 0: if NCVT = NRU = NCC = 0, = 1, a split was marked by a positive value in E = 2, current block of Z not diagonalized after 30*N iterations (in inner while loop) = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks) else NCVT = NRU = NCC = 0, the algorithm did not converge; D and E contain the elements of a bidiagonal matrix which is orthogonally similar to the input matrix B; if INFO = i, i elements of E have not converged to zero.
TOLMUL REAL, default = max(10,min(100,EPS**(-1/8))) TOLMUL controls the convergence criterion of the QR loop. If it is positive, TOLMUL*EPS is the desired relative precision in the computed singular values. If it is negative, abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the computed singular values (corresponds to relative accuracy abs(TOLMUL*EPS) in the largest singular value. abs(TOLMUL) should be between 1 and 1/EPS, and preferably between 10 (for fast convergence) and .1/EPS (for there to be some accuracy in the results). Default is to lose at either one eighth or 2 of the available decimal digits in each computed singular value (whichever is smaller). MAXITR INTEGER, default = 6 MAXITR controls the maximum number of passes of the algorithm through its inner loop. The algorithms stops (and so fails to converge) if the number of passes through the inner loop exceeds MAXITR*N**2.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine sdisna (characterJOB, integerM, integerN, real, dimension( * )D, real, dimension( * )SEP, integerINFO)¶
SDISNA Purpose:SDISNA computes the reciprocal condition numbers for the eigenvectors of a real symmetric or complex Hermitian matrix or for the left or right singular vectors of a general m-by-n matrix. The reciprocal condition number is the 'gap' between the corresponding eigenvalue or singular value and the nearest other one. The bound on the error, measured by angle in radians, in the I-th computed vector is given by SLAMCH( 'E' ) * ( ANORM / SEP( I ) ) where ANORM = 2-norm(A) = max( abs( D(j) ) ). SEP(I) is not allowed to be smaller than SLAMCH( 'E' )*ANORM in order to limit the size of the error bound. SDISNA may also be used to compute error bounds for eigenvectors of the generalized symmetric definite eigenproblem.
JOB
Author:
JOB is CHARACTER*1 Specifies for which problem the reciprocal condition numbers should be computed: = 'E': the eigenvectors of a symmetric/Hermitian matrix; = 'L': the left singular vectors of a general matrix; = 'R': the right singular vectors of a general matrix.M
M is INTEGER The number of rows of the matrix. M >= 0.N
N is INTEGER If JOB = 'L' or 'R', the number of columns of the matrix, in which case N >= 0. Ignored if JOB = 'E'.D
D is REAL array, dimension (M) if JOB = 'E' dimension (min(M,N)) if JOB = 'L' or 'R' The eigenvalues (if JOB = 'E') or singular values (if JOB = 'L' or 'R') of the matrix, in either increasing or decreasing order. If singular values, they must be non-negative.SEP
SEP is REAL array, dimension (M) if JOB = 'E' dimension (min(M,N)) if JOB = 'L' or 'R' The reciprocal condition numbers of the vectors.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine slaed0 (integerICOMPQ, integerQSIZ, integerN, real, dimension( * )D, real, dimension( * )E, real, dimension( ldq, * )Q, integerLDQ, real, dimension( ldqs, * )QSTORE, integerLDQS, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
SLAED0 used by sstedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method. Purpose:SLAED0 computes all eigenvalues and corresponding eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method.
ICOMPQ
Author:
ICOMPQ is INTEGER = 0: Compute eigenvalues only. = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form. = 2: Compute eigenvalues and eigenvectors of tridiagonal matrix.QSIZ
QSIZ is INTEGER The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.D
D is REAL array, dimension (N) On entry, the main diagonal of the tridiagonal matrix. On exit, its eigenvalues.E
E is REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix. On exit, E has been destroyed.Q
Q is REAL array, dimension (LDQ, N) On entry, Q must contain an N-by-N orthogonal matrix. If ICOMPQ = 0 Q is not referenced. If ICOMPQ = 1 On entry, Q is a subset of the columns of the orthogonal matrix used to reduce the full matrix to tridiagonal form corresponding to the subset of the full matrix which is being decomposed at this time. If ICOMPQ = 2 On entry, Q will be the identity matrix. On exit, Q contains the eigenvectors of the tridiagonal matrix.LDQ
LDQ is INTEGER The leading dimension of the array Q. If eigenvectors are desired, then LDQ >= max(1,N). In any case, LDQ >= 1.QSTORE
QSTORE is REAL array, dimension (LDQS, N) Referenced only when ICOMPQ = 1. Used to store parts of the eigenvector matrix when the updating matrix multiplies take place.LDQS
LDQS is INTEGER The leading dimension of the array QSTORE. If ICOMPQ = 1, then LDQS >= max(1,N). In any case, LDQS >= 1.WORK
WORK is REAL array, If ICOMPQ = 0 or 1, the dimension of WORK must be at least 1 + 3*N + 2*N*lg N + 3*N**2 ( lg( N ) = smallest integer k such that 2^k >= N ) If ICOMPQ = 2, the dimension of WORK must be at least 4*N + N**2.IWORK
IWORK is INTEGER array, If ICOMPQ = 0 or 1, the dimension of IWORK must be at least 6 + 6*N + 5*N*lg N. ( lg( N ) = smallest integer k such that 2^k >= N ) If ICOMPQ = 2, the dimension of IWORK must be at least 3 + 5*N.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1).
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine slaed1 (integerN, real, dimension( * )D, real, dimension( ldq, * )Q, integerLDQ, integer, dimension( * )INDXQ, realRHO, integerCUTPNT, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
SLAED1 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal. Purpose:SLAED1 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and eigenvectors of a tridiagonal matrix. SLAED7 handles the case in which eigenvalues only or eigenvalues and eigenvectors of a full symmetric matrix (which was reduced to tridiagonal form) are desired. T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) where Z = Q**T*u, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLAED2. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine SLAED4 (as called by SLAED3). This routine also calculates the eigenvectors of the current problem. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.
N
Author:
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.D
D is REAL array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.Q
Q is REAL array, dimension (LDQ,N) On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).INDXQ
INDXQ is INTEGER array, dimension (N) On entry, the permutation which separately sorts the two subproblems in D into ascending order. On exit, the permutation which will reintegrate the subproblems back into sorted order, i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.RHO
RHO is REAL The subdiagonal entry used to create the rank-1 modification.CUTPNT
CUTPNT is INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N/2.WORK
WORK is REAL array, dimension (4*N + N**2)IWORK
IWORK is INTEGER array, dimension (4*N)INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified by Francoise Tisseur, University of Tennessee
subroutine slaed2 (integerK, integerN, integerN1, real, dimension( * )D, real, dimension( ldq, * )Q, integerLDQ, integer, dimension( * )INDXQ, realRHO, real, dimension( * )Z, real, dimension( * )DLAMDA, real, dimension( * )W, real, dimension( * )Q2, integer, dimension( * )INDX, integer, dimension( * )INDXC, integer, dimension( * )INDXP, integer, dimension( * )COLTYP, integerINFO)¶
SLAED2 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal. Purpose:SLAED2 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.
K
Author:
K is INTEGER The number of non-deflated eigenvalues, and the order of the related secular equation. 0 <= K <=N.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.N1
N1 is INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= N1 <= N/2.D
D is REAL array, dimension (N) On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.Q
Q is REAL array, dimension (LDQ, N) On entry, Q contains the eigenvectors of two submatrices in the two square blocks with corners at (1,1), (N1,N1) and (N1+1, N1+1), (N,N). On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).INDXQ
INDXQ is INTEGER array, dimension (N) The permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have N1 added to their values. Destroyed on exit.RHO
RHO is REAL On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by SLAED3.Z
Z is REAL array, dimension (N) On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z have been destroyed by the updating process.DLAMDA
DLAMDA is REAL array, dimension (N) A copy of the first K eigenvalues which will be used by SLAED3 to form the secular equation.W
W is REAL array, dimension (N) The first k values of the final deflation-altered z-vector which will be passed to SLAED3.Q2
Q2 is REAL array, dimension (N1**2+(N-N1)**2) A copy of the first K eigenvectors which will be used by SLAED3 in a matrix multiply (SGEMM) to solve for the new eigenvectors.INDX
INDX is INTEGER array, dimension (N) The permutation used to sort the contents of DLAMDA into ascending order.INDXC
INDXC is INTEGER array, dimension (N) The permutation used to arrange the columns of the deflated Q matrix into three groups: the first group contains non-zero elements only at and above N1, the second contains non-zero elements only below N1, and the third is dense.INDXP
INDXP is INTEGER array, dimension (N) The permutation used to place deflated values of D at the end of the array. INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues.COLTYP
COLTYP is INTEGER array, dimension (N) During execution, a label which will indicate which of the following types a column in the Q2 matrix is: 1 : non-zero in the upper half only; 2 : dense; 3 : non-zero in the lower half only; 4 : deflated. On exit, COLTYP(i) is the number of columns of type i, for i=1 to 4 only.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified by Francoise Tisseur, University of Tennessee
subroutine slaed3 (integerK, integerN, integerN1, real, dimension( * )D, real, dimension( ldq, * )Q, integerLDQ, realRHO, real, dimension( * )DLAMDA, real, dimension( * )Q2, integer, dimension( * )INDX, integer, dimension( * )CTOT, real, dimension( * )W, real, dimension( * )S, integerINFO)¶
SLAED3 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal. Purpose:SLAED3 finds the roots of the secular equation, as defined by the values in D, W, and RHO, between 1 and K. It makes the appropriate calls to SLAED4 and then updates the eigenvectors by multiplying the matrix of eigenvectors of the pair of eigensystems being combined by the matrix of eigenvectors of the K-by-K system which is solved here. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
K
Author:
K is INTEGER The number of terms in the rational function to be solved by SLAED4. K >= 0.N
N is INTEGER The number of rows and columns in the Q matrix. N >= K (deflation may result in N>K).N1
N1 is INTEGER The location of the last eigenvalue in the leading submatrix. min(1,N) <= N1 <= N/2.D
D is REAL array, dimension (N) D(I) contains the updated eigenvalues for 1 <= I <= K.Q
Q is REAL array, dimension (LDQ,N) Initially the first K columns are used as workspace. On output the columns 1 to K contain the updated eigenvectors.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).RHO
RHO is REAL The value of the parameter in the rank one update equation. RHO >= 0 required.DLAMDA
DLAMDA is REAL array, dimension (K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation. May be changed on output by having lowest order bit set to zero on Cray X-MP, Cray Y-MP, Cray-2, or Cray C-90, as described above.Q2
Q2 is REAL array, dimension (LDQ2, N) The first K columns of this matrix contain the non-deflated eigenvectors for the split problem.INDX
INDX is INTEGER array, dimension (N) The permutation used to arrange the columns of the deflated Q matrix into three groups (see SLAED2). The rows of the eigenvectors found by SLAED4 must be likewise permuted before the matrix multiply can take place.CTOT
CTOT is INTEGER array, dimension (4) A count of the total number of the various types of columns in Q, as described in INDX. The fourth column type is any column which has been deflated.W
W is REAL array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating vector. Destroyed on output.S
S is REAL array, dimension (N1 + 1)*K Will contain the eigenvectors of the repaired matrix which will be multiplied by the previously accumulated eigenvectors to update the system.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified by Francoise Tisseur, University of Tennessee
subroutine slaed4 (integerN, integerI, real, dimension( * )D, real, dimension( * )Z, real, dimension( * )DELTA, realRHO, realDLAM, integerINFO)¶
SLAED4 used by sstedc. Finds a single root of the secular equation. Purpose:This subroutine computes the I-th updated eigenvalue of a symmetric rank-one modification to a diagonal matrix whose elements are given in the array d, and that D(i) < D(j) for i < j and that RHO > 0. This is arranged by the calling routine, and is no loss in generality. The rank-one modified system is thus diag( D ) + RHO * Z * Z_transpose. where we assume the Euclidean norm of Z is 1. The method consists of approximating the rational functions in the secular equation by simpler interpolating rational functions.
N
Internal Parameters:
N is INTEGER The length of all arrays.I
I is INTEGER The index of the eigenvalue to be computed. 1 <= I <= N.D
D is REAL array, dimension (N) The original eigenvalues. It is assumed that they are in order, D(I) < D(J) for I < J.Z
Z is REAL array, dimension (N) The components of the updating vector.DELTA
DELTA is REAL array, dimension (N) If N .GT. 2, DELTA contains (D(j) - lambda_I) in its j-th component. If N = 1, then DELTA(1) = 1. If N = 2, see SLAED5 for detail. The vector DELTA contains the information necessary to construct the eigenvectors by SLAED3 and SLAED9.RHO
RHO is REAL The scalar in the symmetric updating formula.DLAM
DLAM is REAL The computed lambda_I, the I-th updated eigenvalue.INFO
INFO is INTEGER = 0: successful exit > 0: if INFO = 1, the updating process failed.
Logical variable ORGATI (origin-at-i?) is used for distinguishing whether D(i) or D(i+1) is treated as the origin. ORGATI = .true. origin at i ORGATI = .false. origin at i+1 Logical variable SWTCH3 (switch-for-3-poles?) is for noting if we are working with THREE poles! MAXIT is the maximum number of iterations allowed for each eigenvalue.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
subroutine slaed5 (integerI, real, dimension( 2 )D, real, dimension( 2 )Z, real, dimension( 2 )DELTA, realRHO, realDLAM)¶
SLAED5 used by sstedc. Solves the 2-by-2 secular equation. Purpose:This subroutine computes the I-th eigenvalue of a symmetric rank-one modification of a 2-by-2 diagonal matrix diag( D ) + RHO * Z * transpose(Z) . The diagonal elements in the array D are assumed to satisfy D(i) < D(j) for i < j . We also assume RHO > 0 and that the Euclidean norm of the vector Z is one.
I
Author:
I is INTEGER The index of the eigenvalue to be computed. I = 1 or I = 2.D
D is REAL array, dimension (2) The original eigenvalues. We assume D(1) < D(2).Z
Z is REAL array, dimension (2) The components of the updating vector.DELTA
DELTA is REAL array, dimension (2) The vector DELTA contains the information necessary to construct the eigenvectors.RHO
RHO is REAL The scalar in the symmetric updating formula.DLAM
DLAM is REAL The computed lambda_I, the I-th updated eigenvalue.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
subroutine slaed6 (integerKNITER, logicalORGATI, realRHO, real, dimension( 3 )D, real, dimension( 3 )Z, realFINIT, realTAU, integerINFO)¶
SLAED6 used by sstedc. Computes one Newton step in solution of the secular equation. Purpose:SLAED6 computes the positive or negative root (closest to the origin) of z(1) z(2) z(3) f(x) = rho + --------- + ---------- + --------- d(1)-x d(2)-x d(3)-x It is assumed that if ORGATI = .true. the root is between d(2) and d(3); otherwise it is between d(1) and d(2) This routine will be called by SLAED4 when necessary. In most cases, the root sought is the smallest in magnitude, though it might not be in some extremely rare situations.
KNITER
Author:
KNITER is INTEGER Refer to SLAED4 for its significance.ORGATI
ORGATI is LOGICAL If ORGATI is true, the needed root is between d(2) and d(3); otherwise it is between d(1) and d(2). See SLAED4 for further details.RHO
RHO is REAL Refer to the equation f(x) above.D
D is REAL array, dimension (3) D satisfies d(1) < d(2) < d(3).Z
Z is REAL array, dimension (3) Each of the elements in z must be positive.FINIT
FINIT is REAL The value of f at 0. It is more accurate than the one evaluated inside this routine (if someone wants to do so).TAU
TAU is REAL The root of the equation f(x).INFO
INFO is INTEGER = 0: successful exit > 0: if INFO = 1, failure to converge
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Further Details:
10/02/03: This version has a few statements commented out for thread safety (machine parameters are computed on each entry). SJH. 05/10/06: Modified from a new version of Ren-Cang Li, use Gragg-Thornton-Warner cubic convergent scheme for better stability.
Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
subroutine slaed7 (integerICOMPQ, integerN, integerQSIZ, integerTLVLS, integerCURLVL, integerCURPBM, real, dimension( * )D, real, dimension( ldq, * )Q, integerLDQ, integer, dimension( * )INDXQ, realRHO, integerCUTPNT, real, dimension( * )QSTORE, integer, dimension( * )QPTR, integer, dimension( * )PRMPTR, integer, dimension( * )PERM, integer, dimension( * )GIVPTR, integer, dimension( 2, * )GIVCOL, real, dimension( 2, * )GIVNUM, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
SLAED7 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense. Purpose:SLAED7 computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. This routine is used only for the eigenproblem which requires all eigenvalues and optionally eigenvectors of a dense symmetric matrix that has been reduced to tridiagonal form. SLAED1 handles the case in which all eigenvalues and eigenvectors of a symmetric tridiagonal matrix are desired. T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out) where Z = Q**Tu, u is a vector of length N with ones in the CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. The eigenvectors of the original matrix are stored in Q, and the eigenvalues are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple eigenvalues or if there is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLAED8. The second stage consists of calculating the updated eigenvalues. This is done by finding the roots of the secular equation via the routine SLAED4 (as called by SLAED9). This routine also calculates the eigenvectors of the current problem. The final stage consists of computing the updated eigenvectors directly using the updated eigenvalues. The eigenvectors for the current problem are multiplied with the eigenvectors from the overall problem.
ICOMPQ
Author:
ICOMPQ is INTEGER = 0: Compute eigenvalues only. = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.QSIZ
QSIZ is INTEGER The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.TLVLS
TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree.CURLVL
CURLVL is INTEGER The current level in the overall merge routine, 0 <= CURLVL <= TLVLS.CURPBM
CURPBM is INTEGER The current problem in the current level in the overall merge routine (counting from upper left to lower right).D
D is REAL array, dimension (N) On entry, the eigenvalues of the rank-1-perturbed matrix. On exit, the eigenvalues of the repaired matrix.Q
Q is REAL array, dimension (LDQ, N) On entry, the eigenvectors of the rank-1-perturbed matrix. On exit, the eigenvectors of the repaired tridiagonal matrix.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).INDXQ
INDXQ is INTEGER array, dimension (N) The permutation which will reintegrate the subproblem just solved back into sorted order, i.e., D( INDXQ( I = 1, N ) ) will be in ascending order.RHO
RHO is REAL The subdiagonal element used to create the rank-1 modification.CUTPNT
CUTPNT is INTEGER Contains the location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.QSTORE
QSTORE is REAL array, dimension (N**2+1) Stores eigenvectors of submatrices encountered during divide and conquer, packed together. QPTR points to beginning of the submatrices.QPTR
QPTR is INTEGER array, dimension (N+2) List of indices pointing to beginning of submatrices stored in QSTORE. The submatrices are numbered starting at the bottom left of the divide and conquer tree, from left to right and bottom to top.PRMPTR
PRMPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and also the size of the full, non-deflated problem.PERM
PERM is INTEGER array, dimension (N lg N) Contains the permutations (from deflation and sorting) to be applied to each eigenblock.GIVPTR
GIVPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.GIVCOL
GIVCOL is INTEGER array, dimension (2, N lg N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation.GIVNUM
GIVNUM is REAL array, dimension (2, N lg N) Each number indicates the S value to be used in the corresponding Givens rotation.WORK
WORK is REAL array, dimension (3*N+2*QSIZ*N)IWORK
IWORK is INTEGER array, dimension (4*N)INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge
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NAG Ltd.
Date:
June 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine slaed8 (integerICOMPQ, integerK, integerN, integerQSIZ, real, dimension( * )D, real, dimension( ldq, * )Q, integerLDQ, integer, dimension( * )INDXQ, realRHO, integerCUTPNT, real, dimension( * )Z, real, dimension( * )DLAMDA, real, dimension( ldq2, * )Q2, integerLDQ2, real, dimension( * )W, integer, dimension( * )PERM, integerGIVPTR, integer, dimension( 2, * )GIVCOL, real, dimension( 2, * )GIVNUM, integer, dimension( * )INDXP, integer, dimension( * )INDX, integerINFO)¶
SLAED8 used by sstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense. Purpose:SLAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more eigenvalues are close together or if there is a tiny element in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.
ICOMPQ
Author:
ICOMPQ is INTEGER = 0: Compute eigenvalues only. = 1: Compute eigenvectors of original dense symmetric matrix also. On entry, Q contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.K
K is INTEGER The number of non-deflated eigenvalues, and the order of the related secular equation.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.QSIZ
QSIZ is INTEGER The dimension of the orthogonal matrix used to reduce the full matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.D
D is REAL array, dimension (N) On entry, the eigenvalues of the two submatrices to be combined. On exit, the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.Q
Q is REAL array, dimension (LDQ,N) If ICOMPQ = 0, Q is not referenced. Otherwise, on entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N).INDXQ
INDXQ is INTEGER array, dimension (N) The permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have CUTPNT added to their values in order to be accurate.RHO
RHO is REAL On entry, the off-diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. On exit, RHO has been modified to the value required by SLAED3.CUTPNT
CUTPNT is INTEGER The location of the last eigenvalue in the leading sub-matrix. min(1,N) <= CUTPNT <= N.Z
Z is REAL array, dimension (N) On entry, Z contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). On exit, the contents of Z are destroyed by the updating process.DLAMDA
DLAMDA is REAL array, dimension (N) A copy of the first K eigenvalues which will be used by SLAED3 to form the secular equation.Q2
Q2 is REAL array, dimension (LDQ2,N) If ICOMPQ = 0, Q2 is not referenced. Otherwise, a copy of the first K eigenvectors which will be used by SLAED7 in a matrix multiply (SGEMM) to update the new eigenvectors.LDQ2
LDQ2 is INTEGER The leading dimension of the array Q2. LDQ2 >= max(1,N).W
W is REAL array, dimension (N) The first k values of the final deflation-altered z-vector and will be passed to SLAED3.PERM
PERM is INTEGER array, dimension (N) The permutations (from deflation and sorting) to be applied to each eigenblock.GIVPTR
GIVPTR is INTEGER The number of Givens rotations which took place in this subproblem.GIVCOL
GIVCOL is INTEGER array, dimension (2, N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation.GIVNUM
GIVNUM is REAL array, dimension (2, N) Each number indicates the S value to be used in the corresponding Givens rotation.INDXP
INDXP is INTEGER array, dimension (N) The permutation used to place deflated values of D at the end of the array. INDXP(1:K) points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues.INDX
INDX is INTEGER array, dimension (N) The permutation used to sort the contents of D into ascending order.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
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NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine slaed9 (integerK, integerKSTART, integerKSTOP, integerN, real, dimension( * )D, real, dimension( ldq, * )Q, integerLDQ, realRHO, real, dimension( * )DLAMDA, real, dimension( * )W, real, dimension( lds, * )S, integerLDS, integerINFO)¶
SLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense. Purpose:SLAED9 finds the roots of the secular equation, as defined by the values in D, Z, and RHO, between KSTART and KSTOP. It makes the appropriate calls to SLAED4 and then stores the new matrix of eigenvectors for use in calculating the next level of Z vectors.
K
Author:
K is INTEGER The number of terms in the rational function to be solved by SLAED4. K >= 0.KSTART
KSTART is INTEGERKSTOP
KSTOP is INTEGER The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP are to be computed. 1 <= KSTART <= KSTOP <= K.N
N is INTEGER The number of rows and columns in the Q matrix. N >= K (delation may result in N > K).D
D is REAL array, dimension (N) D(I) contains the updated eigenvalues for KSTART <= I <= KSTOP.Q
Q is REAL array, dimension (LDQ,N)LDQ
LDQ is INTEGER The leading dimension of the array Q. LDQ >= max( 1, N ).RHO
RHO is REAL The value of the parameter in the rank one update equation. RHO >= 0 required.DLAMDA
DLAMDA is REAL array, dimension (K) The first K elements of this array contain the old roots of the deflated updating problem. These are the poles of the secular equation.W
W is REAL array, dimension (K) The first K elements of this array contain the components of the deflation-adjusted updating vector.S
S is REAL array, dimension (LDS, K) Will contain the eigenvectors of the repaired matrix which will be stored for subsequent Z vector calculation and multiplied by the previously accumulated eigenvectors to update the system.LDS
LDS is INTEGER The leading dimension of S. LDS >= max( 1, K ).INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = 1, an eigenvalue did not converge
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NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine slaeda (integerN, integerTLVLS, integerCURLVL, integerCURPBM, integer, dimension( * )PRMPTR, integer, dimension( * )PERM, integer, dimension( * )GIVPTR, integer, dimension( 2, * )GIVCOL, real, dimension( 2, * )GIVNUM, real, dimension( * )Q, integer, dimension( * )QPTR, real, dimension( * )Z, real, dimension( * )ZTEMP, integerINFO)¶
SLAEDA used by sstedc. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense. Purpose:SLAEDA computes the Z vector corresponding to the merge step in the CURLVLth step of the merge process with TLVLS steps for the CURPBMth problem.
N
Author:
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.TLVLS
TLVLS is INTEGER The total number of merging levels in the overall divide and conquer tree.CURLVL
CURLVL is INTEGER The current level in the overall merge routine, 0 <= curlvl <= tlvls.CURPBM
CURPBM is INTEGER The current problem in the current level in the overall merge routine (counting from upper left to lower right).PRMPTR
PRMPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in PERM a level's permutation is stored. PRMPTR(i+1) - PRMPTR(i) indicates the size of the permutation and incidentally the size of the full, non-deflated problem.PERM
PERM is INTEGER array, dimension (N lg N) Contains the permutations (from deflation and sorting) to be applied to each eigenblock.GIVPTR
GIVPTR is INTEGER array, dimension (N lg N) Contains a list of pointers which indicate where in GIVCOL a level's Givens rotations are stored. GIVPTR(i+1) - GIVPTR(i) indicates the number of Givens rotations.GIVCOL
GIVCOL is INTEGER array, dimension (2, N lg N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation.GIVNUM
GIVNUM is REAL array, dimension (2, N lg N) Each number indicates the S value to be used in the corresponding Givens rotation.Q
Q is REAL array, dimension (N**2) Contains the square eigenblocks from previous levels, the starting positions for blocks are given by QPTR.QPTR
QPTR is INTEGER array, dimension (N+2) Contains a list of pointers which indicate where in Q an eigenblock is stored. SQRT( QPTR(i+1) - QPTR(i) ) indicates the size of the block.Z
Z is REAL array, dimension (N) On output this vector contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix).ZTEMP
ZTEMP is REAL array, dimension (N)INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.
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NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
subroutine slagtf (integerN, real, dimension( * )A, realLAMBDA, real, dimension( * )B, real, dimension( * )C, realTOL, real, dimension( * )D, integer, dimension( * )IN, integerINFO)¶
SLAGTF computes an LU factorization of a matrix T-λI, where T is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges. Purpose:SLAGTF factorizes the matrix (T - lambda*I), where T is an n by n tridiagonal matrix and lambda is a scalar, as T - lambda*I = PLU, where P is a permutation matrix, L is a unit lower tridiagonal matrix with at most one non-zero sub-diagonal elements per column and U is an upper triangular matrix with at most two non-zero super-diagonal elements per column. The factorization is obtained by Gaussian elimination with partial pivoting and implicit row scaling. The parameter LAMBDA is included in the routine so that SLAGTF may be used, in conjunction with SLAGTS, to obtain eigenvectors of T by inverse iteration.
N
Author:
N is INTEGER The order of the matrix T.A
A is REAL array, dimension (N) On entry, A must contain the diagonal elements of T. On exit, A is overwritten by the n diagonal elements of the upper triangular matrix U of the factorization of T.LAMBDA
LAMBDA is REAL On entry, the scalar lambda.B
B is REAL array, dimension (N-1) On entry, B must contain the (n-1) super-diagonal elements of T. On exit, B is overwritten by the (n-1) super-diagonal elements of the matrix U of the factorization of T.C
C is REAL array, dimension (N-1) On entry, C must contain the (n-1) sub-diagonal elements of T. On exit, C is overwritten by the (n-1) sub-diagonal elements of the matrix L of the factorization of T.TOL
TOL is REAL On entry, a relative tolerance used to indicate whether or not the matrix (T - lambda*I) is nearly singular. TOL should normally be chose as approximately the largest relative error in the elements of T. For example, if the elements of T are correct to about 4 significant figures, then TOL should be set to about 5*10**(-4). If TOL is supplied as less than eps, where eps is the relative machine precision, then the value eps is used in place of TOL.D
D is REAL array, dimension (N-2) On exit, D is overwritten by the (n-2) second super-diagonal elements of the matrix U of the factorization of T.IN
IN is INTEGER array, dimension (N) On exit, IN contains details of the permutation matrix P. If an interchange occurred at the kth step of the elimination, then IN(k) = 1, otherwise IN(k) = 0. The element IN(n) returns the smallest positive integer j such that abs( u(j,j) ).le. norm( (T - lambda*I)(j) )*TOL, where norm( A(j) ) denotes the sum of the absolute values of the jth row of the matrix A. If no such j exists then IN(n) is returned as zero. If IN(n) is returned as positive, then a diagonal element of U is small, indicating that (T - lambda*I) is singular or nearly singular,INFO
INFO is INTEGER = 0 : successful exit .lt. 0: if INFO = -k, the kth argument had an illegal value
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Date:
December 2016
subroutine slamrg (integerN1, integerN2, real, dimension( * )A, integerSTRD1, integerSTRD2, integer, dimension( * )INDEX)¶
SLAMRG creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order. Purpose:SLAMRG will create a permutation list which will merge the elements of A (which is composed of two independently sorted sets) into a single set which is sorted in ascending order.
N1
Author:
N1 is INTEGERN2
N2 is INTEGER These arguments contain the respective lengths of the two sorted lists to be merged.A
A is REAL array, dimension (N1+N2) The first N1 elements of A contain a list of numbers which are sorted in either ascending or descending order. Likewise for the final N2 elements.STRD1
STRD1 is INTEGERSTRD2
STRD2 is INTEGER These are the strides to be taken through the array A. Allowable strides are 1 and -1. They indicate whether a subset of A is sorted in ascending (STRDx = 1) or descending (STRDx = -1) order.INDEX
INDEX is INTEGER array, dimension (N1+N2) On exit this array will contain a permutation such that if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be sorted in ascending order.
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Date:
June 2016
subroutine slartgs (realX, realY, realSIGMA, realCS, realSN)¶
SLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem. Purpose:SLARTGS generates a plane rotation designed to introduce a bulge in Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD problem. X and Y are the top-row entries, and SIGMA is the shift. The computed CS and SN define a plane rotation satisfying [ CS SN ] . [ X^2 - SIGMA ] = [ R ], [ -SN CS ] [ X * Y ] [ 0 ] with R nonnegative. If X^2 - SIGMA and X * Y are 0, then the rotation is by PI/2.
X
Author:
X is REAL The (1,1) entry of an upper bidiagonal matrix.Y
Y is REAL The (1,2) entry of an upper bidiagonal matrix.SIGMA
SIGMA is REAL The shift.CS
CS is REAL The cosine of the rotation.SN
SN is REAL The sine of the rotation.
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Date:
December 2016
subroutine slasq1 (integerN, real, dimension( * )D, real, dimension( * )E, real, dimension( * )WORK, integerINFO)¶
SLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr. Purpose:SLASQ1 computes the singular values of a real N-by-N bidiagonal matrix with diagonal D and off-diagonal E. The singular values are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. The algorithm was first presented in "Accurate singular values and differential qd algorithms" by K. V. Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, 1994, and the present implementation is described in "An implementation of the dqds Algorithm (Positive Case)", LAPACK Working Note.
N
Author:
N is INTEGER The number of rows and columns in the matrix. N >= 0.D
D is REAL array, dimension (N) On entry, D contains the diagonal elements of the bidiagonal matrix whose SVD is desired. On normal exit, D contains the singular values in decreasing order.E
E is REAL array, dimension (N) On entry, elements E(1:N-1) contain the off-diagonal elements of the bidiagonal matrix whose SVD is desired. On exit, E is overwritten.WORK
WORK is REAL array, dimension (4*N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm failed = 1, a split was marked by a positive value in E = 2, current block of Z not diagonalized after 100*N iterations (in inner while loop) On exit D and E represent a matrix with the same singular values which the calling subroutine could use to finish the computation, or even feed back into SLASQ1 = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)
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Date:
December 2016
subroutine slasq2 (integerN, real, dimension( * )Z, integerINFO)¶
SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr. Purpose:SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd array Z to high relative accuracy are computed to high relative accuracy, in the absence of denormalization, underflow and overflow. To see the relation of Z to the tridiagonal matrix, let L be a unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and let U be an upper bidiagonal matrix with 1's above and diagonal Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the symmetric tridiagonal to which it is similar. Note : SLASQ2 defines a logical variable, IEEE, which is true on machines which follow ieee-754 floating-point standard in their handling of infinities and NaNs, and false otherwise. This variable is passed to SLASQ3.
N
Author:
N is INTEGER The number of rows and columns in the matrix. N >= 0.Z
Z is REAL array, dimension ( 4*N ) On entry Z holds the qd array. On exit, entries 1 to N hold the eigenvalues in decreasing order, Z( 2*N+1 ) holds the trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 ) holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of shifts that failed.INFO
INFO is INTEGER = 0: successful exit < 0: if the i-th argument is a scalar and had an illegal value, then INFO = -i, if the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j) > 0: the algorithm failed = 1, a split was marked by a positive value in E = 2, current block of Z not diagonalized after 100*N iterations (in inner while loop). On exit Z holds a qd array with the same eigenvalues as the given Z. = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)
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Date:
December 2016
Further Details:
Local Variables: I0:N0 defines a current unreduced segment of Z. The shifts are accumulated in SIGMA. Iteration count is in ITER. Ping-pong is controlled by PP (alternates between 0 and 1).
subroutine slasq3 (integerI0, integerN0, real, dimension( * )Z, integerPP, realDMIN, realSIGMA, realDESIG, realQMAX, integerNFAIL, integerITER, integerNDIV, logicalIEEE, integerTTYPE, realDMIN1, realDMIN2, realDN, realDN1, realDN2, realG, realTAU)¶
SLASQ3 checks for deflation, computes a shift and calls dqds. Used by sbdsqr. Purpose:SLASQ3 checks for deflation, computes a shift (TAU) and calls dqds. In case of failure it changes shifts, and tries again until output is positive.
I0
Author:
I0 is INTEGER First index.N0
N0 is INTEGER Last index.Z
Z is REAL array, dimension ( 4*N0 ) Z holds the qd array.PP
PP is INTEGER PP=0 for ping, PP=1 for pong. PP=2 indicates that flipping was applied to the Z array and that the initial tests for deflation should not be performed.DMIN
DMIN is REAL Minimum value of d.SIGMA
SIGMA is REAL Sum of shifts used in current segment.DESIG
DESIG is REAL Lower order part of SIGMAQMAX
QMAX is REAL Maximum value of q.NFAIL
NFAIL is INTEGER Increment NFAIL by 1 each time the shift was too big.ITER
ITER is INTEGER Increment ITER by 1 for each iteration.NDIV
NDIV is INTEGER Increment NDIV by 1 for each division.IEEE
IEEE is LOGICAL Flag for IEEE or non IEEE arithmetic (passed to SLASQ5).TTYPE
TTYPE is INTEGER Shift type.DMIN1
DMIN1 is REALDMIN2
DMIN2 is REALDN
DN is REALDN1
DN1 is REALDN2
DN2 is REALG
G is REALTAU
TAU is REAL These are passed as arguments in order to save their values between calls to SLASQ3.
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Date:
June 2016
subroutine slasq4 (integerI0, integerN0, real, dimension( * )Z, integerPP, integerN0IN, realDMIN, realDMIN1, realDMIN2, realDN, realDN1, realDN2, realTAU, integerTTYPE, realG)¶
SLASQ4 computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by sbdsqr. Purpose:SLASQ4 computes an approximation TAU to the smallest eigenvalue using values of d from the previous transform.
I0
Author:
I0 is INTEGER First index.N0
N0 is INTEGER Last index.Z
Z is REAL array, dimension ( 4*N0 ) Z holds the qd array.PP
PP is INTEGER PP=0 for ping, PP=1 for pong.N0IN
N0IN is INTEGER The value of N0 at start of EIGTEST.DMIN
DMIN is REAL Minimum value of d.DMIN1
DMIN1 is REAL Minimum value of d, excluding D( N0 ).DMIN2
DMIN2 is REAL Minimum value of d, excluding D( N0 ) and D( N0-1 ).DN
DN is REAL d(N)DN1
DN1 is REAL d(N-1)DN2
DN2 is REAL d(N-2)TAU
TAU is REAL This is the shift.TTYPE
TTYPE is INTEGER Shift type.G
G is REAL G is passed as an argument in order to save its value between calls to SLASQ4.
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NAG Ltd.
Date:
June 2016
Further Details:
CNST1 = 9/16
subroutine slasq5 (integerI0, integerN0, real, dimension( * )Z, integerPP, realTAU, realSIGMA, realDMIN, realDMIN1, realDMIN2, realDN, realDNM1, realDNM2, logicalIEEE, realEPS)¶
SLASQ5 computes one dqds transform in ping-pong form. Used by sbdsqr and sstegr. Purpose:SLASQ5 computes one dqds transform in ping-pong form, one version for IEEE machines another for non IEEE machines.
I0
Author:
I0 is INTEGER First index.N0
N0 is INTEGER Last index.Z
Z is REAL array, dimension ( 4*N ) Z holds the qd array. EMIN is stored in Z(4*N0) to avoid an extra argument.PP
PP is INTEGER PP=0 for ping, PP=1 for pong.TAU
TAU is REAL This is the shift.SIGMA
SIGMA is REAL This is the accumulated shift up to this step.DMIN
DMIN is REAL Minimum value of d.DMIN1
DMIN1 is REAL Minimum value of d, excluding D( N0 ).DMIN2
DMIN2 is REAL Minimum value of d, excluding D( N0 ) and D( N0-1 ).DN
DN is REAL d(N0), the last value of d.DNM1
DNM1 is REAL d(N0-1).DNM2
DNM2 is REAL d(N0-2).IEEE
IEEE is LOGICAL Flag for IEEE or non IEEE arithmetic.EPS
EPS is REAL This is the value of epsilon used.
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Date:
December 2016
subroutine slasq6 (integerI0, integerN0, real, dimension( * )Z, integerPP, realDMIN, realDMIN1, realDMIN2, realDN, realDNM1, realDNM2)¶
SLASQ6 computes one dqd transform in ping-pong form. Used by sbdsqr and sstegr. Purpose:SLASQ6 computes one dqd (shift equal to zero) transform in ping-pong form, with protection against underflow and overflow.
I0
Author:
I0 is INTEGER First index.N0
N0 is INTEGER Last index.Z
Z is REAL array, dimension ( 4*N ) Z holds the qd array. EMIN is stored in Z(4*N0) to avoid an extra argument.PP
PP is INTEGER PP=0 for ping, PP=1 for pong.DMIN
DMIN is REAL Minimum value of d.DMIN1
DMIN1 is REAL Minimum value of d, excluding D( N0 ).DMIN2
DMIN2 is REAL Minimum value of d, excluding D( N0 ) and D( N0-1 ).DN
DN is REAL d(N0), the last value of d.DNM1
DNM1 is REAL d(N0-1).DNM2
DNM2 is REAL d(N0-2).
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Date:
December 2016
subroutine slasrt (characterID, integerN, real, dimension( * )D, integerINFO)¶
SLASRT sorts numbers in increasing or decreasing order. Purpose:Sort the numbers in D in increasing order (if ID = 'I') or in decreasing order (if ID = 'D' ). Use Quick Sort, reverting to Insertion sort on arrays of size <= 20. Dimension of STACK limits N to about 2**32.
ID
Author:
ID is CHARACTER*1 = 'I': sort D in increasing order; = 'D': sort D in decreasing order.N
N is INTEGER The length of the array D.D
D is REAL array, dimension (N) On entry, the array to be sorted. On exit, D has been sorted into increasing order (D(1) <= ... <= D(N) ) or into decreasing order (D(1) >= ... >= D(N) ), depending on ID.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value
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Date:
June 2016
subroutine spttrf (integerN, real, dimension( * )D, real, dimension( * )E, integerINFO)¶
SPTTRF Purpose:SPTTRF computes the L*D*L**T factorization of a real symmetric positive definite tridiagonal matrix A. The factorization may also be regarded as having the form A = U**T*D*U.
N
Author:
N is INTEGER The order of the matrix A. N >= 0.D
D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, the n diagonal elements of the diagonal matrix D from the L*D*L**T factorization of A.E
E is REAL array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A. On exit, the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**T factorization of A. E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the U**T*D*U factorization of A.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, the leading minor of order k is not positive definite; if k < N, the factorization could not be completed, while if k = N, the factorization was completed, but D(N) <= 0.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine sstebz (characterRANGE, characterORDER, integerN, realVL, realVU, integerIL, integerIU, realABSTOL, real, dimension( * )D, real, dimension( * )E, integerM, integerNSPLIT, real, dimension( * )W, integer, dimension( * )IBLOCK, integer, dimension( * )ISPLIT, real, dimension( * )WORK, integer, dimension( * )IWORK, integerINFO)¶
SSTEBZ Purpose:SSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. To avoid overflow, the matrix must be scaled so that its largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest accuracy, it should not be much smaller than that. See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal Matrix", Report CS41, Computer Science Dept., Stanford University, July 21, 1966.
RANGE
Internal Parameters:
RANGE is CHARACTER*1 = 'A': ("All") all eigenvalues will be found. = 'V': ("Value") all eigenvalues in the half-open interval (VL, VU] will be found. = 'I': ("Index") the IL-th through IU-th eigenvalues (of the entire matrix) will be found.ORDER
ORDER is CHARACTER*1 = 'B': ("By Block") the eigenvalues will be grouped by split-off block (see IBLOCK, ISPLIT) and ordered from smallest to largest within the block. = 'E': ("Entire matrix") the eigenvalues for the entire matrix will be ordered from smallest to largest.N
N is INTEGER The order of the tridiagonal matrix T. N >= 0.VL
VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.VU
VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. Eigenvalues less than or equal to VL, or greater than VU, will not be returned. VL < VU. Not referenced if RANGE = 'A' or 'I'.IL
IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.IU
IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.ABSTOL
ABSTOL is REAL The absolute tolerance for the eigenvalues. An eigenvalue (or cluster) is considered to be located if it has been determined to lie in an interval whose width is ABSTOL or less. If ABSTOL is less than or equal to zero, then ULP*|T| will be used, where |T| means the 1-norm of T. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not zero.D
D is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix T.E
E is REAL array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix T.M
M is INTEGER The actual number of eigenvalues found. 0 <= M <= N. (See also the description of INFO=2,3.)NSPLIT
NSPLIT is INTEGER The number of diagonal blocks in the matrix T. 1 <= NSPLIT <= N.W
W is REAL array, dimension (N) On exit, the first M elements of W will contain the eigenvalues. (SSTEBZ may use the remaining N-M elements as workspace.)IBLOCK
IBLOCK is INTEGER array, dimension (N) At each row/column j where E(j) is zero or small, the matrix T is considered to split into a block diagonal matrix. On exit, if INFO = 0, IBLOCK(i) specifies to which block (from 1 to the number of blocks) the eigenvalue W(i) belongs. (SSTEBZ may use the remaining N-M elements as workspace.)ISPLIT
ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to ISPLIT(1), the second of rows/columns ISPLIT(1)+1 through ISPLIT(2), etc., and the NSPLIT-th consists of rows/columns ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N. (Only the first NSPLIT elements will actually be used, but since the user cannot know a priori what value NSPLIT will have, N words must be reserved for ISPLIT.)WORK
WORK is REAL array, dimension (4*N)IWORK
IWORK is INTEGER array, dimension (3*N)INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: some or all of the eigenvalues failed to converge or were not computed: =1 or 3: Bisection failed to converge for some eigenvalues; these eigenvalues are flagged by a negative block number. The effect is that the eigenvalues may not be as accurate as the absolute and relative tolerances. This is generally caused by unexpectedly inaccurate arithmetic. =2 or 3: RANGE='I' only: Not all of the eigenvalues IL:IU were found. Effect: M < IU+1-IL Cause: non-monotonic arithmetic, causing the Sturm sequence to be non-monotonic. Cure: recalculate, using RANGE='A', and pick out eigenvalues IL:IU. In some cases, increasing the PARAMETER "FUDGE" may make things work. = 4: RANGE='I', and the Gershgorin interval initially used was too small. No eigenvalues were computed. Probable cause: your machine has sloppy floating-point arithmetic. Cure: Increase the PARAMETER "FUDGE", recompile, and try again.
RELFAC REAL, default = 2.0e0 The relative tolerance. An interval (a,b] lies within "relative tolerance" if b-a < RELFAC*ulp*max(|a|,|b|), where "ulp" is the machine precision (distance from 1 to the next larger floating point number.) FUDGE REAL, default = 2 A "fudge factor" to widen the Gershgorin intervals. Ideally, a value of 1 should work, but on machines with sloppy arithmetic, this needs to be larger. The default for publicly released versions should be large enough to handle the worst machine around. Note that this has no effect on accuracy of the solution.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
June 2016
subroutine sstedc (characterCOMPZ, integerN, real, dimension( * )D, real, dimension( * )E, real, dimension( ldz, * )Z, integerLDZ, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)¶
SSTEDC Purpose:SSTEDC computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method. The eigenvectors of a full or band real symmetric matrix can also be found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to tridiagonal form. This code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. See SLAED3 for details.
COMPZ
Author:
COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'I': Compute eigenvectors of tridiagonal matrix also. = 'V': Compute eigenvectors of original dense symmetric matrix also. On entry, Z contains the orthogonal matrix used to reduce the original matrix to tridiagonal form.N
N is INTEGER The dimension of the symmetric tridiagonal matrix. N >= 0.D
D is REAL array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order.E
E is REAL array, dimension (N-1) On entry, the subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.Z
Z is REAL array, dimension (LDZ,N) On entry, if COMPZ = 'V', then Z contains the orthogonal matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original symmetric matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If eigenvectors are desired, then LDZ >= max(1,N).WORK
WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK
LWORK is INTEGER The dimension of the array WORK. If COMPZ = 'N' or N <= 1 then LWORK must be at least 1. If COMPZ = 'V' and N > 1 then LWORK must be at least ( 1 + 3*N + 2*N*lg N + 4*N**2 ), where lg( N ) = smallest integer k such that 2**k >= N. If COMPZ = 'I' and N > 1 then LWORK must be at least ( 1 + 4*N + N**2 ). Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LWORK need only be max(1,2*(N-1)). 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.IWORK
IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.LIWORK
LIWORK is INTEGER The dimension of the array IWORK. If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1. If COMPZ = 'V' and N > 1 then LIWORK must be at least ( 6 + 6*N + 5*N*lg N ). If COMPZ = 'I' and N > 1 then LIWORK must be at least ( 3 + 5*N ). Note that for COMPZ = 'I' or 'V', then if N is less than or equal to the minimum divide size, usually 25, then LIWORK need only be 1. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.INFO
INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns INFO/(N+1) through mod(INFO,N+1).
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Contributors:
Jeff Rutter, Computer Science Division, University of
California at Berkeley, USA
Modified by Francoise Tisseur, University of Tennessee
Modified by Francoise Tisseur, University of Tennessee
subroutine ssteqr (characterCOMPZ, integerN, real, dimension( * )D, real, dimension( * )E, real, dimension( ldz, * )Z, integerLDZ, real, dimension( * )WORK, integerINFO)¶
SSTEQR Purpose:SSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band symmetric matrix can also be found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to tridiagonal form.
COMPZ
Author:
COMPZ is CHARACTER*1 = 'N': Compute eigenvalues only. = 'V': Compute eigenvalues and eigenvectors of the original symmetric matrix. On entry, Z must contain the orthogonal matrix used to reduce the original matrix to tridiagonal form. = 'I': Compute eigenvalues and eigenvectors of the tridiagonal matrix. Z is initialized to the identity matrix.N
N is INTEGER The order of the matrix. N >= 0.D
D is REAL array, dimension (N) On entry, the diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order.E
E is REAL array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.Z
Z is REAL array, dimension (LDZ, N) On entry, if COMPZ = 'V', then Z contains the orthogonal matrix used in the reduction to tridiagonal form. On exit, if INFO = 0, then if COMPZ = 'V', Z contains the orthonormal eigenvectors of the original symmetric matrix, and if COMPZ = 'I', Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix. If COMPZ = 'N', then Z is not referenced.LDZ
LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if eigenvectors are desired, then LDZ >= max(1,N).WORK
WORK is REAL array, dimension (max(1,2*N-2)) If COMPZ = 'N', then WORK is not referenced.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm has failed to find all the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero; on exit, D and E contain the elements of a symmetric tridiagonal matrix which is orthogonally similar to the original matrix.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
subroutine ssterf (integerN, real, dimension( * )D, real, dimension( * )E, integerINFO)¶
SSTERF Purpose:SSTERF computes all eigenvalues of a symmetric tridiagonal matrix using the Pal-Walker-Kahan variant of the QL or QR algorithm.
N
Author:
N is INTEGER The order of the matrix. N >= 0.D
D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix. On exit, if INFO = 0, the eigenvalues in ascending order.E
E is REAL array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.INFO
INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: the algorithm failed to find all of the eigenvalues in a total of 30*N iterations; if INFO = i, then i elements of E have not converged to zero.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
December 2016
Author¶
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