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Notes

1.
This index lists related pairs of real and complex routines together, in the same style as in Appendix A.

2.
Routines are listed in alphanumeric order of the real (single precision) routine name (which always begins with S-). (See subsection 2.2.3 for details of the LAPACK naming scheme.)

3.
A few complex routines have no real equivalents, and they are listed first; routines listed in italics (for example, CROT), have real equivalents in the Level 1 or Level 2 BLAS.

4.
Double precision routines are not listed here; they have names beginning with D- instead of S-, or Z- instead of C-. The only exceptions to this simple rule are that the double precision versions of ICMAX1, SCSUM1 and CSRSCL are named IZMAX1, DZSUM1 and ZDRSCL.

5.
A few routines in the list have names that are independent of data type: ILAENV, LSAME, LSAMEN and XERBLA.

6.
This index gives only a brief description of the purpose of each routine. For a precise description consult the leading comments in the code, which have been written in the same style as for the driver and computational routines.

Routine Description
real complex  
  CLACGV Conjugates a complex vector.
  CLACRM Performs a matrix multiplication $C~=~A \ast B$, where A is complex, B is real, and C is complex.
  CLACRT Performs the transformation $\left( \begin{array}{cc} c & s \\ -s & c \end{array} \right) \; \left( \begin{array}{c} x \\ y \end{array} \right) $, where c, s, x, and y are complex.
  CLAESY Computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix, and checks that the norm of the matrix of eigenvectors is larger than a threshold value.
  CROT Applies a plane rotation with real cosine and complex sine to a pair of complex vectors.
  CSPMV Computes the matrix-vector product $y = \alpha Ax + \beta y$, where $\alpha$ and $\beta$ are complex scalars, x and y are complex vectors and A is a complex symmetric matrix in packed storage.
  CSPR Performs the symmetric rank-1 update $A = \alpha x x^T + A$, where $\alpha$ is a complex scalar, x is a complex vector and A is a complex symmetric matrix in packed storage.
  CSROT Applies a plane rotation with real cosine and sine to a pair of complex vectors.
  CSYMV Computes the matrix-vector product $y = \alpha Ax + \beta y$, where $\alpha$ and $\beta$ are complex scalars, x and y are complex vectors and A is a complex symmetric matrix.
  CSYR Performs the symmetric rank-1 update $A = \alpha x x^T + A$, where $\alpha$ is a complex scalar, x is a complex vector and A is a complex symmetric matrix.
  ICMAX1 Finds the index of the element whose real part has maximum absolute value (similar to the Level 1 BLAS ICAMAX, but using the absolute value of the real part).
ILAENV   Environmental enquiry function which returns values for tuning algorithmic performance.
LSAME   Tests two characters for equality regardless of case.
LSAMEN   Tests two character strings for equality regardless of case.
  SCSUM1 Forms the 1-norm of a complex vector (similar to the Level 1 BLAS SCASUM, but using the true absolute value).
SGBTF2 CGBTF2 Computes an LU factorization of a general band matrix, using partial pivoting with row interchanges (unblocked algorithm).
SGEBD2 CGEBD2 Reduces a general rectangular matrix to real bidiagonal form by an orthogonal/unitary transformation (unblocked algorithm).
SGEHD2 CGEHD2 Reduces a general matrix to upper Hessenberg form by an orthogonal/unitary similarity transformation (unblocked algorithm).
SGELQ2 CGELQ2 Computes an LQ factorization of a general rectangular matrix (unblocked algorithm).
Routine Description
real complex  
SGEQL2 CGEQL2 Computes a QL factorization of a general rectangular matrix (unblocked algorithm).
SGEQR2 CGEQR2 Computes a QR factorization of a general rectangular matrix (unblocked algorithm).
SGERQ2 CGERQ2 Computes an RQ factorization of a general rectangular matrix (unblocked algorithm).
SGESC2 CGESC2 Solves a system of linear equations $A \ast X = scale \ast RHS$ using the LU factorization with complete pivoting computed by xGETC2.
SGETC2 CGETC2 Computes an LU factorization with complete pivoting of the general n-by-n matrix A
SGETF2 CGETF2 Computes an LU factorization of a general matrix, using partial pivoting with row interchanges (unblocked algorithm).
SGTTS2 CGTTS2 Solves one of the systems of equations $A \ast X = B$ or $A^H \ast X = B$, with a tridiagonal matrix A using the LU factorization computed by SGTTRF/CGTTRF.
SLABAD   Returns the square root of the underflow and overflow thresholds if the exponent-range is very large.
SLABRD CLABRD Reduces the first nb rows and columns of a general rectangular matrix A to real bidiagonal form by an orthogonal/unitary transformation, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
SLACON CLACON Estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
SLACPY CLACPY Copies all or part of one two-dimensional array to another.
SLADIV CLADIV Performs complex division in real arithmetic, avoiding unnecessary overflow.
SLAE2   Computes the eigenvalues of a 2-by-2 symmetric matrix.
SLAEBZ   Computes the number of eigenvalues of a real symmetric tridiagonal matrix which are less than or equal to a given value, and performs other tasks required by the routine SSTEBZ.
SLAED0 CLAED0 Used by xSTEDC. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
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.
SLAED2   Used by SSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
SLAED3   Used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
SLAED4   Used by SSTEDC. Finds a single root of the secular equation.
Routine Description
real complex  
SLAED5   Used by SSTEDC. Solves the 2-by-2 secular equation.
SLAED6   Used by SSTEDC. Computes one Newton step in solution of secular equation.
SLAED7 CLAED7 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.
SLAED8 CLAED8 Used by xSTEDC. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
SLAED9   Used by SSTEDC. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
SLAEDA   Used by SSTEDC. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
SLAEIN CLAEIN Computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
SLAEV2 CLAEV2 Computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
SLAEXC   Swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.
SLAG2   Computes the eigenvalues of a 2-by-2 generalized eigenvalue problem $A~-~w\ast B$, with scaling as necessary to avoid over-/underflow.
SLAGS2   Computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel.
SLAGTF   Computes an LU factorization of a matrix $(T - \lambda I)$, where T is a general tridiagonal matrix, and $\lambda$ a scalar, using partial pivoting with row interchanges.
SLAGTM CLAGTM Performs a matrix-matrix product of the form $C = \alpha A B + \beta C$, where A is a tridiagonal matrix, B and C are rectangular matrices, and $\alpha$ and $\beta$ are scalars, which may be 0, 1, or -1.
SLAGTS   Solves the system of equations $(T - \lambda I) x = y$ or $(T - \lambda I)^T x = y$, where T is a general tridiagonal matrix and $\lambda$ a scalar, using the LU factorization computed by SLAGTF.
SLAGV2   Computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.
SLAHQR CLAHQR Computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
SLAHRD CLAHRD Reduces the first nb columns of a general rectangular matrix A so that elements below the kth subdiagonal are zero, by an orthogonal/unitary transformation, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
SLAIC1 CLAIC1 Applies one step of incremental condition estimation.
Routine Description
real complex  
SLALN2   Solves a 1-by-1 or 2-by-2 system of equations of the form $(\gamma A - \lambda D ) x = \sigma b$ or $(\gamma A^T - \lambda D) x = \sigma b$, where D is a diagonal matrix, $\lambda$, b and x may be complex, and $\sigma$ is a scale factor set to avoid overflow.
SLALS0 CLALS0 Used by xGELSD. Applies back the multiplying factors of either the left or the right singular vector matrix of a diagonal matrix appended by a row to the right hand side matrix B in solving the least squares problem using the divide and conquer SVD approach.
SLALSA CLALSA Used by xGELSD. An intermediate step in solving the least squares problem by computing the SVD of the coefficient matrix in compact form.
SLALSD CLALSD Used by xGELSD. Uses the singular value decomposition of A to solve the least squares problem of finding X to minimize the Euclidean norm of each column of $A \ast X-B$.
SLAMCH   Determines machine parameters for floating-point arithmetic.
SLAMRG   Creates a permutation list which will merge the entries of two independently sorted sets into a single set which is sorted in ascending order.
SLANGB CLANGB Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a general band matrix.
SLANGE CLANGE Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a general rectangular matrix.
SLANGT CLANGT Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a general tridiagonal matrix.
SLANHS CLANHS Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of an upper Hessenberg matrix.
SLANSB CLANSB CLANHB Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a real symmetric/complex symmetric/complex Hermitian band matrix.
SLANSP CLANSP CLANHP Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a real symmetric/complex symmetric/complex Hermitian matrix in packed storage.
SLANST CLANHT Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a symmetric/Hermitian tridiagonal matrix.
SLANSY CLANSY CLANHE Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a real symmetric/complex symmetric/complex Hermitian matrix.
SLANTB CLANTB Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a triangular band matrix.
Routine Description
real complex  
SLANTP CLANTP Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a triangular matrix in packed storage.
SLANTR CLANTR Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element, of a triangular matrix.
SLANV2   Computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in Schur canonical form.
SLAPLL CLAPLL Measures the linear dependence of two vectors X and Y.
SLAPMT CLAPMT Performs a forward or backward permutation of the columns of a matrix.
SLAPY2   Returns $ \sqrt{x^2 + y^2}$, avoiding unnecessary overflow or harmful underflow.
SLAPY3   Returns $ \sqrt{x^2 + y^2 + z^2}$, avoiding unnecessary overflow or harmful underflow.
SLAQGB CLAQGB Scales a general band matrix, using row and column scaling factors computed by SGBEQU/CGBEQU.
SLAQGE CLAQGE Scales a general rectangular matrix, using row and column scaling factors computed by SGEEQU/CGEEQU.
SLAQP2 CLAQP2 Computes a QR factorization with column pivoting of the block A(OFFSET+1:M,1:N). The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
SLAQPS CLAQPS Computes a step of QR factorization with column pivoting of a real M-by-N matrix A by using Level 3 Blas.
SLAQSB CLAQSB Scales a symmetric/Hermitian band matrix, using scaling factors computed by SPBEQU/CPBEQU.
SLAQSP CLAQSP Scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by SPPEQU/CPPEQU.
SLAQSY CLAQSY Scales a symmetric/Hermitian matrix, using scaling factors computed by SPOEQU/CPOEQU.
SLAQTR   Solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.
SLAR1V CLAR1V Computes the (scaled) rth column of the inverse of the sumbmatrix in rows B1 through BN of the tridiagonal matrix $L D L^T - \sigma I$.
SLAR2V CLAR2V Applies a vector of plane rotations with real cosines and real/complex sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
SLARF CLARF Applies an elementary reflector to a general rectangular matrix.
SLARFB CLARFB Applies a block reflector or its transpose/conjugate-transpose to a general rectangular matrix.
SLARFG CLARFG Generates an elementary reflector (Householder matrix).
SLARFT CLARFT Forms the triangular factor T of a block reflector H = I - V T VH.
Routine Description
real complex  
SLARFX CLARFX Applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order $\leq 10$.
SLARGV CLARGV Generates a vector of plane rotations with real cosines and real/complex sines.
SLARNV CLARNV Returns a vector of random numbers from a uniform or normal distribution.
SLARRB   Given the relatively robust representation(RRR) L D LT, SLARRB does ``limited'' bisection to locate the eigenvalues of L D LT, W(IFIRST) through W(ILAST), to more accuracy.
SLARRE   Given the tridiagonal matrix T, SLARRE sets ``small'' off-diagonal elements to zero, and for each unreduced block Ti, it finds the numbers $\sigma _ i $, the base $T_i - \sigma_i I~=~L_i D_i L_i^T$ representations and the eigenvalues of each Li Di LiT.
SLARRF   Finds a new relatively robust representation $L D L^T - \Sigma I~=~L(+) D(+) L(+)^T$ such that at least one of the eigenvalues of L(+) D(+) L(+)T is relatively isolated.
SLARRV CLARRV Computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.
SLARTG CLARTG Generates a plane rotation with real cosine and real/complex sine.
SLARTV CLARTV Applies a vector of plane rotations with real cosines and real/complex sines to the elements of a pair of vectors.
SLARUV   Returns a vector of n random real numbers from a uniform (0,1) distribution ($n \leq 128$).
SLARZ CLARZ Applies an elementary reflector (as returned by xTZRZF) to a general matrix.
SLARZB CLARZB Applies a block reflector or its transpose/conjugate-transpose to a general matrix.
SLARZT CLARZT Forms the triangular factor T of a block reflector H = I - V T VH.
SLAS2   Computes the singular values of a 2-by-2 triangular matrix.
SLASCL CLASCL Multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
SLASD0   Used by SBDSDC. Computes via a divide and conquer method the singular values of a real upper bidiagonal n-by-m matrix with diagonal D and offdiagonal E, where M = N + SQRE.
SLASD1   Used by SBDSDC. Computes the SVD of an upper bidiagonal N-by-M matrix, where N = NL + NR + 1 and M = N + SQRE.
SLASD2   Used by SBDSDC. Merges the two sets of singular values together into a single sorted set, and then it tries to deflate the size of the problem.
SLASD3   Used by SBDSDC. Finds all the square roots of the roots of the secular equation, as defined by the values in D and Z, and then updates the singular vectors by matrix multiplication.
Routine Description
real complex  
SLASD4   Used by SBDSDC. Computes the square root of the I-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix.
SLASD5   Used by SBDSDC. Computes the square root of the I-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix.
SLASD6   Used by SBDSDC. Computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row.
SLASD7   Used by SBDSDC. Merges the two sets of singular values together into a single sorted set, and then it tries to deflate the size of the problem.
SLASD8   Used by SBDSDC. Finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles (elements in DSIGMA).
SLASD9   Used by SBDSDC. Finds the square roots of the roots of the secular equation, and stores, for each element in D, the distance to its two nearest poles (elements in DSIGMA).
SLASDA   Used by SBDSDC. Computes the singular value decomposition (SVD) of a real upper bidiagonal N-by-M matrix with diagonal D and offdiagonal E, where M = N + SQRE.
SLASDQ   Used by SBDSDC. Computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal matrix with diagonal D and offdiagonal E, accumulating the transformations if desired.
SLASDT   Used by SBDSDC. Creates a tree of subproblems for bidiagonal divide and conquer.
SLASET CLASET Initializes the off-diagonal elements of a matrix to $\alpha$ and the diagonal elements to $\beta$.
SLASQ1   Used by SBDSQR. Computes the singular values of a real n-by-n bidiagonal matrix with diagonal D and offdiagonal E.
SLASQ2   Used by SBDSQR and SSTEGR. Computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd array Z to high relative accuracy.
SLASQ3   Used by SBDSQR. Checks for deflation, computes a shift (TAU) and calls dqds.
SLASQ4   Used by SBDSQR. Computes an approximation TAU to the smallest eigenvalue using values of d from the previous transform.
SLASQ5   Used by SBDSQR and SSTEGR. Computes one dqds transform in ping-pong form.
SLASQ6   Used by SBDSQR and SSTEGR. computes one dqds transform in ping-pong form.
SLASR CLASR Applies a sequence of plane rotations to a general rectangular matrix.

Routine Description
real complex  
SLASRT   Sorts numbers in increasing or decreasing order using Quick Sort, reverting to Insertion sort on arrays of size $\leq$ 20.
SLASSQ CLASSQ Updates a sum of squares represented in scaled form.
SLASV2   Computes the singular value decomposition of a 2-by-2 triangular matrix.
SLASWP CLASWP Performs a sequence of row interchanges on a general rectangular matrix.
SLASY2   Solves the Sylvester matrix equation $A X \pm X B = \sigma C$ where A and B are of order 1 or 2, and may be transposed, and $\sigma$ is a scale factor.
SLASYF CLASYF CLAHEF Computes a partial factorization of a real symmetric/complex symmetric/complex Hermitian indefinite matrix, using the diagonal pivoting method.
SLATBS CLATBS Solves a triangular banded system of equations $A x = \sigma b$, $A^T x = \sigma b$, or $A^H x = \sigma b$, where $\sigma$ is a scale factor set to prevent overflow.
SLATDF CLATDF Uses the LU factorization of the n-by-n matrix computed by SGETC2 and computes a contribution to the reciprocal Dif-estimate.
SLATPS CLATPS Solves a triangular system of equations $A x = \sigma b$, $A^T x = \sigma b$, or $A^H x = \sigma b$, where A is held in packed storage, and $\sigma$ is a scale factor set to prevent overflow.
SLATRD CLATRD Reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal/unitary similarity transformation, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
SLATRS CLATRS Solves a triangular system of equations $A x = \sigma b$, $A^T x = \sigma b$, or $A^H x = \sigma b$, where $\sigma$ is a scale factor set to prevent overflow.
SLATRZ CLATRZ Factors an upper trapezoidal matrix by means of orthogonal/unitary transformations.
SLAUU2 CLAUU2 Computes the product U UH or LH L, where U and L are upper or lower triangular matrices (unblocked algorithm).
SLAUUM CLAUUM Computes the product U UH or LH L, where U and L are upper or lower triangular matrices.
SORG2L CUNG2L Generates all or part of the orthogonal/unitary matrix Q from a QL factorization determined by SGEQLF/CGEQLF (unblocked algorithm).
SORG2R CUNG2R Generates all or part of the orthogonal/unitary matrix Q from a QR factorization determined by SGEQRF/CGEQRF (unblocked algorithm).
SORGL2 CUNGL2 Generates all or part of the orthogonal/unitary matrix Q from an LQ factorization determined by SGELQF/CGELQF (unblocked algorithm).

Routine Description
real complex  
SORGR2 CUNGR2 Generates all or part of the orthogonal/unitary matrix Q from an RQ factorization determined by SGERQF/CGERQF (unblocked algorithm).
SORM2L CUNM2L Multiplies a general matrix by the orthogonal/unitary matrix from a QL factorization determined by SGEQLF/CGEQLF (unblocked algorithm).
SORM2R CUNM2R Multiplies a general matrix by the orthogonal/unitary matrix from a QR factorization determined by SGEQRF/CGEQRF (unblocked algorithm).
SORML2 CUNML2 Multiplies a general matrix by the orthogonal/unitary matrix from an LQ factorization determined by SGELQF/CGELQF (unblocked algorithm).
SORMR2 CUNMR2 Multiplies a general matrix by the orthogonal/unitary matrix from an RQ factorization determined by SGERQF/CGERQF (unblocked algorithm).
SORMR3 CUNMR3 Multiplies a general matrix by the orthogonal/unitary matrix from an RZ factorization determined by STZRZF/CTZRZF (unblocked algorithm).
SPBTF2 CPBTF2 Computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).
SPOTF2 CPOTF2 Computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
SPTTS2 CPTTS2 Solves a tridiagonal system of the form $A \ast X = B$ using the $L \ast D \ast L^H$ factorization of A computed by SPTTRF/CPTTRF.
SRSCL CSRSCL Multiplies a vector by the reciprocal of a real scalar.
SSYGS2 CHEGS2 Reduces a symmetric/Hermitian definite generalized eigenproblem $Ax = \lambda Bx$, $ABx=\lambda x$, or $BAx=\lambda x$, to standard form, where B has been factorized by SPOTRF/CPOTRF (unblocked algorithm).
SSYTD2 CHETD2 Reduces a symmetric/Hermitian matrix to real symmetric tridiagonal form by an orthogonal/unitary similarity transformation (unblocked algorithm).
SSYTF2 CSYTF2 CHETF2 Computes the factorization of a real symmetric/complex symmetric/complex Hermitian indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
STGEX2 CTGEX2 Swaps adjacent diagonal blocks (A11, B11) and (A22, B22) of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair (A, B) by an orthogonal/unitary equivalence transformation.
STGSY2 CTGSY2 Solves the generalized Sylvester equation (unblocked algorithm).
STRTI2 CTRTI2 Computes the inverse of a triangular matrix (unblocked algorithm).
XERBLA   Error handling routine called by LAPACK routines if an input parameter has an invalid value.





















next up previous contents index
Next: Quick Reference Guide to Up: Index of Auxiliary Routines Previous: Index of Auxiliary Routines   Contents   Index
Susan Blackford
1999-10-01