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.