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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 , where A is complex, B is real, and C is complex.
| CLACRT | Performs the transformation , 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 , where and 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 , where 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 , where and are complex scalars, x and y are complex vectors and A is a complex symmetric matrix.
| CSYR | Performs the symmetric rank-1 update , where 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 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 or , 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 , 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 , where T is a general tridiagonal matrix, and a scalar, using partial pivoting with row interchanges.
SLAGTM | CLAGTM | Performs a matrix-matrix product of the form , where A is a tridiagonal matrix, B and C are rectangular matrices, and and are scalars, which may be 0, 1, or -1.
SLAGTS |  | Solves the system of equations or , where T is a general tridiagonal matrix and 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 or , where D is a diagonal matrix, , b and x may be complex, and 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 .
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 , avoiding unnecessary overflow or harmful underflow.
SLAPY3 |  | Returns , 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 .
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 .
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 , the base representations and the eigenvalues of each Li Di LiT.
SLARRF |  | Finds a new relatively robust representation 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 ().
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 and the diagonal elements to .
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 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 where A and B are of order 1 or 2, and may be transposed, and 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 , , or , where 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 , , or , where A is held in packed storage, and 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 , , or , where 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 using the 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 , , or , 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.

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