LAPACK Auxiliary Routines

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 11/07/2023

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LAPACK Auxiliary Routines

Routine naming conventions, mathematical notation, and matrix storage schemes used for LAPACK auxiliary routines are the same as for the driver and computational routines described in previous chapters.

The table below summarizes information about the available LAPACK auxiliary routines.

LAPACK Auxiliary Routines
Routine Name	Data Types	Description
?lacgv	c, z	Conjugates a complex vector.
?lacrm	c, z	Multiplies a complex matrix by a square real matrix.
?lacrt	c, z	Performs a linear transformation of a pair of complex vectors.
?laesy	c, z	Computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.
?rot	c, z	Applies a plane rotation with real cosine and complex sine to a pair of complex vectors.
?spmv	c, z	Computes a matrix-vector product for complex vectors using a complex symmetric packed matrix
?spr	c, z	Performs the symmetrical rank-1 update of a complex symmetric packed matrix.
?syconv	s, c, d, z	Converts a symmetric matrix given by a triangular matrix factorization into two matrices and vice versa.
?symv	c, z	Computes a matrix-vector product for a complex symmetric matrix.
?syr	c, z	Performs the symmetric rank-1 update of a complex symmetric matrix.
i?max1	c, z	Finds the index of the vector element whose real part has maximum absolute value.
?sum1	sc, dz	Forms the 1-norm of the complex vector using the true absolute value.
?gbtf2	s, d, c, z	Computes the LU factorization of a general band matrix using the unblocked version of the algorithm.
?gebd2	s, d, c, z	Reduces a general matrix to bidiagonal form using an unblocked algorithm.
?gehd2	s, d, c, z	Reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
?gelq2	s, d, c, z	Computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
?gelqt3	s, d, c, z	Recursively computes the LQ factorization of a general matrix using the compact WY representation of Q.
?geql2	s, d, c, z	Computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
?geqr2	s, d, c, z	Computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
?geqr2p	s, d, c, z	Computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
?geqrt2	s, d, c, z	Computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
?geqrt3	s, d, c, z	Recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
?gerq2	s, d, c, z	Computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
?gesc2	s, d, c, z	Solves a system of linear equations using the LU factorization with complete pivoting computed by ?getc2.
?getc2	s, d, c, z	Computes the LU factorization with complete pivoting of the general `n`-by-`n` matrix.
?getf2	s, d, c, z	Computes the LU factorization of a general `m`-by-`n` matrix using partial pivoting with row interchanges (unblocked algorithm).
?gtts2	s, d, c, z	Solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by ?gttrf.
?isnan	s, d,	Tests input for NaN.
?laisnan	s, d,	Tests input for NaN by comparing two arguments for inequality.
?labrd	s, d, c, z	Reduces the first `nb` rows and columns of a general matrix to a bidiagonal form.
?lacn2	s, d, c, z	Estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
?lacon	s, d, c, z	Estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
?lacpy	s, d, c, z	Copies all or part of one two-dimensional array to another.
?ladiv	s, d, c, z	Performs complex division in real arithmetic, avoiding unnecessary overflow.
?lae2	s, d	Computes the eigenvalues of a 2-by-2 symmetric matrix.
?laebz	s, d	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 ?stebz.
?laed0	s, d, c, z	Used by ?stedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
?laed1	s, d	Used by sstedc/dstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
?laed2	s, d	Used by sstedc/dstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
?laed3	s, d	Used by sstedc/dstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
?laed4	s, d	Used by sstedc/dstedc. Finds a single root of the secular equation.
?laed5	s, d	Used by sstedc/dstedc. Solves the 2-by-2 secular equation.
?laed6	s, d	Used by sstedc/dstedc. Computes one Newton step in solution of the secular equation.
?laed7	s, d, c, z	Used by ?stedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
?laed8	s, d, c, z	Used by ?stedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
?laed9	s, d	Used by sstedc/dstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
?laeda	s, d	Used by ?stedc. Computes the `Z` vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
?laein	s, d, c, z	Computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
?laev2	s, d, c, z	Computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
?laexc	s, d	Swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.
?lag2	s, d	Computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.
?lags2	s, d	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.
?lagtf	s, d	Computes an LU factorization of a matrix `T`-λ`I`, where `T` is a general tridiagonal matrix, and λ a scalar, using partial pivoting with row interchanges.
?lagtm	s, d, c, z	Performs a matrix-matrix product of the form `C` = α`AB`+β`C`, where `A` is a tridiagonal matrix, `B` and `C` are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.
?lagts	s, d	Solves the system of equations (`T`-λ`I`)`x` = `y` or (`T`-λ`I`)^T`x` = `y`,where `T` is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by ?lagtf.
?lagv2	s, d	Computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (`A`,`B`) where `B` is upper triangular.
?lahqr	s, d, c, z	Computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
?lahrd	s, d, c, z	Reduces the first `nb` columns of a general rectangular matrix `A` so that elements below the `k`-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of `A`.
?lahr2	s, d, c, z	Reduces the specified number of first columns of a general rectangular matrix `A` so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of `A`.
?laic1	s, d, c, z	Applies one step of incremental condition estimation.
?lakf2	s, d, c, z	Forms a matrix containing Kronecker products between the given matrices.
?lals0	s, d, c, z	Applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by ?gelsd.
?lalsa	s, d, c, z	Computes the SVD of the coefficient matrix in compact form. Used by ?gelsd.
?lalsd	s, d, c, z	Uses the singular value decomposition of `A` to solve the least squares problem.
?lamrg	s, d	Creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order.
?lamswlq	s, d, c, z	Multiplies a general real matrix by a real orthogonal matrix defined as the product of blocked elementary reflectors computed by short wide LQ factorization.
?lamtsqr	s, d, c, z	Multiplies a general matrix by the product of blocked elementary reflectors computed by tall skinny QR factorization.
?laneg	s, d	Computes the Sturm count.
?langb	s, d, c, z	Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of general band matrix.
?lange	s, d, c, z	Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.
?langt	s, d, c, z	Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.
?lanhs	s, d, c, z	Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.
?lansb	s, d, c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.
?lanhb	c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix.
?lansp	s, d, c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.
?lanhp	c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix supplied in packed form.
?lanst/?lanht	s, d/c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric or complex Hermitian tridiagonal matrix.
?lansy	s, d, c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real/complex symmetric matrix.
?lanhe	c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix.
?lantb	s, d, c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.
?lantp	s, d, c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.
?lantr	s, d, c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.
?lanv2	s, d	Computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
?lapll	s, d, c, z	Measures the linear dependence of two vectors.
?lapmr	s, d, c, z	Rearranges rows of a matrix as specified by a permutation vector.
?lapmt	s, d, c, z	Performs a forward or backward permutation of the columns of a matrix.
?lapy2	s, d	Returns sqrt(`x`²+`y`²).
?lapy3	s, d	Returns sqrt(`x`²+`y`²+`z`²).
?laqgb	s, d, c, z	Scales a general band matrix, using row and column scaling factors computed by ?gbequ.
?laqge	s, d, c, z	Scales a general rectangular matrix, using row and column scaling factors computed by ?geequ.
?laqhb	c, z	Scales a Hermitian band matrix, using scaling factors computed by ?pbequ.
?laqp2	s, d, c, z	Computes a QR factorization with column pivoting of the matrix block.
?laqps	s, d, c, z	Computes a step of QR factorization with column pivoting of a real m-by-n matrix `A` by using BLAS level 3.
?laqr0	s, d, c, z	Computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
?laqr1	s, d, c, z	Sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix `H` and specified shifts.
?laqr2	s, d, c, z	Performs the orthogonal/unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
?laqr3	s, d, c, z	Performs the orthogonal/unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
?laqr4	s, d, c, z	Computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
?laqr5	s, d, c, z	Performs a single small-bulge multi-shift QR sweep.
?laqsb	s, d, c, z	Scales a symmetric/Hermitian band matrix, using scaling factors computed by ?pbequ.
?laqsp	s, d, c, z	Scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by ?ppequ.
?laqsy	s, d, c, z	Scales a symmetric/Hermitian matrix, using scaling factors computed by ?poequ.
?laqtr	s, d	Solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.
?lar1v	s, d, c, z	Computes the (scaled) `r`-th column of the inverse of the submatrix in rows `b1` through `bn` of the tridiagonal matrix `LDL`^T - λ`I`.
?lar2v	s, d, c, z	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.
?laran	s, d	Returns a random real number from a uniform distribution.
?larf	s, d, c, z	Applies an elementary reflector to a general rectangular matrix.
?larfb	s, d, c, z	Applies a block reflector or its transpose/conjugate-transpose to a general rectangular matrix.
?larfg	s, d, c, z	Generates an elementary reflector (Householder matrix).
?larfgp	s, d, c, z	Generates an elementary reflector (Householder matrix) with non-negatibe beta.
?larft	s, d, c, z	Forms the triangular factor `T` of a block reflector `H` = `I` - `vtv`^H
?larfx	s, d, c, z	Applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.
?large	s, d, c, z	Pre- and post-multiplies a real general matrix with a random orthogonal matrix.
?larnd	s, d, c, z	Returns a random real number from a uniform or normal distribution.
?largv	s, d, c, z	Generates a vector of plane rotations with real cosines and real/complex sines.
?larnv	s, d, c, z	Returns a vector of random numbers from a uniform or normal distribution.
?laror	s, d, c, z	Pre- or post-multiplies an `m`-by-`n` matrix by a random orthogonal/unitary matrix.
?larot	s, d, c, z	Applies a Givens rotation to two adjacent rows or columns.
?larra	s, d	Computes the splitting points with the specified threshold.
?larrb	s, d	Provides limited bisection to locate eigenvalues for more accuracy.
?larrc	s, d	Computes the number of eigenvalues of the symmetric tridiagonal matrix.
?larrd	s, d	Computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.
?larre	s, d	Given the tridiagonal matrix `T`, sets small off-diagonal elements to zero and for each unreduced block `T`_i, finds base representations and eigenvalues.
?larrf	s, d	Finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
?larrj	s, d	Performs refinement of the initial estimates of the eigenvalues of the matrix `T`.
?larrk	s, d	Computes one eigenvalue of a symmetric tridiagonal matrix `T` to suitable accuracy.
?larrr	s, d	Performs tests to decide whether the symmetric tridiagonal matrix `T` warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
?larrv	s, d, c, z	Computes the eigenvectors of the tridiagonal matrix `T` = `LDL`^T given `L`, `D` and the eigenvalues of `LDL`^T.
?lartg	s, d, c, z	Generates a plane rotation with real cosine and real/complex sine.
?lartgp	s, d	Generates a plane rotation so that the diagonal is nonnegative.
?lartgs	s, d	Generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.
?lartv	s, d, c, z	Applies a vector of plane rotations with real cosines and real/complex sines to the elements of a pair of vectors.
?laruv	s, d	Returns a vector of n random real numbers from a uniform distribution.
?larz	s, d, c, z	Applies an elementary reflector (as returned by ?tzrzf) to a general matrix.
?larzb	s, d, c, z	Applies a block reflector or its transpose/conjugate-transpose to a general matrix.
?larzt	s, d, c, z	Forms the triangular factor `T` of a block reflector `H` = `I` - `vtv`^H.
?las2	s, d	Computes singular values of a 2-by-2 triangular matrix.
?lascl	s, d, c, z	Multiplies a general rectangular matrix by a real scalar defined as `c`_to/`c`_from.
?lasd0	s, d	Computes the singular values of a real upper bidiagonal n-by-m matrix `B` with diagonal `d` and off-diagonal `e`. Used by ?bdsdc.
?lasd1	s, d	Computes the SVD of an upper bidiagonal matrix `B` of the specified size. Used by ?bdsdc.
?lasd2	s, d	Merges the two sets of singular values together into a single sorted set. Used by ?bdsdc.
?lasd3	s, d	Finds all 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. Used by ?bdsdc.
?lasd4	s, d	Computes the square root of the `i`-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by ?bdsdc.
?lasd5	s, d	Computes the square root of the `i`-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by ?bdsdc.
?lasd6	s, d	Computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by ?bdsdc.
?lasd7	s, d	Merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by ?bdsdc.
?lasd8	s, d	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. Used by ?bdsdc.
?lasda	s, d	Computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal `d` and off-diagonal `e`. Used by ?bdsdc.
?lasdq	s, d	Computes the SVD of a real bidiagonal matrix with diagonal `d` and off-diagonal `e`. Used by ?bdsdc.
?lasdt	s, d	Creates a tree of subproblems for bidiagonal divide and conquer. Used by ?bdsdc.
?laset	s, d, c, z	Initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
?lasq1	s, d	Computes the singular values of a real square bidiagonal matrix. Used by ?bdsqr.
?lasq2	s, d	Computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the `qd` Array `Z` to high relative accuracy. Used by ?bdsqr and ?stegr.
?lasq3	s, d	Checks for deflation, computes a shift and calls `dqds`. Used by ?bdsqr.
?lasq4	s, d	Computes an approximation to the smallest eigenvalue using values of `d` from the previous transform. Used by ?bdsqr.
?lasq5	s, d	Computes one `dqds` transform in ping-pong form. Used by ?bdsqr and ?stegr.
?lasq6	s, d	Computes one `dqd` transform in ping-pong form. Used by ?bdsqr and ?stegr.
?lasr	s, d, c, z	Applies a sequence of plane rotations to a general rectangular matrix.
?lasrt	s, d	Sorts numbers in increasing or decreasing order.
?lassq	s, d, c, z	Updates a sum of squares represented in scaled form.
?lasv2	s, d	Computes the singular value decomposition of a 2-by-2 triangular matrix.
?laswp	s, d, c, z	Performs a series of row interchanges on a general rectangular matrix.
?laswlq	s, d, c, z	Computes blocked short-wide LQ matrix factorization.
?lasy2	s, d	Solves the Sylvester matrix equation where the matrices are of order 1 or 2.
?lasyf	s, d, c, z	Computes a partial factorization of a real/complex symmetric matrix, using the diagonal pivoting method.
?lasyf_aa	s, d, c, z	Factorizes a panel of a symmetric matrix using Aasen's algorithm.
?lasyf_rook	s, d, c, z	Computes a partial factorization of a real/complex symmetric matrix, using the bounded Bunch-Kaufman diagonal pivoting method.
?lahef	c, z	Computes a partial factorization of a complex Hermitian indefinite matrix, using the diagonal pivoting method.
?lahef_rook	c, z	Computes a partial factorization of a complex Hermitian indefinite matrix, using the bounded Bunch-Kaufman diagonal pivoting method.
?lahef_aa	c, z	Computes a partial factorization of a complex Hermitian matrix, using Aasen's algorithm.
?latbs	s, d, c, z	Solves a triangular banded system of equations.
?latdf	s, d, c, z	Uses the LU factorization of the `n`-by-`n` matrix computed by ?getc2 and computes a contribution to the reciprocal Dif-estimate.
?latm1	s, d, c, z	Computes the entries of a matrix as specified.
?latm2	s, d, c, z	Returns an entry of a random matrix.
?latm3	s, d, c, z	Returns set entry of a random matrix.
?latm5	s, d, c, z	Generates matrices involved in the Generalized Sylvester equation.
?latm6	s, d, c, z	Generates test matrices for the generalized eigenvalue problem, their corresponding right and left eigenvector matrices, and also reciprocal condition numbers for all eigenvalues and the reciprocal condition numbers of eigenvectors corresponding to the 1th and 5th eigenvalues.
?latme	s, d, c, z	Generates random non-symmetric square matrices with specified eigenvalues.
?latmr	s, d, c, z	Generates random matrices of various types.
?latps	s, d, c, z	Solves a triangular system of equations with the matrix held in packed storage.
?latrd	s, d, c, z	Reduces the first `nb` rows and columns of a symmetric/Hermitian matrix `A` to real tridiagonal form by an orthogonal/unitary similarity transformation.
?latrs	s, d, c, z	Solves a triangular system of equations with the scale factor set to prevent overflow.
?latrz	s, d, c, z	Factors an upper trapezoidal matrix by means of orthogonal/unitary transformations.
?latsqr	s, d, c, z	Computes a blocked tall-skinny QR matrix factorization.
?lauu2	s, d, c, z	Computes the product `UU`^H or `L`^H`L`, where `U` and `L` are upper or lower triangular matrices (unblocked algorithm).
?lauum	s, d, c, z	Computes the product `UU`^H or `L`^H`L`, where `U` and `L` are upper or lower triangular matrices (blocked algorithm).
?orbdb1/?unbdb1 ?orbdb2/?unbdb2 ?orbdb3/?unbdb3 ?orbdb4/?unbdb4	s, d, c, z	Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns.
?orbdb5/?unbdb5 ?orbdb6/?unbdb6	s, d, c, z	Orthogonalizes a column vector with respect to the orthonormal basis matrix.
?org2l/?ung2l	s, d/c, z	Generates all or part of the orthogonal/unitary matrix `Q` from a QL factorization determined by ?geqlf (unblocked algorithm).
?org2r/?ung2r	s, d/c, z	Generates all or part of the orthogonal/unitary matrix `Q` from a QR factorization determined by ?geqrf (unblocked algorithm).
?orgl2/?ungl2	s, d/c, z	Generates all or part of the orthogonal/unitary matrix `Q` from an LQ factorization determined by ?gelqf (unblocked algorithm).
?orgr2/?ungr2	s, d/c, z	Generates all or part of the orthogonal/unitary matrix `Q` from an RQ factorization determined by ?gerqf (unblocked algorithm).
?orm2l/?unm2l	s, d/c, z	Multiplies a general matrix by the orthogonal/unitary matrix from a QL factorization determined by ?geqlf (unblocked algorithm).
?orm2r/?unm2r	s, d/c, z	Multiplies a general matrix by the orthogonal/unitary matrix from a QR factorization determined by ?geqrf (unblocked algorithm).
?orml2/?unml2	s, d/c, z	Multiplies a general matrix by the orthogonal/unitary matrix from a LQ factorization determined by ?gelqf (unblocked algorithm).
?ormr2/?unmr2	s, d/c, z	Multiplies a general matrix by the orthogonal/unitary matrix from a RQ factorization determined by ?gerqf (unblocked algorithm).
?ormr3/?unmr3	s, d/c, z	Multiplies a general matrix by the orthogonal/unitary matrix from a RZ factorization determined by ?tzrzf (unblocked algorithm).
?pbtf2	s, d, c, z	Computes the Cholesky factorization of a symmetric/ Hermitian positive definite band matrix (unblocked algorithm).
?potf2	s, d, c, z	Computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
?ptts2	s, d, c, z	Solves a tridiagonal system of the form `AX`=`B` using the `LDL`^H factorization computed by ?pttrf.
?rscl	s, d, cs, zd	Multiplies a vector by the reciprocal of a real scalar.
?syswapr	s, d, c, z	Applies an elementary permutation on the rows and columns of a symmetric matrix.
?heswapr	c, z	Applies an elementary permutation on the rows and columns of a Hermitian matrix.
?sygs2/?hegs2	s, d/c, z	Reduces a symmetric/Hermitian positive-definite generalized eigenproblem to standard form, using the factorization results obtained from ?potrf (unblocked algorithm).
?sytd2/?hetd2	s, d/c, z	Reduces a symmetric/Hermitian matrix to real symmetric tridiagonal form by an orthogonal/unitary similarity transformation (unblocked algorithm).
?sytf2	s, d, c, z	Computes the factorization of a real/complex symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
?sytf2_rook	s, d, c, z	Computes the factorization of a real/complex symmetric indefinite matrix, using the bounded Bunch-Kaufman diagonal pivoting method (unblocked algorithm).
?hetf2	c, z	Computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm).
?hetf2_rook	c, z	Computes the factorization of a complex Hermitian matrix, using the bounded Bunch-Kaufman diagonal pivoting method (unblocked algorithm).
?tgex2	s, d, c, z	Swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal/unitary equivalence transformation.
?tgsy2	s, d, c, z	Solves the generalized Sylvester equation (unblocked algorithm).
?trti2	s, d, c, z	Computes the inverse of a triangular matrix (unblocked algorithm).
clag2z	c → z	Converts a complex single precision matrix to a complex double precision matrix.
dlag2s	d → s	Converts a double precision matrix to a single precision matrix.
slag2d	s → d	Converts a single precision matrix to a double precision matrix.
zlag2c	z → c	Converts a complex double precision matrix to a complex single precision matrix.
?larfp	s, d, c, z	Generates a real or complex elementary reflector.
ila?lc	s, d, c, z	Scans a matrix for its last non-zero column.
ila?lr	s, d, c, z	Scans a matrix for its last non-zero row.
?gsvj0	s, d	Pre-processor for the routine ?gesvj.
?gsvj1	s, d	Pre-processor for the routine ?gesvj, applies Jacobi rotations targeting only particular pivots.
?sfrk	s, d	Performs a symmetric rank-k operation for matrix in RFP format.
?hfrk	c, z	Performs a Hermitian rank-k operation for matrix in RFP format.
?tfsm	s, d, c, z	Solves a matrix equation (one operand is a triangular matrix in RFP format).
?lansf	s, d	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix in RFP format.
?lanhf	c, z	Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.
?tfttp	s, d, c, z	Copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP).
?tfttr	s, d, c, z	Copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR).
?tplqt2	s, d, c, z	Computes an LQ factorization of a triangular-pentagonal matrix using the compact WY representation for Q.
?tpqrt2	s, d, c, z	Computes a QR factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
?tprfb	s, d, c, z	Applies a real or complex "triangular-pentagonal" blocked reflector to a real or complex matrix, which is composed of two blocks.
?tpttf	s, d, c, z	Copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).
?tpttr	s, d, c, z	Copies a triangular matrix from the standard packed format (TP) to the standard full format (TR).
?trttf	s, d, c, z	Copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).
?trttp	s, d, c, z	Copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
?pstf2	s, d, c, z	Computes the Cholesky factorization with complete pivoting of a real symmetric or complex Hermitian positive semi-definite matrix.
dlat2s	d → s	Converts a double-precision triangular matrix to a single-precision triangular matrix.
zlat2c	z → c	Converts a double complex triangular matrix to a complex triangular matrix.
?lacp2	c, z	Copies all or part of a real two-dimensional array to a complex array.
?la_gbamv	s, d, c, z	Performs a matrix-vector operation to calculate error bounds.
?la_gbrcond	s, d	Estimates the Skeel condition number for a general banded matrix.
?la_gbrcond_c	c, z	Computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices.
?la_gbrcond_x	c, z	Computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.
?la_gbrfsx_extended	s, d, c, z	Improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_gbrpvgrw	s, d, c, z	Computes the reciprocal pivot growth factor norm(A)/norm(U) for a general banded matrix.
?la_geamv	s, d, c, z	Computes a matrix-vector product using a general matrix to calculate error bounds.
?la_gercond	s, d	Estimates the Skeel condition number for a general matrix.
?la_gercond_c	c, z	Computes the infinity norm condition number of op(A)*inv(diag(c)) for general matrices.
?la_gercond_x	c, z	Computes the infinity norm condition number of op(A)*diag(x) for general matrices.
?la_gerfsx_extended	s, d	Improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_heamv	c, z	Computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bounds.
?la_hercond_c	c, z	Computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefinite matrices.
?la_hercond_x	c, z	Computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite matrices.
?la_herfsx_extended	c, z	Improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_lin_berr	s, d, c, z	Computes a component-wise relative backward error.
?la_porcond	s, d	Estimates the Skeel condition number for a symmetric positive-definite matrix.
?la_porcond_c	c, z	Computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian positive-definite matrices.
?la_porcond_x	c, z	Computes the infinity norm condition number of op(A)*diag(x) for Hermitian positive-definite matrices.
?la_porfsx_extended	s, d, c, z	Improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_porpvgrw	s, d, c, z	Computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
?laqhe	c, z	Scales a Hermitian matrix.
?laqhp	c, z	Scales a Hermitian matrix stored in packed form.
?larcm	c, z	Copies all or part of a real two-dimensional array to a complex array.
?la_rpvgrw	c, z	Multiplies a square real matrix by a complex matrix.
?larscl2	s, d, c, z	Performs reciprocal diagonal scaling on a vector.
?lascl2	s, d, c, z	Performs diagonal scaling on a vector.
?la_syamv	s, d, c, z	Computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.
?la_syrcond	s, d	Estimates the Skeel condition number for a symmetric indefinite matrix.
?la_syrcond_c	c, z	Computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.
?la_syrcond_x	c, z	Computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.
?la_syrfsx_extended	s, d, c, z	Improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_syrpvgrw	s, d, c, z	Computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
?la_wwaddw	s, d, c, z	Adds a vector into a doubled-single vector.
?larfy	s, d, c, z	Applies an elementary reflector, or Householder matrix, H, to an `n` by `n` symmetric or Hermitian matrix C, from both the left and the right.
mkl_?tppack	s, d, c, z	Copies a triangular/symmetric matrix or submatrix from standard full format to standard packed format.
mkl_?tpunpack	s, d, c, z	Copies a triangular/symmetric matrix or submatrix from standard packed format to full format.

Parent topic: LAPACK Routines

?lacgv
Conjugates a complex vector.
?lacrm
Multiplies a complex matrix by a square real matrix.
?lacrt
Performs a linear transformation of a pair of complex vectors.
?laesy
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.
?rot
Applies a plane rotation with real cosine and complex sine to a pair of complex vectors.
?spmv
Computes a matrix-vector product for complex vectors using a complex symmetric packed matrix.
?spr
Performs the symmetrical rank-1 update of a complex symmetric packed matrix.
?syconv
Converts a symmetric matrix given by a triangular matrix factorization into two matrices and vice versa.
?symv
Computes a matrix-vector product for a complex symmetric matrix.
?syr
Performs the symmetric rank-1 update of a complex symmetric matrix.
i?max1
Finds the index of the vector element whose real part has maximum absolute value.
?sum1
Forms the 1-norm of the complex vector using the true absolute value.
?gbtf2
Computes the LU factorization of a general band matrix using the unblocked version of the algorithm.
?gebd2
Reduces a general matrix to bidiagonal form using an unblocked algorithm.
?gehd2
Reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
?gelq2
Computes the LQ factorization of a general rectangular matrix using an unblocked algorithm.
?gelqt3
?gelqt3 recursively computes a LQ factorization of a general real or complex M-by-N matrix A, using the compact WY representation of Q.
?geql2
Computes the QL factorization of a general rectangular matrix using an unblocked algorithm.
?geqr2
Computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
?geqr2p
Computes the QR factorization of a general rectangular matrix with non-negative diagonal elements using an unblocked algorithm.
?geqrt2
Computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
?geqrt3
Recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q.
?gerq2
Computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
?gesc2
Solves a system of linear equations using the LU factorization with complete pivoting computed by ?getc2.
?getc2
Computes the LU factorization with complete pivoting of the general n-by-n matrix.
?getf2
Computes the LU factorization of a general m-by-n matrix using partial pivoting with row interchanges (unblocked algorithm).
?gtts2
Solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by ?gttrf.
?isnan
Tests input for NaN.
?laisnan
Tests input for NaN.
?labrd
Reduces the first nb rows and columns of a general matrix to a bidiagonal form.
?lacn2
Estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
?lacon
Estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
?lacpy
Copies all or part of one two-dimensional array to another.
?ladiv
Performs complex division in real arithmetic, avoiding unnecessary overflow.
?lae2
Computes the eigenvalues of a 2-by-2 symmetric matrix.
?laebz
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 ?stebz.
?laed0
Used by ?stedc. Computes all eigenvalues and corresponding eigenvectors of an unreduced symmetric tridiagonal matrix using the divide and conquer method.
?laed1
Used by sstedc/dstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
?laed2
Used by sstedc/dstedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is tridiagonal.
?laed3
Used by sstedc/dstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is tridiagonal.
?laed4
Used by sstedc/dstedc. Finds a single root of the secular equation.
?laed5
Used by sstedc/dstedc. Solves the 2-by-2 secular equation.
?laed6
Used by sstedc/dstedc. Computes one Newton step in solution of the secular equation.
?laed7
Used by ?stedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is dense.
?laed8
Used by ?stedc. Merges eigenvalues and deflates secular equation. Used when the original matrix is dense.
?laed9
Used by sstedc/dstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
?laeda
Used by ?stedc. Computes the Z vector determining the rank-one modification of the diagonal matrix. Used when the original matrix is dense.
?laein
Computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
?laev2
Computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
?laexc
Swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation.
?lag2
Computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.
?lags2
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.
?lagtf
Computes an LU factorization of a matrix T-λ*I, where T is a general tridiagonal matrix, and λ is a scalar, using partial pivoting with row interchanges.
?lagtm
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.
?lagts
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 is a scalar, using the LU factorization computed by ?lagtf.
?lagv2
Computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular.
?lahqr
Computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
?lahrd
Reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A (deprecated).
?lahr2
Reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
?laic1
Applies one step of incremental condition estimation.
?lakf2
Forms a matrix containing Kronecker products between the given matrices.
?laln2
Solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
?lals0
Applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by ?gelsd.
?lalsa
Computes the SVD of the coefficient matrix in compact form. Used by ?gelsd.
?lalsd
Uses the singular value decomposition of A to solve the least squares problem.
?lamrg
Creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in acsending order.
?lamswlq
Multiplies a general real matrix by a real orthogonal matrix defined as the product of blocked elementary reflectors computed by short wide LQ factorization.
?lamtsqr
Multiplies a general matrix by the product of blocked elementary reflectors computed by tall skinny QR factorization (?latsqr)
?laneg
Computes the Sturm count, the number of negative pivots encountered while factoring tridiagonal T-sigma*I = L*D*L^T.
?langb
Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of general band matrix.
?lange
Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general rectangular matrix.
?langt
Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.
?lanhs
Returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.
?lansb
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.
?lanhb
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix.
?lansp
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.
?lanhp
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix supplied in packed form.
?lanst/?lanht
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric or complex Hermitian tridiagonal matrix.
?lansy
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real/complex symmetric matrix.
?lanhe
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix.
?lantb
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.
?lantp
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.
?lantr
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.
?lanv2
Computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form.
?lapll
Measures the linear dependence of two vectors.
?lapmr
Rearranges rows of a matrix as specified by a permutation vector.
?lapmt
Performs a forward or backward permutation of the columns of a matrix.
?lapy2
Returns sqrt(x²+y²).
?lapy3
Returns sqrt(x²+y²+z²).
?laqgb
Scales a general band matrix, using row and column scaling factors computed by ?gbequ.
?laqge
Scales a general rectangular matrix, using row and column scaling factors computed by ?geequ.
?laqhb
Scales a Hermetian band matrix, using scaling factors computed by ?pbequ.
?laqp2
Computes a QR factorization with column pivoting of the matrix block.
?laqps
Computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.
?laqr0
Computes the eigenvalues of a Hessenberg matrix, and optionally the marixes from the Schur decomposition.
?laqr1
Sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
?laqr2
Performs the orthogonal/unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
?laqr3
Performs the orthogonal/unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
?laqr4
Computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
?laqr5
Performs a single small-bulge multi-shift QR sweep.
?laqsb
Scales a symmetric band matrix, using scaling factors computed by ?pbequ.
?laqsp
Scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by ?ppequ.
?laqsy
Scales a symmetric/Hermitian matrix, using scaling factors computed by ?syequ, ?syequb, ?poequ , or ?poequb.
?laqtr
Solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.
?laqz0
Implements the multishift QZ method with aggressive early deflation for finding the generalized eigenvalues of the matrix pair (A,B).
?lar1v
Computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of tridiagonal matrix.
?lar2v
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.
?laran
Returns a random real number from a uniform distribution.
?larf
Applies an elementary reflector to a general rectangular matrix.
?larfb
Applies a block reflector or its transpose/conjugate-transpose to a general rectangular matrix.
?larfg
Generates an elementary reflector (Householder matrix).
?larfgp
Generates an elementary reflector (Householder matrix) with non-negative beta .
?larft
Forms the triangular factor T of a block reflector H = I - V*T*V**H.
?larfx
Applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order less than or equal to 10.
?larfy
Applies an elementary reflector, or Householder matrix, H, to an n by n symmetric or Hermitian matrix C, from both the left and the right.
?large
Pre- and post-multiplies a real general matrix with a random orthogonal matrix.
?largv
Generates a vector of plane rotations with real cosines and real/complex sines.
?larnd
Returns a random real number from a uniform or normal distribution.
?larnv
Returns a vector of random numbers from a uniform or normal distribution.
?laror
Pre- or post-multiplies an m-by-n matrix by a random orthogonal/unitary matrix.
?larot
Applies a Givens rotation to two adjacent rows or columns.
?larra
Computes the splitting points with the specified threshold.
?larrb
Provides limited bisection to locate eigenvalues for more accuracy.
?larrc
Computes the number of eigenvalues of the symmetric tridiagonal matrix.
?larrd
Computes the eigenvalues of a symmetric tridiagonal matrix to suitable accuracy.
?larre
Given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block T_i, finds base representations and eigenvalues.
?larrf
Finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
?larrj
Performs refinement of the initial estimates of the eigenvalues of the matrix T.
?larrk
Computes one eigenvalue of a symmetric tridiagonal matrix T to suitable accuracy.
?larrr
Performs tests to decide whether the symmetric tridiagonal matrix T warrants expensive computations which guarantee high relative accuracy in the eigenvalues.
?larrv
Computes the eigenvectors of the tridiagonal matrix T = L*D*L^T given L, D and the eigenvalues of L*D*L^T.
?lartg
Generates a plane rotation with real cosine and real/complex sine.
?lartgp
Generates a plane rotation.
?lartgs
Generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.
?lartv
Applies a vector of plane rotations with real cosines and real/complex sines to the elements of a pair of vectors.
?laruv
Returns a vector of n random real numbers from a uniform distribution.
?larz
Applies an elementary reflector (as returned by ?tzrzf) to a general matrix.
?larzb
Applies a block reflector or its transpose/conjugate-transpose to a general matrix.
?larzt
Forms the triangular factor T of a block reflector H = I - V*T*V^H.
?las2
Computes singular values of a 2-by-2 triangular matrix.
?lascl
Multiplies a general rectangular matrix by a real scalar defined as c_to/c_from.
?lasd0
Computes the singular values of a real upper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e. Used by ?bdsdc.
?lasd1
Computes the SVD of an upper bidiagonal matrix B of the specified size. Used by ?bdsdc.
?lasd2
Merges the two sets of singular values together into a single sorted set. Used by ?bdsdc.
?lasd3
Finds all 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. Used by ?bdsdc.
?lasd4
Computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by ?bdsdc.
?lasd5
Computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix.Used by ?bdsdc.
?lasd6
Computes the SVD of an updated upper bidiagonal matrix obtained by merging two smaller ones by appending a row. Used by ?bdsdc.
?lasd7
Merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by ?bdsdc.
?lasd8
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. Used by ?bdsdc.
?lasd9
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. Used by ?bdsdc.
?lasda
Computes the singular value decomposition (SVD) of a real upper bidiagonal matrix with diagonal d and off-diagonal e. Used by ?bdsdc.
?lasdq
Computes the SVD of a real bidiagonal matrix with diagonal d and off-diagonal e. Used by ?bdsdc.
?lasdt
Creates a tree of subproblems for bidiagonal divide and conquer. Used by ?bdsdc.
?laset
Initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
?lasq1
Computes the singular values of a real square bidiagonal matrix. Used by ?bdsqr.
?lasq2
Computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the quotient difference array z to high relative accuracy. Used by ?bdsqr and ?stegr.
?lasq3
Checks for deflation, computes a shift and calls dqds. Used by ?bdsqr.
?lasq4
Computes an approximation to the smallest eigenvalue using values of d from the previous transform. Used by ?bdsqr.
?lasq5
Computes one dqds transform in ping-pong form. Used by ?bdsqr and ?stegr.
?lasq6
Computes one dqd transform in ping-pong form. Used by ?bdsqr and ?stegr.
?lasr
Applies a sequence of plane rotations to a general rectangular matrix.
?lasrt
Sorts numbers in increasing or decreasing order.
?lassq
Updates a sum of squares represented in scaled form.
?lasv2
Computes the singular value decomposition of a 2-by-2 triangular matrix.
?laswlq
Computes blocked Short-Wide LQ matrix factorization.
?laswp
Performs a series of row interchanges on a general rectangular matrix.
?lasy2
Solves the Sylvester matrix equation where the matrices are of order 1 or 2.
?lasyf
Computes a partial factorization of a symmetric matrix, using the diagonal pivoting method.
?lasyf_aa
Factorizes a panel of a real or complex symmetric matrix using Aasen's algorithm.
?lasyf_rook
Computes a partial factorization of a complex symmetric matrix, using the bounded Bunch-Kaufman diagonal pivoting method.
?lahef
Computes a partial factorization of a complex Hermitian indefinite matrix, using the diagonal pivoting method.
?lahef_aa
Factorizes a panel of a complex Hermitian matrix A using Aasen's algorithm.
?lahef_rook
Computes a partial factorization of a complex Hermitian indefinite matrix, using the bounded Bunch-Kaufman diagonal pivoting method.
?latbs
Solves a triangular banded system of equations.
?latm1
Computes the entries of a matrix as specified.
?latm2
Returns an entry of a random matrix.
?latm3
Returns set entry of a random matrix.
?latm5
Generates matrices involved in the Generalized Sylvester equation.
?latm6
Generates test matrices for the generalized eigenvalue problem, their corresponding right and left eigenvector matrices, and also reciprocal condition numbers for all eigenvalues and the reciprocal condition numbers of eigenvectors corresponding to the 1th and 5th eigenvalues.
?latme
Generates random non-symmetric square matrices with specified eigenvalues.
?latmr
Generates random matrices of various types.
?latdf
Uses the LU factorization of the n-by-n matrix computed by ?getc2 and computes a contribution to the reciprocal Dif-estimate.
?latps
Solves a triangular system of equations with the matrix held in packed storage.
?latrd
Reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal/unitary similarity transformation.
?latrs
Solves a triangular system of equations with the scale factor set to prevent overflow.
?latrz
Factors an upper trapezoidal matrix by means of orthogonal/unitary transformations.
?latsqr
Computes a blocked Tall-Skinny QR matrix factorization.
?lauu2
Computes the product U*U^T(U*U^H) or L^T*L (L^H*L), where U and L are upper or lower triangular matrices (unblocked algorithm).
?lauum
Computes the product U*U^T(U*U^H) or L^T*L (L^H*L), where U and L are upper or lower triangular matrices (blocked algorithm).
?orbdb1/?unbdb1
Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns.
?orbdb2/?unbdb2
Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns.
?orbdb3/?unbdb3
Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns.
?orbdb4/?unbdb4
Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns.
?orbdb5/?unbdb5
Orthogonalizes a column vector with respect to the orthonormal basis matrix.
?orbdb6/?unbdb6
Orthogonalizes a column vector with respect to the orthonormal basis matrix.
?org2l/?ung2l
Generates all or part of the orthogonal/unitary matrix Q from a QL factorization determined by ?geqlf (unblocked algorithm).
?org2r/?ung2r
Generates all or part of the orthogonal/unitary matrix Q from a QR factorization determined by ?geqrf (unblocked algorithm).
?orgl2/?ungl2
Generates all or part of the orthogonal/unitary matrix Q from an LQ factorization determined by ?gelqf (unblocked algorithm).
?orgr2/?ungr2
Generates all or part of the orthogonal/unitary matrix Q from an RQ factorization determined by ?gerqf (unblocked algorithm).
?orm2l/?unm2l
Multiplies a general matrix by the orthogonal/unitary matrix from a QL factorization determined by ?geqlf (unblocked algorithm).
?orm2r/?unm2r
Multiplies a general matrix by the orthogonal/unitary matrix from a QR factorization determined by ?geqrf (unblocked algorithm).
?orml2/?unml2
Multiplies a general matrix by the orthogonal/unitary matrix from a LQ factorization determined by ?gelqf (unblocked algorithm).
?ormr2/?unmr2
Multiplies a general matrix by the orthogonal/unitary matrix from a RQ factorization determined by ?gerqf (unblocked algorithm).
?ormr3/?unmr3
Multiplies a general matrix by the orthogonal/unitary matrix from a RZ factorization determined by ?tzrzf (unblocked algorithm).
?pbtf2
Computes the Cholesky factorization of a symmetric/ Hermitian positive-definite band matrix (unblocked algorithm).
?potf2
Computes the Cholesky factorization of a symmetric/Hermitian positive-definite matrix (unblocked algorithm).
?ptts2
Solves a tridiagonal system of the form A*X=B using the L*D*L^H/L*D*L^H factorization computed by ?pttrf.
?rscl
Multiplies a vector by the reciprocal of a real scalar.
?syswapr
Applies an elementary permutation on the rows and columns of a symmetric matrix.
?heswapr
Applies an elementary permutation on the rows and columns of a Hermitian matrix.
?syswapr1
Applies an elementary permutation on the rows and columns of a symmetric matrix.
?sygs2/?hegs2
Reduces a symmetric/Hermitian positive-definite generalized eigenproblem to standard form, using the factorization results obtained from ?potrf (unblocked algorithm).
?sytd2/?hetd2
Reduces a symmetric/Hermitian matrix to real symmetric tridiagonal form by an orthogonal/unitary similarity transformation(unblocked algorithm).
?sytf2
Computes the factorization of a real/complex symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm).
?sytf2_rook
Computes the factorization of a real/complex symmetric indefinite matrix, using the bounded Bunch-Kaufman diagonal pivoting method (unblocked algorithm).
?hetf2
Computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm).
?hetf2_rook
Computes the factorization of a complex Hermitian matrix, using the bounded Bunch-Kaufman diagonal pivoting method (unblocked algorithm).
?tgex2
Swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal/unitary equivalence transformation.
?tgsy2
Solves the generalized Sylvester equation (unblocked algorithm).
?trti2
Computes the inverse of a triangular matrix (unblocked algorithm).
clag2z
Converts a complex single precision matrix to a complex double precision matrix.
dlag2s
Converts a double precision matrix to a single precision matrix.
slag2d
Converts a single precision matrix to a double precision matrix.
zlag2c
Converts a complex double precision matrix to a complex single precision matrix.
?larfp
Generates a real or complex elementary reflector.
ila?lc
Scans a matrix for its last non-zero column.
ila?lr
Scans a matrix for its last non-zero row.
?gsvj0
Pre-processor for the routine ?gesvj.
?gsvj1
Pre-processor for the routine ?gesvj, applies Jacobi rotations targeting only particular pivots.
?sfrk
Performs a symmetric rank-k operation for matrix in RFP format.
?hfrk
Performs a Hermitian rank-k operation for matrix in RFP format.
?tfsm
Solves a matrix equation (one operand is a triangular matrix in RFP format).
?lansf
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix in RFP format.
?lanhf
Returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.
?tfttp
Copies a triangular matrix from the rectangular full packed format (TF) to the standard packed format (TP) .
?tfttr
Copies a triangular matrix from the rectangular full packed format (TF) to the standard full format (TR) .
?tplqt2
?tplqt2 computes an LQ factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal using the compact WY representation for Q.
?tpqrt2
Computes a QR factorization of a real or complex "triangular-pentagonal" matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
?tprfb
Applies a real or complex "triangular-pentagonal" blocked reflector to a real or complex matrix, which is composed of two blocks.
?tpttf
Copies a triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF).
?tpttr
Copies a triangular matrix from the standard packed format (TP) to the standard full format (TR) .
?trttf
Copies a triangular matrix from the standard full format (TR) to the rectangular full packed format (TF).
?trttp
Copies a triangular matrix from the standard full format (TR) to the standard packed format (TP) .
?pstf2
Computes the Cholesky factorization with complete pivoting of a real symmetric or complex Hermitian positive semi-definite matrix.
dlat2s
Converts a double-precision triangular matrix to a single-precision triangular matrix.
zlat2c
Converts a double complex triangular matrix to a complex triangular matrix.
?lacp2
Copies all or part of a real two-dimensional array to a complex array.
?la_gbamv
Performs a matrix-vector operation to calculate error bounds.
?la_gbrcond
Estimates the Skeel condition number for a general banded matrix.
?la_gbrcond_c
Computes the infinity norm condition number of op(A)*inv(diag(c)) for general banded matrices.
?la_gbrcond_x
Computes the infinity norm condition number of op(A)*diag(x) for general banded matrices.
?la_gbrfsx_extended
Improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_gbrpvgrw
Computes the reciprocal pivot growth factor norm(A)/norm(U) for a general band matrix.
?la_geamv
Computes a matrix-vector product using a general matrix to calculate error bounds.
?la_gercond
Estimates the Skeel condition number for a general matrix.
?la_gercond_c
Computes the infinity norm condition number of op(A)*inv(diag(c)) for general matrices.
?la_gercond_x
Computes the infinity norm condition number of op(A)*diag(x) for general matrices.
?la_gerfsx_extended
Improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_heamv
Computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bounds.
?la_hercond_c
Computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian indefinite matrices.
?la_hercond_x
Computes the infinity norm condition number of op(A)*diag(x) for Hermitian indefinite matrices.
?la_herfsx_extended
Improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_herpvgrw
Computes the reciprocal pivot growth factor norm(A)/norm(U) for a Hermitian indefinite matrix.
?la_lin_berr
Computes component-wise relative backward error.
?la_porcond
Estimates the Skeel condition number for a symmetric positive-definite matrix.
?la_porcond_c
Computes the infinity norm condition number of op(A)*inv(diag(c)) for Hermitian positive-definite matrices.
?la_porcond_x
Computes the infinity norm condition number of op(A)*diag(x) for Hermitian positive-definite matrices.
?la_porfsx_extended
Improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_porpvgrw
Computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
?laqhe
Scales a Hermitian matrix.
?laqhp
Scales a Hermitian matrix stored in packed form.
?larcm
Multiplies a square real matrix by a complex matrix.
?la_gerpvgrw
Computes the reciprocal pivot growth factor norm(A)/norm(U) for a general matrix.
?larscl2
Performs reciprocal diagonal scaling on a vector.
?lascl2
Performs diagonal scaling on a vector.
?la_syamv
Computes a matrix-vector product using a symmetric indefinite matrix to calculate error bounds.
?la_syrcond
Estimates the Skeel condition number for a symmetric indefinite matrix.
?la_syrcond_c
Computes the infinity norm condition number of op(A)*inv(diag(c)) for symmetric indefinite matrices.
?la_syrcond_x
Computes the infinity norm condition number of op(A)*diag(x) for symmetric indefinite matrices.
?la_syrfsx_extended
Improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
?la_syrpvgrw
Computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
?la_wwaddw
Adds a vector into a doubled-single vector.
mkl_?tppack
Copies a triangular/symmetric matrix or submatrix from standard full format to standard packed format.
mkl_?tpunpack
Copies a triangular/symmetric matrix or submatrix from standard packed format to full format.
Additional LAPACK Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

LAPACK Auxiliary Routines