Module Lacaml.C

module C: Lacaml__C

Single precision complex BLAS and LAPACK functions.

type prec = Bigarray.complex32_elt

type num_type = Stdlib.Complex.t

type vec = (Stdlib.Complex.t, Bigarray.complex32_elt, Bigarray.fortran_layout)
       Bigarray.Array1.t

Complex vectors (precision: complex32).

type rvec = (float, Bigarray.float32_elt, Bigarray.fortran_layout) Bigarray.Array1.t

Vectors of reals (precision: float32).

type mat = (Stdlib.Complex.t, Bigarray.complex32_elt, Bigarray.fortran_layout)
       Bigarray.Array2.t

Complex matrices (precision: complex32).

type trans3 = [ `C | `N | `T ]

Transpose parameter (conjugate transposed, normal, or transposed).

val prec : (Stdlib.Complex.t, Bigarray.complex32_elt) Bigarray.kind

Precision for this submodule C. Allows to write precision independent code.

module Vec: sig .. end

module Mat: sig .. end

val pp_num : Stdlib.Format.formatter -> Stdlib.Complex.t -> unit

pp_num ppf el is equivalent to fprintf ppf "(%G, %Gi)" el.re el.im.

val pp_vec : (Stdlib.Complex.t, 'a) Lacaml__io.pp_vec

Pretty-printer for column vectors.

val pp_mat : (Stdlib.Complex.t, 'a) Lacaml__io.pp_mat

Pretty-printer for matrices.

BLAS-1 interface

val dotu : ?n:int ->
       ?ofsx:int ->
       ?incx:int ->
       vec -> ?ofsy:int -> ?incy:int -> vec -> Stdlib.Complex.t

dotu ?n ?ofsx ?incx x ?ofsy ?incy y see BLAS documentation!

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

ofsx : default = 1

incx : default = 1

ofsy : default = 1

incy : default = 1

val dotc : ?n:int ->
       ?ofsx:int ->
       ?incx:int ->
       vec -> ?ofsy:int -> ?incy:int -> vec -> Stdlib.Complex.t

dotc ?n ?ofsx ?incx x ?ofsy ?incy y see BLAS documentation!

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

ofsx : default = 1

incx : default = 1

ofsy : default = 1

incy : default = 1

LAPACK interface

val lansy_min_lwork : int -> Lacaml__common.norm4 -> int

lansy_min_lwork m norm

Returns the minimum length of the work array used by the lansy-function.

val lansy : ?n:int ->
       ?up:bool ->
       ?norm:Lacaml__common.norm4 ->
       ?work:rvec -> ?ar:int -> ?ac:int -> mat -> float

lansy ?n ?up ?norm ?work ?ar ?ac a see LAPACK documentation!

n : default = number of columns of matrix a

up : default = true (reference upper triangular part of a)

norm : default = `O

work : default = allocated work space for norm `I

val gecon_min_lwork : int -> int

gecon_min_lwork n

Returns the minimum length of the work array used by the gecon-function.

val gecon_min_lrwork : int -> int

gecon_min_lrwork n

Returns the minimum length of the rwork array used by the gecon-function.

val gecon : ?n:int ->
       ?norm:Lacaml__common.norm2 ->
       ?anorm:float ->
       ?work:vec ->
       ?rwork:rvec -> ?ar:int -> ?ac:int -> mat -> float

gecon ?n ?norm ?anorm ?work ?rwork ?ar ?ac a

Returns estimate of the reciprocal of the condition number of matrix a

n : default = available number of columns of matrix a

norm : default = 1-norm

anorm : default = norm of the matrix a as returned by lange

work : default = automatically allocated workspace

rwork : default = automatically allocated workspace

ar : default = 1

ac : default = 1

val sycon_min_lwork : int -> int

sycon_min_lwork n

Returns the minimum length of the work array used by the sycon-function.

val sycon : ?n:int ->
       ?up:bool ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?anorm:float ->
       ?work:vec -> ?ar:int -> ?ac:int -> mat -> float

sycon ?n ?up ?ipiv ?anorm ?work ?ar ?ac a

Returns estimate of the reciprocal of the condition number of symmetric matrix a

n : default = available number of columns of matrix a

up : default = upper triangle of the factorization of a is stored

ipiv : default = vec of length n

anorm : default = 1-norm of the matrix a as returned by lange

work : default = automatically allocated workspace

val pocon_min_lwork : int -> int

pocon_min_lwork n

Returns the minimum length of the work array used by the pocon-function.

val pocon_min_lrwork : int -> int

pocon_min_lrwork n

Returns the minimum length of the rwork array used by the pocon-function.

val pocon : ?n:int ->
       ?up:bool ->
       ?anorm:float ->
       ?work:vec ->
       ?rwork:rvec -> ?ar:int -> ?ac:int -> mat -> float

pocon ?n ?up ?anorm ?work ?rwork ?ar ?ac a

Returns estimate of the reciprocal of the condition number of complex Hermitian positive definite matrix a

n : default = available number of columns of matrix a

up : default = upper triangle of Cholesky factorization of a is stored

anorm : default = 1-norm of the matrix a as returned by lange

work : default = automatically allocated workspace

rwork : default = automatically allocated workspace

General Schur factorization

val gees : ?n:int ->
       ?jobvs:Lacaml__common.schur_vectors ->
       ?sort:Lacaml__common.eigen_value_sort ->
       ?w:vec ->
       ?vsr:int ->
       ?vsc:int ->
       ?vs:mat ->
       ?work:vec ->
       ?ar:int -> ?ac:int -> mat -> int * vec * mat

gees ?n ?jobvs ?sort ?w ?vsr ?vsc ?vs ?work ?ar ?ac a See gees-function for details about arguments.

Returns (sdim, w, vs)

General SVD routines

val gesvd_min_lwork : m:int -> n:int -> int

gesvd_min_lwork ~m ~n

Returns the minimum length of the work array used by the gesvd-function for matrices with m rows and n columns.

val gesvd_lrwork : m:int -> n:int -> int

gesvd_lrwork m n

Returns the (minimum) length of the rwork array used by the gesvd-function.

val gesvd_opt_lwork : ?m:int ->
       ?n:int ->
       ?jobu:Lacaml__common.svd_job ->
       ?jobvt:Lacaml__common.svd_job ->
       ?s:rvec ->
       ?ur:int ->
       ?uc:int ->
       ?u:mat ->
       ?vtr:int ->
       ?vtc:int -> ?vt:mat -> ?ar:int -> ?ac:int -> mat -> int

val gesvd : ?m:int ->
       ?n:int ->
       ?jobu:Lacaml__common.svd_job ->
       ?jobvt:Lacaml__common.svd_job ->
       ?s:rvec ->
       ?ur:int ->
       ?uc:int ->
       ?u:mat ->
       ?vtr:int ->
       ?vtc:int ->
       ?vt:mat ->
       ?work:vec ->
       ?rwork:rvec ->
       ?ar:int ->
       ?ac:int -> mat -> rvec * mat * mat

General eigenvalue problem (simple drivers)

val geev_min_lwork : int -> int

geev_min_lwork n

Returns the minimum length of the work array used by the geev-function.

val geev_min_lrwork : int -> int

geev_min_lrwork n

Returns the minimum length of the rwork array used by the geev-function.

val geev_opt_lwork : ?n:int ->
       ?vlr:int ->
       ?vlc:int ->
       ?vl:mat option ->
       ?vrr:int ->
       ?vrc:int ->
       ?vr:mat option ->
       ?ofsw:int -> ?w:vec -> ?ar:int -> ?ac:int -> mat -> int

geev ?work ?rwork ?n ?vlr ?vlc ?vl ?vrr ?vrc ?vr ?ofsw w ?ar ?ac a See geev-function for details about arguments.

Returns "optimal" work size

val geev : ?n:int ->
       ?work:vec ->
       ?rwork:vec ->
       ?vlr:int ->
       ?vlc:int ->
       ?vl:mat option ->
       ?vrr:int ->
       ?vrc:int ->
       ?vr:mat option ->
       ?ofsw:int ->
       ?w:vec ->
       ?ar:int ->
       ?ac:int -> mat -> mat * vec * mat

geev ?work ?rwork ?n ?vlr ?vlc ?vl ?vrr ?vrc ?vr ?ofsw w ?ar ?ac a

Raises Failure if the function fails to converge
Returns (lv, w, rv), where lv and rv correspond to the left and right eigenvectors respectively, w to the eigenvalues. lv (rv) is the empty matrix if vl (vr) is set to None.

n : default = available number of columns of matrix a

work : default = automatically allocated workspace

rwork : default = automatically allocated workspace

vl : default = Automatically allocated left eigenvectors. Pass None if you do not want to compute them, Some lv if you want to provide the storage. You can set vlr, vlc in the last case. (See LAPACK GEEV docs for details about storage of complex eigenvectors)

vr : default = Automatically allocated right eigenvectors. Pass None if you do not want to compute them, Some rv if you want to provide the storage. You can set vrr, vrc in the last case.

w : default = automatically allocate eigenvalues

BLAS-1 interface

val swap : ?n:int ->
       ?ofsx:int ->
       ?incx:int -> vec -> ?ofsy:int -> ?incy:int -> vec -> unit

swap ?n ?ofsx ?incx x ?ofsy ?incy y see BLAS documentation!

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

ofsx : default = 1

incx : default = 1

ofsy : default = 1

incy : default = 1

val scal : ?n:int -> Stdlib.Complex.t -> ?ofsx:int -> ?incx:int -> vec -> unit

scal ?n alpha ?ofsx ?incx x see BLAS documentation!

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

ofsx : default = 1

incx : default = 1

val copy : ?n:int ->
       ?ofsy:int ->
       ?incy:int ->
       ?y:vec -> ?ofsx:int -> ?incx:int -> vec -> vec

copy ?n ?ofsy ?incy ?y ?ofsx ?incx x see BLAS documentation!

Returns vector y, which is overwritten.

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

ofsy : default = 1

incy : default = 1

y : default = new vector with ofsy+(n-1)(abs incy) rows

ofsx : default = 1

incx : default = 1

val nrm2 : ?n:int -> ?ofsx:int -> ?incx:int -> vec -> float

nrm2 ?n ?ofsx ?incx x see BLAS documentation!

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

ofsx : default = 1

incx : default = 1

val axpy : ?alpha:Stdlib.Complex.t ->
       ?n:int ->
       ?ofsx:int ->
       ?incx:int -> vec -> ?ofsy:int -> ?incy:int -> vec -> unit

axpy ?alpha ?n ?ofsx ?incx x ?ofsy ?incy y see BLAS documentation!

alpha : default = { re = 1.; im = 0. }

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

ofsx : default = 1

incx : default = 1

ofsy : default = 1

incy : default = 1

val iamax : ?n:int -> ?ofsx:int -> ?incx:int -> vec -> int

iamax ?n ?ofsx ?incx x see BLAS documentation!

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

ofsx : default = 1

incx : default = 1

val amax : ?n:int -> ?ofsx:int -> ?incx:int -> vec -> Stdlib.Complex.t

amax ?n ?ofsx ?incx x

Returns the greater of the absolute values of the elements of the vector x.

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

ofsx : default = 1

incx : default = 1

BLAS-2 interface

val gemv : ?m:int ->
       ?n:int ->
       ?beta:Stdlib.Complex.t ->
       ?ofsy:int ->
       ?incy:int ->
       ?y:vec ->
       ?trans:trans3 ->
       ?alpha:Stdlib.Complex.t ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?ofsx:int -> ?incx:int -> vec -> vec

gemv ?m ?n ?beta ?ofsy ?incy ?y ?trans ?alpha ?ar ?ac a ?ofsx ?incx x performs the operation y := alpha * op(a) * x + beta * y where op(a) = a or aᵀ according to the value of trans. See BLAS documentation for more information. BEWARE that the 1988 BLAS-2 specification mandates that this function has no effect when n=0 while the mathematically expected behavior is y ← beta * y.

Returns vector y, which is overwritten.

m : default = number of available rows in matrix a

n : default = available columns in matrix a

beta : default = { re = 0.; im = 0. }

ofsy : default = 1

incy : default = 1

y : default = vector with minimal required length (see BLAS)

trans : default = `N

alpha : default = { re = 1.; im = 0. }

ar : default = 1

ac : default = 1

ofsx : default = 1

incx : default = 1

val gbmv : ?m:int ->
       ?n:int ->
       ?beta:Stdlib.Complex.t ->
       ?ofsy:int ->
       ?incy:int ->
       ?y:vec ->
       ?trans:trans3 ->
       ?alpha:Stdlib.Complex.t ->
       ?ar:int ->
       ?ac:int ->
       mat ->
       int -> int -> ?ofsx:int -> ?incx:int -> vec -> vec

gbmv ?m ?n ?beta ?ofsy ?incy ?y ?trans ?alpha ?ar ?ac a kl ku ?ofsx ?incx x see BLAS documentation!

Returns vector y, which is overwritten.

m : default = same as n (i.e., a is a square matrix)

n : default = available number of columns in matrix a

beta : default = { re = 0.; im = 0. }

ofsy : default = 1

incy : default = 1

y : default = vector with minimal required length (see BLAS)

trans : default = `N

alpha : default = { re = 1.; im = 0. }

ar : default = 1

ac : default = 1

ofsx : default = 1

incx : default = 1

val symv : ?n:int ->
       ?beta:Stdlib.Complex.t ->
       ?ofsy:int ->
       ?incy:int ->
       ?y:vec ->
       ?up:bool ->
       ?alpha:Stdlib.Complex.t ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?ofsx:int -> ?incx:int -> vec -> vec

symv ?n ?beta ?ofsy ?incy ?y ?up ?alpha ?ar ?ac a ?ofsx ?incx x see BLAS documentation!

Returns vector y, which is overwritten.

n : default = dimension of symmetric matrix a

beta : default = { re = 0.; im = 0. }

ofsy : default = 1

incy : default = 1

y : default = vector with minimal required length (see BLAS)

up : default = true (upper triangular portion of a is accessed)

alpha : default = { re = 1.; im = 0. }

ar : default = 1

ac : default = 1

ofsx : default = 1

incx : default = 1

val trmv : ?n:int ->
       ?trans:trans3 ->
       ?diag:Lacaml__common.diag ->
       ?up:bool ->
       ?ar:int ->
       ?ac:int -> mat -> ?ofsx:int -> ?incx:int -> vec -> unit

trmv ?n ?trans ?diag ?up ?ar ?ac a ?ofsx ?incx x see BLAS documentation!

n : default = dimension of triangular matrix a

trans : default = `N

diag : default = false (not a unit triangular matrix)

up : default = true (upper triangular portion of a is accessed)

ar : default = 1

ac : default = 1

ofsx : default = 1

incx : default = 1

val trsv : ?n:int ->
       ?trans:trans3 ->
       ?diag:Lacaml__common.diag ->
       ?up:bool ->
       ?ar:int ->
       ?ac:int -> mat -> ?ofsx:int -> ?incx:int -> vec -> unit

trsv ?n ?trans ?diag ?up ?ar ?ac a ?ofsx ?incx x see BLAS documentation!

n : default = dimension of triangular matrix a

trans : default = `N

diag : default = false (not a unit triangular matrix)

up : default = true (upper triangular portion of a is accessed)

ar : default = 1

ac : default = 1

ofsx : default = 1

incx : default = 1

val tpmv : ?n:int ->
       ?trans:trans3 ->
       ?diag:Lacaml__common.diag ->
       ?up:bool ->
       ?ofsap:int ->
       vec -> ?ofsx:int -> ?incx:int -> vec -> unit

tpmv ?n ?trans ?diag ?up ?ofsap ap ?ofsx ?incx x see BLAS documentation!

n : default = dimension of packed triangular matrix ap

trans : default = `N

diag : default = false (not a unit triangular matrix)

up : default = true (upper triangular portion of ap is accessed)

ofsap : default = 1

ofsx : default = 1

incx : default = 1

val tpsv : ?n:int ->
       ?trans:trans3 ->
       ?diag:Lacaml__common.diag ->
       ?up:bool ->
       ?ofsap:int ->
       vec -> ?ofsx:int -> ?incx:int -> vec -> unit

tpsv ?n ?trans ?diag ?up ?ofsap ap ?ofsx ?incx x see BLAS documentation!

n : default = dimension of packed triangular matrix ap

trans : default = `N

diag : default = false (not a unit triangular matrix)

up : default = true (upper triangular portion of ap is accessed)

ofsap : default = 1

ofsx : default = 1

incx : default = 1

BLAS-3 interface

val gemm : ?m:int ->
       ?n:int ->
       ?k:int ->
       ?beta:Stdlib.Complex.t ->
       ?cr:int ->
       ?cc:int ->
       ?c:mat ->
       ?transa:trans3 ->
       ?alpha:Stdlib.Complex.t ->
       ?ar:int ->
       ?ac:int ->
       mat ->
       ?transb:trans3 ->
       ?br:int -> ?bc:int -> mat -> mat

gemm ?m ?n ?k ?beta ?cr ?cc ?c ?transa ?alpha ?ar ?ac a ?transb ?br ?bc b performs the operation c := alpha * op(a) * op(b) + beta * c where op(x) = x or xᵀ depending on transx. See BLAS documentation for more information.

Returns matrix c, which is overwritten.

m : default = number of rows of a (or tr a) and c

n : default = number of columns of b (or tr b) and c

k : default = number of columns of a (or tr a) and number of rows of b (or tr b)

beta : default = { re = 0.; im = 0. }

cr : default = 1

cc : default = 1

c : default = matrix with minimal required dimension

transa : default = `N

alpha : default = { re = 1.; im = 0. }

ar : default = 1

ac : default = 1

transb : default = `N

br : default = 1

bc : default = 1

val symm : ?m:int ->
       ?n:int ->
       ?side:Lacaml__common.side ->
       ?up:bool ->
       ?beta:Stdlib.Complex.t ->
       ?cr:int ->
       ?cc:int ->
       ?c:mat ->
       ?alpha:Stdlib.Complex.t ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?br:int -> ?bc:int -> mat -> mat

symm ?m ?n ?side ?up ?beta ?cr ?cc ?c ?alpha ?ar ?ac a ?br ?bc b see BLAS documentation!

Returns matrix c, which is overwritten.

m : default = number of rows of c

n : default = number of columns of c

side : default = `L (left - multiplication is ab)

up : default = true (upper triangular portion of a is accessed)

beta : default = { re = 0.; im = 0. }

cr : default = 1

cc : default = 1

c : default = matrix with minimal required dimension

alpha : default = { re = 1.; im = 0. }

ar : default = 1

ac : default = 1

br : default = 1

bc : default = 1

val trmm : ?m:int ->
       ?n:int ->
       ?side:Lacaml__common.side ->
       ?up:bool ->
       ?transa:trans3 ->
       ?diag:Lacaml__common.diag ->
       ?alpha:Stdlib.Complex.t ->
       ?ar:int ->
       ?ac:int -> a:mat -> ?br:int -> ?bc:int -> mat -> unit

trmm ?m ?n ?side ?up ?transa ?diag ?alpha ?ar ?ac ~a ?br ?bc b see BLAS documentation!

m : default = number of rows of b

n : default = number of columns of b

side : default = `L (left - multiplication is ab)

up : default = true (upper triangular portion of a is accessed)

transa : default = `N

diag : default = `N (non-unit)

alpha : default = { re = 1.; im = 0. }

ar : default = 1

ac : default = 1

br : default = 1

bc : default = 1

val trsm : ?m:int ->
       ?n:int ->
       ?side:Lacaml__common.side ->
       ?up:bool ->
       ?transa:trans3 ->
       ?diag:Lacaml__common.diag ->
       ?alpha:Stdlib.Complex.t ->
       ?ar:int ->
       ?ac:int -> a:mat -> ?br:int -> ?bc:int -> mat -> unit

trsm ?m ?n ?side ?up ?transa ?diag ?alpha ?ar ?ac ~a ?br ?bc b see BLAS documentation!

Returns matrix b, which is overwritten.

m : default = number of rows of b

n : default = number of columns of b

side : default = `L (left - multiplication is ab)

up : default = true (upper triangular portion of a is accessed)

transa : default = `N

diag : default = `N (non-unit)

alpha : default = { re = 1.; im = 0. }

ar : default = 1

ac : default = 1

br : default = 1

bc : default = 1

val syrk : ?n:int ->
       ?k:int ->
       ?up:bool ->
       ?beta:Stdlib.Complex.t ->
       ?cr:int ->
       ?cc:int ->
       ?c:mat ->
       ?trans:Lacaml__common.trans2 ->
       ?alpha:Stdlib.Complex.t ->
       ?ar:int -> ?ac:int -> mat -> mat

syrk ?n ?k ?up ?beta ?cr ?cc ?c ?trans ?alpha ?ar ?ac a see BLAS documentation!

Returns matrix c, which is overwritten.

n : default = number of rows of a (or a'), c

k : default = number of columns of a (or a')

up : default = true (upper triangular portion of c is accessed)

beta : default = { re = 0.; im = 0. }

cr : default = 1

cc : default = 1

c : default = matrix with minimal required dimension

trans : default = `N

alpha : default = { re = 1.; im = 0. }

ar : default = 1

ac : default = 1

val syr2k : ?n:int ->
       ?k:int ->
       ?up:bool ->
       ?beta:Stdlib.Complex.t ->
       ?cr:int ->
       ?cc:int ->
       ?c:mat ->
       ?trans:Lacaml__common.trans2 ->
       ?alpha:Stdlib.Complex.t ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?br:int -> ?bc:int -> mat -> mat

syr2k ?n ?k ?up ?beta ?cr ?cc ?c ?trans ?alpha ?ar ?ac a ?br ?bc b see BLAS documentation!

Returns matrix c, which is overwritten.

n : default = number of rows of a (or a'), c

k : default = number of columns of a (or a')

up : default = true (upper triangular portion of c is accessed)

beta : default = { re = 0.; im = 0. }

cr : default = 1

cc : default = 1

c : default = matrix with minimal required dimension

trans : default = `N

alpha : default = { re = 1.; im = 0. }

ar : default = 1

ac : default = 1

br : default = 1

bc : default = 1

LAPACK interface

Auxiliary routines

val lacpy : ?uplo:[ `L | `U ] ->
       ?m:int ->
       ?n:int ->
       ?br:int ->
       ?bc:int ->
       ?b:mat -> ?ar:int -> ?ac:int -> mat -> mat

lacpy ?uplo ?m ?n ?br ?bc ?b ?ar ?ac a copy the (triangular) (sub-)matrix a (to an optional (sub-)matrix b) and return it.

uplo : default = whole matrix

b : The target matrix. By default a fresh matrix to accommodate the sizes m and n and the offsets br and bc is created.

val laswp : ?n:int ->
       ?ar:int ->
       ?ac:int ->
       mat ->
       ?k1:int -> ?k2:int -> ?incx:int -> Lacaml__common.int32_vec -> unit

laswp ?n ?ar ?ac a ?k1 ?k2 ?incx ipiv swap rows of a according to ipiv. See LAPACK-documentation for details!

n : default = number of columns of matrix

ar : default = 1

ac : default = 1

k1 : default = 1

k2 : default = dimension of ipiv

incx : default = 1

val lapmt : ?forward:bool ->
       ?m:int ->
       ?n:int ->
       ?ar:int -> ?ac:int -> mat -> Lacaml__common.int32_vec -> unit

lapmt ?forward ?n ?m ?ar ?ac a k swap columns of a according to the permutations in k. See LAPACK-documentation for details!

forward : default = true

m : default = number of rows of matrix

n : default = number of columns of matrix

ar : default = 1

ac : default = 1

val lassq : ?n:int ->
       ?scale:float ->
       ?sumsq:float -> ?ofsx:int -> ?incx:int -> vec -> float * float

lassq ?n ?ofsx ?incx ?scale ?sumsq

Returns (scl, ssq), where scl is a scaling factor and ssq the sum of squares of vector x starting at ofs and using increment incx and initial scale and sumsq. The following equality holds: scl**2. *. ssq = x.{1}**2. +. ... +. x.{n}**2. +. scale**2. *. sumsq. See LAPACK-documentation for details!

n : default = greater n s.t. ofsx+(n-1)(abs incx) <= dim x

scale : default = 0.

sumsq : default = 1.

ofsx : default = 1

incx : default = 1

val larnv : ?idist:[ `Normal | `Uniform0 | `Uniform1 ] ->
       ?iseed:Lacaml__common.int32_vec ->
       ?n:int -> ?ofsx:int -> ?x:vec -> unit -> vec

larnv ?idist ?iseed ?n ?ofsx ?x ()

Returns a random vector with random distribution as specifified by idist, random seed iseed, vector offset ofsx and optional vector x.

idist : default = `Normal

iseed : default = integer vector of size 4 with all ones.

n : default = dim x - ofsx + 1 if x is provided, 1 otherwise.

ofsx : default = 1

x : default = vector of length ofsx - 1 + n if n is provided.

val lange_min_lwork : int -> Lacaml__common.norm4 -> int

lange_min_lwork m norm

Returns the minimum length of the work array used by the lange-function.

val lange : ?m:int ->
       ?n:int ->
       ?norm:Lacaml__common.norm4 ->
       ?work:rvec -> ?ar:int -> ?ac:int -> mat -> float

lange ?m ?n ?norm ?work ?ar ?ac a

Returns the value of the one norm (norm = `O), or the Frobenius norm (norm = `F), or the infinity norm (norm = `I), or the element of largest absolute value (norm = `M) of a real matrix a.

m : default = number of rows of matrix a

n : default = number of columns of matrix a

norm : default = `O

work : default = allocated work space for norm `I

ar : default = 1

ac : default = 1

val lauum : ?up:bool -> ?n:int -> ?ar:int -> ?ac:int -> mat -> unit

lauum ?up ?n ?ar ?ac a computes the product U * U**T or L**T * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array a. The upper or lower part of a is overwritten.

up : default = true

n : default = minimum of available number of rows/columns in matrix a

ar : default = 1

ac : default = 1

Linear equations (computational routines)

val getrf : ?m:int ->
       ?n:int ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?ar:int -> ?ac:int -> mat -> Lacaml__common.int32_vec

getrf ?m ?n ?ipiv ?ar ?ac a computes an LU factorization of a general m-by-n matrix a using partial pivoting with row interchanges. See LAPACK documentation.

Raises Failure if the matrix is singular.
Returns ipiv, the pivot indices.

m : default = number of rows in matrix a

n : default = number of columns in matrix a

ipiv : = vec of length min(m, n)

ar : default = 1

ac : default = 1

val getrs : ?n:int ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?trans:trans3 ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

getrs ?n ?ipiv ?trans ?ar ?ac a ?nrhs ?br ?bc b solves a system of linear equations a * X = b or a' * X = b with a general n-by-n matrix a using the LU factorization computed by Lacaml__C.getrf. Note that matrix a will be passed to Lacaml__C.getrf if ipiv was not provided.

Raises Failure if the matrix is singular.

n : default = number of columns in matrix a

ipiv : default = result from getrf applied to a

trans : default = `N

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val getri_min_lwork : int -> int

getri_min_lwork n

Returns the minimum length of the work array used by the Lacaml__C.getri-function if the matrix has n columns.

val getri_opt_lwork : ?n:int -> ?ar:int -> ?ac:int -> mat -> int

getri_opt_lwork ?n ?ar ?ac a

Returns the optimal size of the work array used by the Lacaml__C.getri-function.

n : default = number of columns of matrix a

ar : default = 1

ac : default = 1

val getri : ?n:int ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?work:vec -> ?ar:int -> ?ac:int -> mat -> unit

getri ?n ?ipiv ?work ?ar ?ac a computes the inverse of a matrix using the LU factorization computed by Lacaml__C.getrf. Note that matrix a will be passed to Lacaml__C.getrf if ipiv was not provided.

Raises Failure if the matrix is singular.

n : default = number of columns in matrix a

ipiv : default = vec of length m from getri

work : default = vec of optimum length

ar : default = 1

ac : default = 1

val sytrf_min_lwork : unit -> int

sytrf_min_lwork ()

Returns the minimum length of the work array used by the Lacaml__C.sytrf-function.

val sytrf_opt_lwork : ?n:int -> ?up:bool -> ?ar:int -> ?ac:int -> mat -> int

sytrf_opt_lwork ?n ?up ?ar ?ac a

Returns the optimal size of the work array used by the Lacaml__C.sytrf-function.

n : default = number of columns of matrix a

up : default = true (store upper triangle in a)

ar : default = 1

ac : default = 1

val sytrf : ?n:int ->
       ?up:bool ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?work:vec ->
       ?ar:int -> ?ac:int -> mat -> Lacaml__common.int32_vec

sytrf ?n ?up ?ipiv ?work ?ar ?ac a computes the factorization of the real symmetric matrix a using the Bunch-Kaufman diagonal pivoting method.

Raises Failure if D in a = U*D*U' or L*D*L' is singular.

n : default = number of columns in matrix a

up : default = true (store upper triangle in a)

ipiv : = vec of length n

work : default = vec of optimum length

ar : default = 1

ac : default = 1

val sytrs : ?n:int ->
       ?up:bool ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

sytrs ?n ?up ?ipiv ?ar ?ac a ?nrhs ?br ?bc b solves a system of linear equations a*X = b with a real symmetric matrix a using the factorization a = U*D*U**T or a = L*D*L**T computed by Lacaml__C.sytrf. Note that matrix a will be passed to Lacaml__C.sytrf if ipiv was not provided.

Raises Failure if the matrix is singular.

n : default = number of columns in matrix a

up : default = true (store upper triangle in a)

ipiv : default = vec of length n

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val sytri_min_lwork : int -> int

sytri_min_lwork n

Returns the minimum length of the work array used by the Lacaml__C.sytri-function if the matrix has n columns.

val sytri : ?n:int ->
       ?up:bool ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?work:vec -> ?ar:int -> ?ac:int -> mat -> unit

sytri ?n ?up ?ipiv ?work ?ar ?ac a computes the inverse of the real symmetric indefinite matrix a using the factorization a = U*D*U**T or a = L*D*L**T computed by Lacaml__C.sytrf. Note that matrix a will be passed to Lacaml__C.sytrf if ipiv was not provided.

Raises Failure if the matrix is singular.

n : default = number of columns in matrix a

up : default = true (store upper triangle in a)

ipiv : default = vec of length n from Lacaml__C.sytrf

work : default = vec of optimum length

ar : default = 1

ac : default = 1

val potrf : ?n:int ->
       ?up:bool ->
       ?ar:int -> ?ac:int -> ?jitter:Stdlib.Complex.t -> mat -> unit

potrf ?n ?up ?ar ?ac ?jitter a factorizes symmetric positive definite matrix a (or the designated submatrix) using Cholesky factorization.

Due to rounding errors ill-conditioned matrices may actually appear as if they were not positive definite, thus leading to an exception. One remedy for this problem is to add a small jitter to the diagonal of the matrix, which will usually allow Cholesky to complete successfully (though at a small bias). For extremely ill-conditioned matrices it is recommended to use (symmetric) eigenvalue decomposition instead of this function for a numerically more stable factorization.

Raises Failure if the matrix is singular.

n : default = number of columns in matrix a

up : default = true (store upper triangle in a)

ar : default = 1

ac : default = 1

jitter : default = nothing

val potrs : ?n:int ->
       ?up:bool ->
       ?ar:int ->
       ?ac:int ->
       mat ->
       ?nrhs:int ->
       ?br:int ->
       ?bc:int ->
       ?factorize:bool -> ?jitter:Stdlib.Complex.t -> mat -> unit

potrs ?n ?up ?ar ?ac a ?nrhs ?br ?bc ?factorize ?jitter b solves a system of linear equations a*X = b, where a is symmetric positive definite matrix, using the Cholesky factorization a = U**T*U or a = L*L**T computed by Lacaml__C.potrf.

Raises Failure if the matrix is singular.

n : default = number of columns in matrix a

up : default = true

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

factorize : default = true (calls Lacaml__C.potrf implicitly)

jitter : default = nothing

val potri : ?n:int ->
       ?up:bool ->
       ?ar:int ->
       ?ac:int ->
       ?factorize:bool -> ?jitter:Stdlib.Complex.t -> mat -> unit

potri ?n ?up ?ar ?ac ?factorize ?jitter a computes the inverse of the real symmetric positive definite matrix a using the Cholesky factorization a = U**T*U or a = L*L**T computed by Lacaml__C.potrf.

Raises Failure if the matrix is singular.

n : default = number of columns in matrix a

up : default = true (upper triangle stored in a)

ar : default = 1

ac : default = 1

factorize : default = true (calls Lacaml__C.potrf implicitly)

jitter : default = nothing

val trtrs : ?n:int ->
       ?up:bool ->
       ?trans:trans3 ->
       ?diag:Lacaml__common.diag ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

trtrs ?n ?up ?trans ?diag ?ar ?ac a ?nrhs ?br ?bc b solves a triangular system of the form a * X = b or a**T * X = n, where a is a triangular matrix of order n, and b is an n-by-nrhs matrix.

Raises Failure if the matrix a is singular.

n : default = number of columns in matrix a

up : default = true

trans : default = `N

diag : default = `N

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val tbtrs : ?n:int ->
       ?kd:int ->
       ?up:bool ->
       ?trans:trans3 ->
       ?diag:Lacaml__common.diag ->
       ?abr:int ->
       ?abc:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

tbtrs ?n ?kd ?up ?trans ?diag ?abr ?abc ab ?nrhs ?br ?bc b solves a triangular system of the form a * X = b or a**T * X = b, where a is a triangular band matrix of order n, and b is an n-by-nrhs matrix.

Raises Failure if the matrix a is singular.

n : default = number of columns in matrix ab

kd : default = number of rows in matrix ab - 1

up : default = true

trans : default = `N

diag : default = `N

abr : default = 1

abc : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val trtri : ?n:int ->
       ?up:bool ->
       ?diag:Lacaml__common.diag -> ?ar:int -> ?ac:int -> mat -> unit

trtri ?n ?up ?diag ?ar ?ac a computes the inverse of a real upper or lower triangular matrix a.

Raises Failure if the matrix a is singular.

n : default = number of columns in matrix a

up : default = true (upper triangle stored in a)

diag : default = `N

ar : default = 1

ac : default = 1

val geqrf_opt_lwork : ?m:int -> ?n:int -> ?ar:int -> ?ac:int -> mat -> int

geqrf_opt_lwork ?m ?n ?ar ?ac a

Returns the optimum length of the work-array used by the Lacaml__C.geqrf-function given matrix a and optionally its logical dimensions m and n.

m : default = number of rows in matrix a

n : default = number of columns in matrix a

ar : default = 1

ac : default = 1

val geqrf_min_lwork : n:int -> int

geqrf_min_lwork ~n

Returns the minimum length of the work-array used by the Lacaml__C.geqrf-function if the matrix has n columns.

val geqrf : ?m:int ->
       ?n:int ->
       ?work:vec ->
       ?tau:vec -> ?ar:int -> ?ac:int -> mat -> vec

geqrf ?m ?n ?work ?tau ?ar ?ac a computes a QR factorization of a real m-by-n matrix a. See LAPACK documentation.

Returns tau, the scalar factors of the elementary reflectors.

m : default = number of rows in matrix a

n : default = number of columns in matrix a

work : default = vec of optimum length

tau : default = vec of required length

ar : default = 1

ac : default = 1

Linear equations (simple drivers)

val gesv : ?n:int ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

gesv ?n ?ipiv ?ar ?ac a ?nrhs ?br ?bc b computes the solution to a real system of linear equations a * X = b, where a is an n-by-n matrix and X and b are n-by-nrhs matrices. The LU decomposition with partial pivoting and row interchanges is used to factor a as a = P * L * U, where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of a is then used to solve the system of equations a * X = b. On exit, b contains the solution matrix X.

Raises Failure if the matrix a is singular.

n : default = available number of columns in matrix a

ipiv : default = vec of length n

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val gbsv : ?n:int ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?abr:int ->
       ?abc:int ->
       mat ->
       int -> int -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

gbsv ?n ?ipiv ?abr ?abc ab kl ku ?nrhs ?br ?bc b computes the solution to a real system of linear equations a * X = b, where a is a band matrix of order n with kl subdiagonals and ku superdiagonals, and X and b are n-by-nrhs matrices. The LU decomposition with partial pivoting and row interchanges is used to factor a as a = L * U, where L is a product of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular with kl+ku superdiagonals. The factored form of a is then used to solve the system of equations a * X = b.

Raises Failure if the matrix a is singular.

n : default = available number of columns in matrix ab

ipiv : default = vec of length n

abr : default = 1

abc : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val gtsv : ?n:int ->
       ?ofsdl:int ->
       vec ->
       ?ofsd:int ->
       vec ->
       ?ofsdu:int ->
       vec -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

gtsv ?n ?ofsdl dl ?ofsd d ?ofsdu du ?nrhs ?br ?bc b solves the equation a * X = b where a is an n-by-n tridiagonal matrix, by Gaussian elimination with partial pivoting. Note that the equation A'*X = b may be solved by interchanging the order of the arguments du and dl.

Raises Failure if the matrix is singular.

n : default = available length of vector d

ofsdl : default = 1

ofsd : default = 1

ofsdu : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val posv : ?n:int ->
       ?up:bool ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

posv ?n ?up ?ar ?ac a ?nrhs ?br ?bc b computes the solution to a real system of linear equations a * X = b, where a is an n-by-n symmetric positive definite matrix and X and b are n-by-nrhs matrices. The Cholesky decomposition is used to factor a as a = U**T * U, if up = true, or a = L * L**T, if up = false, where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of a is then used to solve the system of equations a * X = b.

Raises Failure if the matrix is singular.

n : default = available number of columns in matrix a

up : default = true i.e., upper triangle of a is stored

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val ppsv : ?n:int ->
       ?up:bool ->
       ?ofsap:int ->
       vec -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

ppsv ?n ?up ?ofsap ap ?nrhs ?br ?bc b computes the solution to the real system of linear equations a * X = b, where a is an n-by-n symmetric positive definite matrix stored in packed format and X and b are n-by-nrhs matrices. The Cholesky decomposition is used to factor a as a = U**T * U, if up = true, or a = L * L**T, if up = false, where U is an upper triangular matrix and L is a lower triangular matrix. The factored form of a is then used to solve the system of equations a * X = b.

Raises Failure if the matrix is singular.

n : default = the greater n s.t. n(n+1)/2 <= Vec.dim ap

up : default = true i.e., upper triangle of ap is stored

ofsap : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val pbsv : ?n:int ->
       ?up:bool ->
       ?kd:int ->
       ?abr:int ->
       ?abc:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

pbsv ?n ?up ?kd ?abr ?abc ab ?nrhs ?br ?bc b computes the solution to a real system of linear equations a * X = b, where a is an n-by-n symmetric positive definite band matrix and X and b are n-by-nrhs matrices. The Cholesky decomposition is used to factor a as a = U**T * U, if up = true, or a = L * L**T, if up = false, where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as a. The factored form of a is then used to solve the system of equations a * X = b.

Raises Failure if the matrix is singular.

n : default = available number of columns in matrix ab

up : default = true i.e., upper triangle of ab is stored

kd : default = available number of rows in matrix ab - 1

abr : default = 1

abc : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val ptsv : ?n:int ->
       ?ofsd:int ->
       vec ->
       ?ofse:int ->
       vec -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

ptsv ?n ?ofsd d ?ofse e ?nrhs ?br ?bc b computes the solution to the real system of linear equations a*X = b, where a is an n-by-n symmetric positive definite tridiagonal matrix, and X and b are n-by-nrhs matrices. A is factored as a = L*D*L**T, and the factored form of a is then used to solve the system of equations.

Raises Failure if the matrix is singular.

n : default = available length of vector d

ofsd : default = 1

ofse : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val sysv_opt_lwork : ?n:int ->
       ?up:bool ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> int

sysv_opt_lwork ?n ?up ?ar ?ac a ?nrhs ?br ?bc b

Returns the optimum length of the work-array used by the sysv-function given matrix a, optionally its logical dimension n and given right hand side matrix b with an optional number nrhs of vectors.

n : default = available number of columns in matrix a

up : default = true i.e., upper triangle of a is stored

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val sysv : ?n:int ->
       ?up:bool ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?work:vec ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

sysv ?n ?up ?ipiv ?work ?ar ?ac a ?nrhs ?br ?bc b computes the solution to a real system of linear equations a * X = b, where a is an N-by-N symmetric matrix and X and b are n-by-nrhs matrices. The diagonal pivoting method is used to factor a as a = U * D * U**T, if up = true, or a = L * D * L**T, if up = false, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of a is then used to solve the system of equations a * X = b.

Raises Failure if the matrix is singular.

n : default = available number of columns in matrix a

up : default = true i.e., upper triangle of a is stored

ipiv : default = vec of length n

work : default = vec of optimum length (-> sysv_opt_lwork)

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val spsv : ?n:int ->
       ?up:bool ->
       ?ipiv:Lacaml__common.int32_vec ->
       ?ofsap:int ->
       vec -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

spsv ?n ?up ?ipiv ?ofsap ap ?nrhs ?br ?bc b computes the solution to the real system of linear equations a * X = b, where a is an n-by-n symmetric matrix stored in packed format and X and b are n-by-nrhs matrices. The diagonal pivoting method is used to factor a as a = U * D * U**T, if up = true, or a = L * D * L**T, if up = false, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The factored form of a is then used to solve the system of equations a * X = b.

Raises Failure if the matrix is singular.

n : default = the greater n s.t. n(n+1)/2 <= Vec.dim ap

up : default = true i.e., upper triangle of ap is stored

ipiv : default = vec of length n

ofsap : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

Least squares (simple drivers)

val gels_min_lwork : m:int -> n:int -> nrhs:int -> int

gels_min_lwork ~m ~n ~nrhs

Returns the minimum length of the work-array used by the gels-function if the logical dimensions of the matrix are m rows and n columns and if there are nrhs right hand side vectors.

val gels_opt_lwork : ?m:int ->
       ?n:int ->
       ?trans:Lacaml__common.trans2 ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> int

gels_opt_lwork ?m ?n ?trans ?ar ?ac a ?nrhs ?br ?bc b

Returns the optimum length of the work-array used by the gels-function given matrix a, optionally its logical dimensions m and n and given right hand side matrix b with an optional number nrhs of vectors.

m : default = available number of rows in matrix a

n : default = available number of columns in matrix a

trans : default = `N

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1

val gels : ?m:int ->
       ?n:int ->
       ?work:vec ->
       ?trans:Lacaml__common.trans2 ->
       ?ar:int ->
       ?ac:int ->
       mat -> ?nrhs:int -> ?br:int -> ?bc:int -> mat -> unit

gels ?m ?n ?work ?trans ?ar ?ac a ?nrhs ?br ?bc b see LAPACK documentation!

m : default = available number of rows in matrix a

n : default = available number of columns of matrix a

work : default = vec of optimum length (-> Lacaml__C.gels_opt_lwork)

trans : default = `N

ar : default = 1

ac : default = 1

nrhs : default = available number of columns in matrix b

br : default = 1

bc : default = 1