BLASjs (Basic Linear Algebra Subprograms)

This is a 100% Pure Javascript ( TypeScript ) re-write of the reference implementation Basic Linear Algebra SubPrograms (BLAS) numerical library found here. This is a full manual re-write, "emscripten" was not used.

summary

BLASjs contains all the functions (Complex, Real) of the reference implementation capable for 32 bit and 64 bit floating point arithmatic:

👌 100% code coverage
1005 tests
Output off all tests equal to the BLAS FORTRAN reference implementation.
Level 1: all vector-vector operations implemented.
Level 2: all vector-matrix operations implemented.
Level 3: all matrix-matrix operations implemented.
Helper functions to ease the porting of FORTRAN BLAS usage to Javascript.

Node and Web

The resulting bundled blasjs file is an agnostic UMD library, it can be used in a web client as-well as in a server side node environment.

Installation

node

$ npm i blasjs

Usage:

//node
   const blas = require('blasjs');
//or typescript
   import * as blas from 'blasjs';

web

The module directory contains a standalone bundle for use in html <script> insertion. The library assigns window.BLAS after loading.

<!-- <script src="your_server_url/blasjs.min.js"></script> -->
<!-- this example uses unpkg as CDN -->
<script src="https://unpkg.com/blasjs@latest/dist/lib/blasjs.min.js"></script>
<script>
  const blas = window.BLAS; //UMD exposes it as BLAS

  //fetch some level3 complex 64 bit precision matrix-matrix operations
  const {
      level3: { zsyrk, ztrmm, ztrsm }
   } = blas;
</script>

Language differences with FORTRAN/BLAS

FORTRAN language can instrinsicly work with non-zero based multidimensional arrays and complex numbers. Below are some examples from FORTRAN that have no Javascript counterpart. The reference implementation of BLAS functions expect inputs of these types.

The FORTRAN complex scalar, complex array and complex "Matrix"

!    double precision Complex number
     COMPLEX*16 alpha
!
!    double precision Complex array with offset 2
     COMPLEX*16 vector(2,10)
!
!    double precision complex MultiDimensional Array (matrix)
!    rows 1 to 5 , columns 1 to 10
     COMPLEX*16 A(1:5,1:10)

To work with the concept of non-zero based arrays and complex numbers in JS, these FORTRAN constructs have equivalents in the blasjs library.

The blasjs helpers to create complex scalar, complex array and complex "Matrix" objects

  const blas = require('blasjs');

  const {
      helper:{
        /* create complex Object from 2 real numbers */
        complex,

        /* create single precision Real/complex arrays, */
        fortranArrComplex32,

        /* create double precision Real/Complex arrays */
        fortranArrComplex64,

        /* create single precision 2 dimensional Real/Complex arrays */
        fortranMatrixComplex32,

        /* Double precision 2 dimensional Real/Complex arrays */
        fortranMatrixComplex64,
      }
  } = blas;

These functions are extensively documented in the helper functions. It is recommended you read this introductory part of the documentation first. before anything else.

Helper functions

blasjs uses "FORTRAN like" complex number 32/64 bit precision multidimensional complex/real data. These helper functions have been designed to significantly ease the use of working with these data types in JavaScript.

Types

Typescript types/interfaces to mimic FORTRAN native (complex) multidimensional arrays.

`fpArray`

Wraps JS types Float32Array and Float64Array into a single type.

Details (click to show)

decl:

export type fpArray = Float32Array | Float64Array;

`FortranArr`

Abstraction of a 1 dimensional single/double precision complex/real FORTRAN array. Used by level 1 and level 2 blasjs functions. FortranArrobjects should be created by the fortranArrComplex32 and fortranArrComplex64 helper functions.

Details (click to show)

decl:

export declare type FortranArr = {
    base: number;
    r: fpArray;
    i?: fpArray;
    s: (index: number) => (re?: number, im?: number) => number | Complex;
    toArr: () => Complex[] | number[];
};

fields:

base: fortran by default has a 1-value based array. Mimiced by this property.
r: See decl fpArray. The Real part of complex array.
i: (optional). See decl fpArray. The Imaginary part of the complex array.
s: set, get values of the array. Uses FORTRAN style array indexes taking the value of base into account.
toArr generates an JavaScript array from the r and i (optional) data.

Usage:

const blas = require('blasjs');

const { helper: { fortranArrComplex64 } } = blas;

// You can also use the helper "complex" or "muxComplex"
// to generate JS complex arrays
const complexDataArr = [
    { re: 1.8, im: -0.2 },
    { re: 2.3, im: 0.6 }
];

// Create an object that mimics FORTRAN COMPLEX*16 SP(2:3)
//    and fill it with above data
const sp = fortranArrComplex64(complexArr)(2);

// fast! normal JS TypedArray access
let re = sp.r[ 2 - sp.base ];
// 1.8

let  im = sp.i[ 2 - sp.base ];
// -0.2

// not so fast, but easier syntax
let v = sp.s(2)(); // Terse syntax,
// { re: 1.8, im: -0.2 }

// sets the value at index 3 to complex: 0.11 - i0.9
//      and returns the old value: 2.3 + i0.6
let old = sp.s(3)(0.11, -0.9);

sp.toArr();
// [ { re:1.8, im: -0.2 },
//   { re:0.11, im: -0.9 } ]

Usage TypeScript:

import {
    // pure types
    Complex,
    fpArray,
    FortranArr,
    // helper
    helper
} from 'blasjs';

const { fortranArrComplex64 } = helper;

const complexArr: Complex[] [
    { re: 1.8, im: -0.2 },
    { re: 2.3, im: 0.6 }
];

// Create an object that mimics FORTRAN COMPLEX*16 SP(2:3)
//    and fill it with above data
const sp: FortranArr = fortranArrComplex64(complexArr)(2);

let re = sp.r[ 2 - sp.base ]; //fastest! direct TypedArray access
// 1.8

let  im = sp.i[ 2 - sp.base ]; //fastest! direct TypedArray access
// -0.2

// not so fast, but easier syntax
let v = sp.s(2)(); // Terse syntax,
// { re: 1.8, im: -0.2 }

// sets the value at index 3 to complex: 0.11 - i0.9
//      and returns the old value: 2.3 + i0.6
let old = sp.s(3)(0.11, -0.9);
// {re: 2.3, im: 0.6 }

`Type Complex`

Typescript definition of a complex scalar.

Details (click to show)

decl:

declare type Complex = {
   re: number;
   im?: number;
}

Usage:

import { Complex /* pure type */ } from 'blasjs';

const complexArr: Complex[] [
    { re: 1.8, im: -0.2 },
    { re: 2.3, im: 0.6 }
];

`Matrix`

The Matrix object is the input of many level-2 and level-3 blasjs functions. Matrix is created by the helpers fortranMatrixComplex32 and fortranMatrixComplex64. Matrix encapsulates objects of Float32Array or Float64Array, the blasjs.

In this section the internals of Matrix are explained in detail and how blasjs accesses the data in the JS TypesArrays.

Float[32/64]Array Complex number storage for Matrix.

The Matrix object has 2 properties r and i for respectively real and imaginary parts of matrix elements. These are the actual aforementioned JS TypedArrays. The imaginary property part is optional if it is not defined the Matrix represents solely an array of real elements.

Details (click to show)

declare type Matrix = { //Incomplete declaration
    .
    r: Float64Array|Float32Array;
    i: Float64Array|Float32Array;
    .
}

Handling FORTRAN matrices (multidimensional Arrays).

Contrary to languages like JavaScript. FORTRAN defines arrays ( aka DIMENSIONS in FORTRAN lingo ) as 1 based arrays by default.. This can be changed by specifying a different base in the declaration.

Details (click to show)

Some examples:

       DOUBLE PRECISION A1(4)  ! array indexes 1,2,3,4
       DOUBLE PRECISION A2(-1:3)  ! array indexes -1,0,2,3
       DOUBLE PRECISION A3(0:3) ! Javascript like Array with 4 elements

This expands to 2-dimensional arrays (matrices).

! (default) first index loops from 1 to 4(inclusive), second index loops from 1 to 5(inclusive)
       DOUBLE PRECISION A1(4,5)
! first index loops from -2 to 4(inclusive), second index loops from -5 to -7(inclusive)
       DOUBLE PRECISION A2(-2:4,-5:-7)

The values of the FORTRAN array basis are preserved as rowBase (first index) and colBase (second index).

declare type Matrix = { //SHOW PARTIAL TYPE
    .
    rowBase: number;
    colBase: number;
    .
}

JavaScript doesn't have the notion of typed 2-dimensional arrays. The Matrix objects handles this by mapping 2 dimensional arrays to single 1-dimensional array, by serializing data on a column-first basis.

For example the elements 2x2 Matrix will be mapped in a TypedArray as:

matrix A =
 *           *
 | a11  a12  |
 | a21  a22  |
 *           *

# Stored in TypedArray as
At = [a11,a21, a12, a22]

In case of complex values for A, the real part will be stored in r and the imaginary part in i each in the same column-first manner.

Performance

Direct access to TypedArrays within the Matrix object is the preferable way to get/set matrix data. Since BLAS (and therefore blasjs) functions access matrices mostly to iterate over matrix row's first . It was decided to story 2 dimensional an a column-first basis.

To help with the calculation of finding/setting an element A(i,j) in Matrix the following helper member functions have been added to Matrix.

Details (click to show)

declare type Matrix = { //SHOW PARTIAL TYPE
    .
    rowBase: number;
    colBase: number;
    nrCols: number;
    nrRows: number;
    .
    colOfEx(number): number;
    coord(col): (row) => number;
    setCol(col: number, rowStart: number, rowEnd: number, value: number): void;
    .
}

Explanation:

nrRows: The number of rows in the matrix.
nrCols: The number of columns in the matrix.
colofEx: Calculates the physical location of a column offset within the TypedArray. Taking int account the column base colBase and row base colBase. The index of A(i,j) = (j - colBase)*nrRows + i - rowBase.
coord: Curried, emulates non-zero based FORTRAN index values for 2 dimensional Arrays. The index that is iterated over the least (usually) is used as the first to create the curried function.
setCol: Uses underlying TypedArray, fill method to set multiple column elements to a single value.

See Example

Creating new transformed Matrix instances from existing ones

One can create/transform new Matrix instances form existing onces. A copy of all relevant data is made into the new Matrix instance.

`Matrix.prototype.slice`

Slices a rectangular piece of data out of an matrix into a new Matrix instance. All arguments are FORTRAN-style non-zero based indexes.

Details (click to show)

declare type Matrix = { // only "slice" is shown
   .
   slice(rowStart: number, rowEnd: number, colStart: number, colEnd: number): Matrix;
   .
}

rowStart: The row in the matrix to begin slicing.
rowEnd: The last row to include in the slice.
colStart: The column in the matrix to begin slicing.
colEnd: The last column to include in the slice.

See Example

`Matrix.prototype.setLower`

Returns a new Matrix where everything below the matrix diagonal is set to a value. Sets the real (and imaginary part, if it exist) to said value.

Details (click to show)

declare type Matrix = { // only "setLower" is shown.
  .
  setLower(value = 0): Matrix;
  .
}

See Example

`Matrix.prototype.setUpper`

Returns a new Matrix where everything below the matrix diagonal is set to a value. Sets the real (and imaginary part, if it exist) to said value.

Details (click to show)

declare type Matrix = { //only "setUpper" is shown
  .
  setUpper(value = 0): Matrix;
  .
}

See Example

`Matrix.prototype.upperBand`

Returns a new Matrix object where the k super-diagonals are retained into the new copy. The efficient storage format of BLAS band matrices is used.

Details (click to show)

declare type Matrix = { //only "upperBand" is shown
  .
  upperBand(k = nrRows - 1): Matrix;
  .
}

The default value for k is the the maximum size possible for the number of super-diagonals: ( nrRows-1 )

See Example

`Matrix.prototype.lowerBand`

Returns a new Matrix object where the k sub-diagonals are retained into the new copy. The efficient storage format of BLAS band matrices is used.

Details (click to show)

declare type Matrix = { // Only "lowerBand" is shown
  .
  lowerBand(k = nrRows-1): Matrix;
  .
}

The default value for k is the the maximum size possible for the number of sub-diagonals: ( nrRows-1 )

See Example

`Matrix.prototype.real`

Returns a new Matrix object where with only real elements (omits the imaginary part during copy).

Details (click to show)

declare type Matrix = { // Only "real" is shown
  .
  real(): Matrix;
  .
}

See Example

`Matrix.prototype.imaginary`

Returns a new Matrix object where with only imaginary part of the element (omits the real part during copy). If there were now imaginary elements

Details (click to show)

declare type Matrix = { // Only "imaginary" is shown.
  .
  imaginary(): Matrix;
  .
}

See Example

Packed Matrices

BLAS ( and therefore blasjs ) can work with upper/lower-matrices and band-matrices in the most compacted form, aka packed matrices. With packed matrices there are no unused elements in the matrix (no zeros). Packed matrices are instances of FortranArr. BLAS reference implementation in FORTRAN uses 1 dimensional arrays as an analog.

`Matrix.prototype.packedUpper`

Creates a packed array from a normal/upper Matrix only referencing the diagonal and super-diagonals.

Details (click to show)

declare type Matrix = { // Only "packedUpper" is shown.
  .
  packedUpper(k = nrRows-1): FortranArr;
  .
}

See Example

The default value for k is the the maximum size possible for the number of super-diagonals: ( nrRows-1 )

`Matrix.prototype.packedLower`

Creates a packed array from a normal/upper Matrix only referencing the diagonal and sub-diagonals.

Details (click to show)

declare type Matrix = { // Only "packedUpper" is shown.
  .
  packedLower(k = nrRows-1): FortranArr;
  .
}

See Example

The default value for k is the the maximum size possible for the number of sub-diagonals: ( nrRows-1 )

declare type Matrix = { // Only "packedUpper" is shown.
  .
  packedLower(k = nrRows - 1): FortranArr;
  .
}

The default value for k is the the maximum size possible for the number of sub-diagonals: ( nrRows - 1 )

Convert Matrix object to a JS array

The Matrix object can convert the underlying TypedArray(s) to real JavaScript arrays.

`Matrix.prototype.toArr`

Creates a normal JS Array with element of type 'number' or of type Complex

Details (click to show)

declare type Matrix = { // Only "toArr" is shown.
  .
  toArr(): number[]|Complex[];
  .
}

See Example

Summary: Full type declaration of Matrix

Putting it all together, here is the full type declaration of Matrix:

declare type Matrix = {
     rowBase: number;
     colBase: number;
     nrCols: number;
     nrRows: number;
     r: fpArray;
     i?: fpArray; //optional
     //
     // methods
     //
     colOfEx(column: number): void;
     coord(col: number): (row: number): void;
     setCol(col: number, rowStart: number, rowEnd: number, value: number): void;
     //
     slice(rowStart: number, rowEnd: number, colStart: number, colEnd: number): Matrix;
     setLower(value?: number): Matrix;
     setUpper(value?: number): Matrix;
     upperBand(k: number): Matrix;
     lowerBand(k: number): Matrix;
     real(): Matrix;
     imaginary(): Matrix;
     //
     packedUpper(value?: number): FortranArr;
     packedLower(value?: number): FortranArr;
     //
     toArr(): Complex[] | number[];
}

Matrix Examples

Common usage of the Matrix type.

Details (click to show)

const blas = require('../blasjs');
const { fortranMatrixComplex64 } = blas.helper;

// some matrix data 3x3 array  aka a_row_column

const a11 = { re: .2, im: -.11 };
const a21 = { re: .1, im: -.2 };
const a31 = { re: .3, im: .9 };
const a12 = { re: .4, im: .5 };
const a22 = { re: .9, im: -.34 };
const a32 = { re: -.2, im: .45 };
const a13 = { re: -.1, im: .89 };
const a23 = { re: .43, im: .23 };
const a33 = { re: .23, im: .56 };

//create Matrix A
const A = fortranMatrixComplex64([
    a11, a21, a31, a12, a22, a32, a13, a23, a33
])(3, 3);

// get the second column
const columnj = A.colOfEx(3); // formula: (j - colBase )* nrRows

A.r[A.coord(1, 2)] === a12.re // true

A.slice(1, 2,  2, 3);// creates new matrix with elements from A
/*[
    a12 a13
    a22 a23
]*/

A.setLower(0); // creates new Matrix object from A
/*[
    a11 a12 a13
    0   a22 a23
    0   0   a33
]*/

A.setUpper(0); //creates new Matrix object from A
/*[
    a11 0   0
    a21 a22 0
    a31 a32 a33
]*/

A.upperBand(1); // banded array storage for BLAS(js)
/*[
    0   a12   a23
    a11 a22   a33
]*/

A.lowerBand(1); // banded array storage for BLAS(js)
/*[
   a11  a22  a33
   a21  a32  0    
]*/

const Areal = A.real();
// Areal.i is undefined
// Areal.r =
/*[
    0.2 0.4  -0.1
    0.1 0.9   0.43
    0.3 -0.2, 0.23
]*/

const Aimag = A.imaginary();
// imaginary parts are copied to real side in new Matrix
// Aimag.i is undefined
// Aimag.r =
/*[
    -0.11   0.5,  0.89
     -0.2  -0.34  0.23
      0.9   0.45  0.56
]*/

A.packedUpper(1)
/* [ a11 a12 a22 a23 a 33] */

A.packedLower(1)
/* [ a11 a21 a22 a32 a33] */

A.toArr(); // returns JavaScript Array
/*[
  { re: 0.2, im: -0.11 },
  { re: 0.1, im: -0.2 },
  { re: 0.3, im: 0.9 },
  { re: 0.4, im: 0.5 },
  { re: 0.9, im: -0.34 },
  { re: -0.2, im: 0.45 },
  { re: -0.1, im: 0.89 },
  { re: 0.43, im: 0.23 },
  { re: 0.23, im: 0.56 }
]
*/

General Helpers

Collection of helper function to manipulate common JS array and object types in a functional way.

`arrayrify`

Creates a new function from an existing one, to add the ability to accept vectorized input.

Details (click to show)

Example:

const blas = require('blasjs');

const { helper: { arrayrify } } = blas;
const PI = Math.PI;
//
const sin = arrayrify(Math.sin)

sin([PI/3, PI/4, PI/6]); // returns array aswell
// [ 0.866025, 0.7071067811, 0.5 ]

sin(PI/3); // returns scalar
sin( [ PI/3 ] ); // returns scalar
// 0.866025

sin([]) // edge case
// undefined

sin() //
//NaN  same as Math.sin()

`complex`

Mimics the GNU Fortran extension complex. Creates a JS object that represents a complex scalar number. Used by blasjs for scalar input arguments.

Details (click to show)

Example:

const blas = require('blasjs');

const { helper: { complex } } = blas;

const c1 = complex(0.1,0.3);
//c1 = { re: 0.1, im: 0.3 }

const c2 = complex();
//c2 = { re: 0, im: 0 }

const c3 = complex(0.5);
//c3 = { re: 0.5, im:0 }

`each`

Curried functional analog to Array.prototype.forEach, but takes arbitrary input.

Details (click to show)

Example:

const blas = require('blasjs');

const { helper: { each } } = blas;

//Iterates over an object like a map
const curry1 = each( {  hello: 'world', ts: new Date() })
curry1( (val, key) => console.log(`${val} ':'  ${key}`)))
//world: hello
//2018-05-10T13:57:08.923Z : ts

//Handles array also
each( ['a','b','c','d'])( (v,idx) =>console.log(v,idx, typeof idx))
//a 0 number
//b 1 number
//c 2 number
//d 3 number

//Edge cases
each()(console.log)
//nothing happens

each(null)(console.log)
//nothing happens

each([])(console.log)
//nothing happens

`map`

Curried functional analog to Array.prototype.map, but takes arbitrary input.

⚠️ Forces the output to be a an array regardless of the input.

Example (click to show)

Example:

const blas = require('blasjs');

const { helper: { map } } = blas;

//trivial
map([1,2,3])(v=>v*2);
//[ 2, 4, 6 ]

//key properties
map({ a:'A', b:'B' })( (val, key) => key+'='+val);
//[ 'a=A', 'b=B' ]

map(null)( v => '/'+v);
//[]

map()( v => '/'+v);
//[]

map()()
//[]

`muxCmplx`

Creates an array of complex numbers from arrayed input. The result is always an array type.

Example (click to show)

Example:

const blas = require('blasjs');

const { helper: { muxCmplx } } = blas;

const reals = [ 0.1, -0.2, 0.3, 0.45 ];
const imaginary = [ 0.1, -0.2, 0.3, 0.45 ];

// normal usage
muxCmplx(reals, imaginary)
/*[ { re: 0.1, im: 0.1 },
    { re: -0.2, im: -0.2 },
    { re: 0.3, im: 0.3 },
    { re: 0.45, im: 0.45 } ]*/

//R recycling rule is used
muxCmplx([1,2], imaginary)
/*^[ { re: 1, im: 0.1 },
     { re: 2, im: -0.2 },
     { re: 1, im: 0.3 },
     { re: 2, im: 0.45 } ]*/

//dont care about imaginary
muxCmplx(reals)
/*[ { re: 0.1, im: undefined },
    { re: -0.2, im: undefined },
    { re: 0.3, im: undefined },
    { re: 0.45, im: undefined } ]*/

muxCmplx() //
// [ { re: undefined, im: undefined } ]

muxCmplx(1) //
// [ { re: 1, im: undefined } ]

//3 specify real and imaginary
muxCmplx(1,-2)//
//[ { re: 1, im: -2 } ]

`numberPrecision`

Enforces significant figure of a number, or on the properties of a JS object (deep search) with numeric values.

Example (click to show)

Example:

const blas = require('blasjs');

const { helper: { numberPrecision } } = blas;

const _4 = numberPrecision(4);

_4(0.123456789);
//0.1235

_4(123456789)
//123500000

//enforce significance over properties
_4( { car: 'Mazda' , aux: { priceUSD: 24.3253E+3, maxWarpSpeed:3.42111E-4 } } );
//{ car: 'Mazda', aux: { priceUSD: 24330, maxWarpSpeed: 0.0003421 } }

_4([0.123456, 0.78901234]);
//[ 0.1235, 0.789 ]

Vector Constructors

These constructors create the FortranArr object for working with single/double precision complex/real Arrays.

`fortranArrComplex32`

Constructs a FortranArr object using Float32Array as the underlying array(s) (plural in the case of complex) elements.

Details (click to show)

declare function fortranArrComplex32(
    ...rest: (number | number[] | Complex | Complex[])[]
    ): (offset = 1) => FortranArr;

Argument list:

rest: takes as input.
- A single numeric value.
- A single Complex object.
- An array of Complex objects.
- An array of number values.
offset: the Fortran dimension offset (defaults to 1)

See Examples

`fortranArrComplex64`

Constructs a FortranArr object using Float64Array as the underlying array(s) (plural in the case of complex) elements.

Details (click to show)

declare function fortranArrComplex64(
    ...rest: (number | number[] | Complex | Complex[])[]
    ): (offset = 1) => FortranArr;

Argument list:

rest: takes as input.
- A single numeric value.
- A single Complex object.
- An array of Complex objects.
- An array of number values.
offset: the Fortran dimension offset (defaults to 1)

Vector creation examples

Example (click to show)

const blas = require('blasjs');

const { fortranArrComplex64, fortranArrComplex32 } = blas.helper;

const complexDataArr = [
    { re: 1.8, im: -0.2 },
    { re: 2.3, im: 0.6 }
];

const realData = [ 0.1, 2, 0.34, .56 ];

const sp1 = fortranArrComplex32(complexDataArr)();
//sp1.r = [ 1.7999999523162842, 2.299999952316284 ],
//sp1.i = [ -0.20000000298023224, 0.6000000238418579 ],

const sp2 = fortranArrComplex32(realData)();
//sp2.r =  [ 0.10000000149011612, 2, 0.3400000035762787, 0.5600000023841858 ]
//sp2.i = undefined

const sp3 = fortranArrComplex32({re:0.2, im:-0.3})();
//[ 0.20000000298023224 ]
//[ -0.30000001192092896 ]

const sp4 = fortranArrComplex32(123)(4);
/*{
  base: 4,
  r: Float32Array [ 123 ],
  i: undefined,
}*/

const sdp1 = fortranArrComplex64(complexDataArr)();
//sp1.r = [ 1.8, 2.3 ],
//sp1.i = [ -0.2, 0.6 ],

const sdp2 = fortranArrComplex64(realData)();
//sp2.r =  [ 0.1, 2, 0.34, 0.56 ]
//sp2.i = undefined

const sp3 = fortranArrComplex64({re:0.2, im:-0.3})();
//[ 0.2 ]
//[ -0.3 ]

const sp4 = fortranArrComplex64(123)(4);
/*{
  base: 4,
  r: Float32Array [ 123 ],
  i: undefined,
}*/

Matrix Constructors

These constructors create the Matrix object for working with single/double precision complex/real Matrices.

`fortranMatrixComplex32`

Constructs a Matrix object using Float32Array as the underlying array(s) (plural in the case of complex) elements.

Details (click to show)

declare function fortranMatrixComplex32(...rest: (Complex | Complex[])[]):
    (nrRows: number, nrCols: number, rowBase?: number, colBase?: number) => Matrix

Argument list:

rest: takes as input.
- A single numeric value.
- A single Complex object.
- An array of Complex objects.
- An array of number values.
nrRows: where nrRows is equal to n in the matrix A(m,n).
nrCols: where nrCols is equal to m in the matrix A(m,n).
rowBase: FORTRAN offset for the first dimension (rows) as explained in Language differences.
rowBase: FORTRAN offset for the second dimension (columns) as explained in Language differences.

See Examples

`fortranMatrixComplex64`

Constructs a Matrix object using Float64Array as the underlying array(s) (plural in the case of complex) elements.

Details (click to show)

declare function fortranMatrixComplex64(...rest: (Complex | Complex[])[]):
    (nrRows: number, nrCols: number, rowBase?: number, colBase?: number) => Matrix

Argument list:

rest: takes as input.
- A single numeric value.
- A single Complex object.
- An array of Complex objects.
- An array of number values.
nrRows: where rnRows is equal to n in the matrix A(m,n).
nrCols: where nrCols is equal to m in the matrix A(m,n).
rowBase: FORTRAN offset for the first dimension (rows) as explained in Language differences.
rowBase: FORTRAN offset for the second dimension (columns) as explained in Language differences.

Matrix Creation Examples

Details (click to show)

const blas = require('blasjs');
const {
    fortranMatrixComplex64,
    fortranMatrixComplex32
} = blas.helper;

// some matrix data 3x3 array  aka a_row_column

const a11 = { re: .2, im: -.11 };
const a21 = { re: .1, im: -.2 };
const a31 = { re: .3, im: .9 };
const a12 = { re: .4, im: .5 };
const a22 = { re: .9, im: -.34 };
const a32 = { re: -.2, im: .45 };
const a13 = { re: -.1, im: .89 };
const a23 = { re: .43, im: .23 };
const a33 = { re: .23, im: .56 };

const {
    fortranMatrixComplex64,
    fortranMatrixComplex32
} = blas.helper;

// Some matrix data 3x3 array  aka a_row_column

const a11 = { re: .2, im: -.11 };
const a21 = { re: .1, im: -.2 };
const a31 = { re: .3, im: .9 };
const a12 = { re: .4, im: .5 };
const a22 = { re: .9, im: -.34 };
const a32 = { re: -.2, im: .45 };
const a13 = { re: -.1, im: .89 };
const a23 = { re: .43, im: .23 };
const a33 = { re: .23, im: .56 };

//functional curry to prepare for different mappings of A()
const A32 = fortranMatrixComplex64([
    a11, a21, a31, a12, a22, a32, a13, a23, a33
]);

//matrix 1
const m1 = A32(3, 3); // 3x3 matrix with rowBase=1, colBase=1

// mimic FORTRAN  "COMPLEX*8  A(-2:1, -3:0)"
const m2 = A32(3, 3, -2, -3);

//same as FORTRAN default COMPLEX*8 A(3,3) !aka A(1:3,1:3)
const m3 = A32(3, 3, 1, 1)

/* double precision */
/* double precision */
/* double precision */

const A64 = fortranMatrixComplex64([
    a11, a21, a31, a12, a22, a32, a13, a23, a33
]);

// matrix 1 FORTRAN "COMPLEX*16  A(-2:1, -3:0).
const m1 = A64(3, 3); // 3x3 matrix with rowBase=1, colBase=1

// mimic FORTRAN  "COMPLEX*16  A(-2:1, -3:0)"
const m2 = A64(3, 3, -2, -3);

// same as FORTRAN default COMPLEX*16 A(3,3) !aka A(1:3,1:3)
const m3 = A64(3, 3, 1, 1);

A note on numeric precision

In blasjs, contrary to the FORTRAN reference implementation, the numeric precision of a routine, is not determined by its name but by how its arguments like FortranArr and Matrix are constructed before used as arguments in blasjs routines. The original FORTRAN names are kept for backwards compatibility to ease the porting of FORTRAN code toward blasjs.

Mimicking FORTRAN OUT Arguments

In FORTRAN a subroutine can have IN, OUT and IN/OUT scalar arguments. In JavaScript only arguments of type object are passed by reference. To mimic OUT and IN/OUT FORTRAN arguments, scalars are wrapped in a JS object. See Construct a Givens plane rotation for an example.

Level 1 routines

Routines categorized as Level 1 perform scalar-vector and vector-vector operations.

Euclidean norm: √(xᴴ·x) or √(xᵀ·x)

Calculates the norm of a (complex) vector.

xᴴ is the conjugate of x

xᵀ is the transpose of x

scrnm2/dznrm2, snrm2/dnrm2

scrnm2: complex, single or double precision. See blas ref.
dznrm2: complex, (alias for scrnm2). See blas ref.
snrm2: real, single or double precision. See blas ref.
dnrm2: real, (alias for dnrm2). See blas ref.

Details (click to show)

decl

function scnrm2(n: number, x: FortranArr, incx: number): number;
function dznrm2(n: number, x: FortranArr, incx: number): number;
function snrm2(n: number, x: FortranArr, incx: number): number;
function dnrm2(n: number, x: FortranArr, incx: number): number;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { scnrm2, dznrm2, snrm2, dnrm2 } = BLAS.level1;

Construct a Givens plane rotation

See wiki.

$$|c -s| × |a| = |r | |s c| |b| |0 | r = √( a² + b² )$$

srotg/drotg, crotg/zrotg

srotg: real, (alias for drotg). See blas ref.
drotg: real, single or double precision. See blas ref.
crotg: complex, single or double precision. See blas ref.
zrotg: complex, (alias for crotg). See blas ref.

Details (click to show)

decl

function srotg(p: { sa: number, sb: number, c: number, s: number } ): void;
function drotg(p: { sa: number, sb: number, c: number, s: number } ): void;
function crotg(ca: Complex, cb: Complex, c: { val: number }, s: Complex ): void
function zrotg(ca: Complex, cb: Complex, c: { val: number }, s: Complex ): void

Usage:

const BLAS = require('blasjs');
const { srotg, drotg, crotg, zrotg } = BLAS.level1;

Construct the modified Givens rotation matrix `H`

Construct the modified Givens transformation matrix H which zeros the second component of the 2 vector ( sx1√(sd1) , sy1 √(sd2) ) See researchgate.net.

srotmg/drotmg

srotmg: real, (alias for drotmg). See blas ref.
drotmg: real, single or double precision. See blas ref.

Details (click to show)

decl

function srotmg(p: { sd1: number, sd2: number, sx1: number, sy1: number, sparam: FortranArr }): void;
function drotmg(p: { sd1: number, sd2: number, sx1: number, sy1: number, sparam: FortranArr }): void;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { srotmg, drotmg } = BLAS.level1;

Apply the modified Givens Transformation

See wiki.

srotm/drotm

srotm: real, (alias for drotm). See blas ref.
drotm: real, single or double precision. See blas ref.

Details (click to show)

decl

function srotm(n: number, sy: FortranArr, incx: number, sy: FortranArr, incy: number, sparam: FortranArr)): void;

function drotm(n: number, sy: FortranArr, incx: number, sy: FortranArr, incy: number, sparam: FortranArr)): void;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { srotm, drotm } = BLAS.level1;

Applies a plane rotation

See researchgate.net.

srot/drot, csrot/zdrot

srot: real, (alias for drot). See blas ref.
drot: real, single or double precision. See blas ref.
csrot: complex, (alias for zdrot). See blas ref.
zdrot: complex, single or double precision. See blas ref.

Details (click to show)

decl

function srot(n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number, c: number, s: number): void;
function drot(n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number, c: number, s: number): void;

function csrot: (n: number, cx: FortranArr, incx: number, cy: FortranArr, incy: number, c: number, s: number): void;
function zdrot: (n: number, cx: FortranArr, incx: number, cy: FortranArr, incy: number, c: number, s: number): void;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { srot, drot, csrot, zdrot } = BLAS.level1;

Scale a vector by a constant

x ⟵ α·x

sscal/dscal, cscal/zscal, csscal/zdscal

sscal: Alias for dscal. See blas ref.
dscal: by a REAL constant. See blas ref.
cscal: Alias for zscal. See blas ref.
zscal: Scales a COMPLEX vector with a COMPLEX constant. See blas ref.
csscal: Alias for zdscal. blas ref.
zdscal: Scales a COMPLEX vector with a REAL constant. See blas ref.

Details (click to show)

decl

function sscal(n: number, sa: number, sx: FortranArr, incx: number): void;
function dscal(n: number, sa: number, sx: FortranArr, incx: number): void;
function cscal(n: number, ca: Complex,cx: FortranArr, incx: number): void;
function zscal(n: number, ca: Complex,cx: FortranArr, incx: number): void;
function csscal(n: number, sa: number, cx: FortranArr, incx: number): void;
function zdscal(n: number, sa: number, cx: FortranArr, incx: number): void;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { sscal, dscal, cscal, zscal, csscal, zdscal } = BLAS.level1;

Takes the sum of the absolute values of the components of vector

s ⟵ ∑ ∥ Re( x ) ∥ + ∥ Im( x ) ∥

sasum/dasum, scasum/dzasum

sasum: Alias for dasum. See blas ref
dasum: uses REAL vector, ( single or double precision ). See blas-ref.
scasum: Alias for dzasum. See blas ref.
dzasum: uses Complex vector, ( single or double precision ). See blas-ref.

Details (click to show)

decl

function sasum(n: number, sx: FortranArr, incx: number): number;
function dasum(n: number, sx: FortranArr, incx: number): number;
function scasum(n: number, cx: FortranArr, incx: number): number;
function dzasum(n: number, cx: FortranArr, incx: number): number;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { sasum, dasum, scasum, dzasum } = BLAS.level1;

Interchanges 2 vectors

Swap 2 vectors.

sswap/dswap, cswap/zswap

sswap: Alias for dswap. See blas ref.
dswap: REAL vector, ( single or double precision ). See blas ref.
cswap: Alias for zswap. See blas ref.
zswap: REAL vector, ( single or double precision ). See blas ref.

Details (click to show)

decl

function sswap(n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number ): void;
function dswap(n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number ): void;
function cswap(n: number, cx: FortranArr, incx: number, cy: FortranArr, incy: number ): void;
function zswap(n: number, cx: FortranArr, incx: number, cy: FortranArr, incy: number ): void;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { sswap, dswap, cswap, zswap } = BLAS.level1;

Dot product of two complex vectors

xᵀ·y or xᴴ·y

cdotu/cdotc, zdotu/zdotc

cdotu: Alias for zdotu. See blas ref.
cdotc: Alias for zdotc. See blas ref.
zdotu: xᵀ·y. Complex arguments, ( single or double precision ). See blas-ref.
zdotc: xᴴ·y. The fist complex vector argument is made conjugate, ( single or double precision ). See blas-ref.

Details (click to show)

decl

function cdotu(n: number, cx: FortranArr, incx: number, cy: FortranArr, incy: number): Complex;

// first argument sx is made conjugate
function cdotc(n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number ): Complex;

function zdotu(n: number, cx: FortranArr, incx: number, cy: FortranArr, incy: number ): Complex;

// first argument sx is made conjugate
function zdotc(n: number, cx: FortranArr, incx: number, cy: FortranArr, incy: number ): Complex;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { cdotu, cdotc, zdotu, zdotc } = BLAS.level1;

Dot product of two non complex vectors

xᵀ·y

sdot/ddot, sdsdot/dsdot

sdot: Alias for dsdot. See blas ref.
ddot: Alias for dsdot. See blas ref.
sdsdot: Alias for dsdot. See blas ref.
dsdot: xᵀ·y Inner product of 2 vectors ( single or double precision ). See blas ref.

Details (click to show)

decl

function sdot(n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number): number;
function ddot(n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number): number;
function sdsdot(n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number): number;
function dsdot(n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number): number;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { sdot, ddot, sdsdot, dsdot } = BLAS.level1;

Finds the index of the first element having maximum absolut value.

Find k for wich: ∥ xₖ ∥ > ∥ xₜ ∥ for all t ∈ [1, n].

isamax/idamax, icamax/izamax

isamax: Alias for idamax. See [blas ref]:ref-isamax
idamax: Find the index of the maximum element of a REAL vector ( single or double precision ). See blas ref.
icamax: Alias for izamax. See [blas ref]:ref-icamax
izamax: Find the index of the maximum element of a COMPLEX vector ( single or double precision ). See blas ref.

Details (click to show)

decl

function isamax: (n: number, sx: FortranArr, incx: number): number;
function idamax: (n: number, sx: FortranArr, incx: number): number;
function icamax: (n: number, sx: FortranArr, incx: number): number;
function izamax: (n: number, sx: FortranArr, incx: number): number;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { isamax, idamax, icamax, izamax } = BLAS.level1;

Copy a vector x to a vector y

scopy/dcopy, ccopy/zcopy

scopy: Alias for dcopy. See [blas ref]:ref-scopy
dcopy: Copies a REAL vector ( single or double precision ). See blas ref.
ccopy: Alias for zcopy. See [blas ref]:ref-ccopy
zcopy: Copies a COMPLEX vector ( single or double precision ). See blas ref.

Details (click to show)

decl

function scopy (n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number): void;
function dcopy (n: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number): void;
function ccopy (n: number, cx: FortranArr, incx: number, cy: FortranArr, incy: number): void;
function zcopy (n: number, cx: FortranArr, incx: number, cy: FortranArr, incy: number): void;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { scopy, dcopy, ccopy, zcopy } = BLAS.level1;

Constant times a vector plus a vector

y ⟵ y + a·x where y, a and x can be complex or a real number.

saxpy/daxpy, caxpy/zaxpy

saxpy: Alias for daxpy. See [blas ref]:[ref-saxpy].
daxpy: REAL constant used in multiplication with a vector ( single or double precision ). See [blas ref]:ref-daxpy.
caxpy: Alias for zaxpy. See [blas ref]:[ref-saxpy].
zaxpy: Complex constant used in multiplication with a vector ( single or double precision ). See [blas ref]:ref-zaxpy.

Details (click to show)

decl

function saxpy(n: number, sa: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number): void;
function daxpy(n: number, sa: number, sx: FortranArr, incx: number, sy: FortranArr, incy: number): void;
function caxpy(n: number, ca: Complex, cx: FortranArr, incx: number, cy: FortranArr, incy: number): void;
function zaxpy(n: number, ca: Complex, cx: FortranArr, incx: number, cy: FortranArr, incy: number): void;

See: how to create fortranArr.

Usage:

const BLAS = require('blasjs');
const { saxpy, daxpy, caxpy, zaxpy } = BLAS.level1;

Level 2 Routines

Routines categorized as Level 2 perform Matrix-vector operations.

The hermitian rank 2 operation A ⟵ α·x·yᴴ + conjg( α )·y·xᴴ + A

( ᴴ means conjugate transpose )

For the routines chpr2 and zhpr2 the matrix A is in packed form ( a fortranArr ).

For the routines cher2 and zher2 the matrix symmetry is exploited (use only upper/lower triangular part of the matrix).

cher2/zher2, chpr2|zhpr2

cher2: alias for zher2. See blas ref.
zher2: The Matrix A is in upper or lower triangular form ( single or double precision ). See blas ref.
chpr2: alias for zhpr2. See blas ref.
zhpr2: The matrix A is in packed form ( single or double precision ). See blas ref.

Details (click to show)

decl

function cher2|zher2(
     uplo: "u" | "l", 
     n: number, 
     alpha: Complex, 
     x: FortranArr, 
     incx: number, 
     y: FortranArr, 
     incy: number, 
     a: Matrix, 
     lda: number): void;

function chpr2|zhpr2(
     uplo: "u" | "l", 
     n: number, 
     alpha: Complex, 
     x: FortranArr, 
     incx: number, 
     y: FortranArr, 
     incy: number, 
     ap: FortranArr): void;

See: how to create fortranArr.

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { cher2, zher2, chpr, zhpr } = BLAS.level2;

The symmetric rank 2 operation A ⟵ α·x·yᵀ + α·y·xᵀ + A

For the routines sspr2 and dspr2 the matrix A is in packed form ( a fortranArr ).

For the routines ssyr2 and dsyr2 the matrix symmetry is exploited (use only upper/lower triangular part of the matrix).

sspr2/dspr2, ssyr2/dsyr2

sspr2: Alias for dspr2. See blas ref.
dspr2: The matrix A is in packed form ( single or double precision ). See blas ref.
ssyr2: Alias for dsyr2. See blas ref.
dsyr2: The Matrix A is in upper or lower triangular form ( single or double precision ). See blas ref.

Details (click to show)

decl

function sspr2|dspr2(
    uplo: "u" | "l", 
    n: number, 
    alpha: number, 
    x: FortranArr, 
    incx: number, 
    y: FortranArr, 
    incy: number, 
    ap: FortranArr):void;

function ssyr2|dsyr2(
    uplo: 'u' | 'l', 
    n: number, 
    alpha: number, 
    x: FortranArr, 
    incx: number, 
    y: FortranArr, 
    incy: number, 
    A: Matrix, 
    lda: number): void;

See: how to create fortranArr.

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { sspr2, dspr2, ssyr2, dsyr2 } = BLAS.level2;

The rank 1 operation A ⟵ α·x·yᴴ + A or A ⟵ α·x·yᵀ + A

( ᴴ means conjugate transpose )

The subroutines sger and dger perform A ⟵ α·x·yᵀ + A. Where α is a REAL scalar, A, x, y are single or double precision REAL Matrix and vectors.

The subroutines cgerc and zgerc perform A ⟵ α·x·yᴴ + A. Where α is a COMPLEX scalar, A, x, y are single or double precision COMPLEX Matrix and vectors.

The subroutines cgeru and zgeru perform A ⟵ α·x·yᵀ + A. Where α is a COMPLEX scalar, A, x, y are single or double precision COMPLEX Matrix and vectors.

sger/dger, cgerc/zgerc, cgeru/zgeru

sger: alias for dger. See blas ref.
dger: See blas ref.
cgerc: alias for zgerc. See blas ref.
zgerc: See blas ref.
cgeru: alias for zgeru. See blas ref.
zgeru: See blas ref.

Details (click to show)

decl

function sger|dger(
    m: number, 
    n: number, 
    alpha: number, 
    x: FortranArr, 
    incx: number, 
    y: FortranArr, 
    incy: number, 
    a: Matrix, 
    lda: number):void;

function cgerc|zgerc(
    m: number, 
    n: number, 
    alpha: Complex, 
    x: FortranArr, 
    incx: number, 
    y: FortranArr, 
    incy: number, 
    a: Matrix, 
    lda: number): void;

function cgeru|zgeru(
    m: number, 
    n: number, 
    alpha: Complex, 
    x: FortranArr, 
    incx: number, 
    y: FortranArr, 
    incy: number, 
    a: Matrix, 
    lda: number): void;

See: how to create fortranArr.

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { sger, dger, cgerc, zgerc, cgeru, zgeru } = BLAS.level2;

The hermitian rank 1 operation A ⟵ α·x·xᴴ + A

( ᴴ means conjugate transpose )

For the routines cher and zher α is a REAL scalar, the matrix symmetry of A is exploited (use only upper/lower triangular part of the matrix).

For the routines chpr and zhpr α is a REAL scalar, the matrix A is in packed form ( a fortranArr ).

Naming

cher: alias for zher. See blas ref.
zher: For single or double precision complex x and A. See blas ref.
chpr: alias for zher. See blas ref.
zhpr: For single or double precision complex x and A. See blas ref.

Details (click to show)

function cher(
    uplo: "u" | "l", 
    n: number, 
    alpha: number, 
    x: FortranArr, 
    incx: number, 
    a: Matrix, 
    lda: number): void;

function zher(
    uplo: "u" | "l", 
    n: number, 
    alpha: number, 
    x: FortranArr, 
    incx: number, 
    a: Matrix, 
    lda: number): void;

function chpr(u
    uplo: "u" | "l", 
    n: number, 
    alpha: number, 
    x: FortranArr, 
    incx: number, 
    ap: FortranArr): void;

function zhpr(
    uplo: "u" | "l", 
    n: number, 
    alpha: number, 
    x: FortranArr, 
    incx: number, 
    ap: FortranArr): void;

See: how to create fortranArr.

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { cher, zher, chpr, zhpr } = BLAS.level2;

The symmetric rank 1 operation A ⟵ α·x·xᵀ + A

For the routines ssyr and dsyr α is a REAL scalar, the symmetry of the REAL matrix A is exploited (use only upper/lower triangular part of the matrix).

For the routines sspr and dspr α is a REAL scalar, the REAL matrix A is in packed form ( a fortranArr ).

sspr/dspr, ssyr/dsyr

sspr: alias for dspr. See blas ref.
dspr: For single or double precision REAL α , x and A. See blas ref.
ssyr: alias for ssyr. See blas ref.
dsyr: For single or double precision REAL α , x and A. See blas ref.

Details (click to show)

decl

function sspr(
   uplo: "u" | "l", 
   n: number, 
   alpha: number, 
   x: FortranArr, 
   incx: number, 
   ap: FortranArr): void;

function dspr(
   uplo: "u" | "l", 
   n: number, 
   alpha: number, 
   x: FortranArr, 
   incx: number, 
   ap: FortranArr): void;

function ssyr(
    uplo: "u" | "l", 
    n: number, 
    alpha: number, 
    x: FortranArr, 
    incx: number, 
    a: Matrix, 
    lda: number): void;

function dsyr(
    uplo: "u" | "l", 
    n: number, 
    alpha: number, 
    x: FortranArr, 
    incx: number, 
    a: Matrix, 
    lda: number): void;

See: how to create fortranArr.

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { sspr, dspr, ssyr, dsyr } = BLAS.level2;

The matrix-vector operation, y ⟵ α·A·x + β·y, or y ⟵ α·Aᵀ·x + β·y or y ⟵ α·Aᴴ·x + β·y

Aᴴ is the complex conjugate transpose of matrix A

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

cgbmv/zgbmv, chbmv/zhbmv, ssbmv/dsbmv, sgbmv/dgbmv, stbmv/dtbmv, chemv/zhemv, sgemv/dgemv, cgemv/zgemv, chpmv/zhpmv, sspmv/dspmv, ssymv/dsymv

subroutine	operation	complex	real	type of matrix A	blas ref link
cgbmv/zgbmv	y ⟵ α·A·x + β·y , y ⟵ α·Aᵀ·x + β·y, y ⟵ α·Aᴴ·x + β·y	α, A, β	none	upper/lower band	cgbmv/zgbmv
chbmv/zhbmv	y ⟵ α·A·x + β·y	α, A, β	none	upper/lower band	chbmv/zhbmv
ssbmv/dsbmv	y ⟵ α·A·x + β·y	none	α, A, β	upper/lower band	chbmv/zhbmv
sgbmv/dgbmv	y ⟵ α·A·x + β·y , y ⟵ α·Aᵀ·x + β·y	none	α, A, β	upper/lower band	sgbmv/dgbmv
stbmv/dtbmv	y ⟵ α·A·x + β·y	none	α, A, β	upper/lower band	stbmv/dtbmv
chemv/zhemv	y ⟵ α·A·x + β·y	α, A, β	none	triangular upper/lower	chemv/zhemv
sgemv/dgemv	y ⟵ α·A·x + β·y , y ⟵ α·Aᵀ·x + β·y	none	α, A, β	full m x n	sgemv/dgemv
cgemv/zgemv	y ⟵ α·A·x + β·y , y ⟵ α·Aᵀ·x + β·y, y ⟵ α·Aᴴ·x + β·y	α, A, β	none	full m x n	cgemv/zgemv
chpmv/zhpmv	y ⟵ α·A·x + β·y	α, A, β	none	packed upper/lower triangular	cgemv/zgemv
sspmv/dspmv	y ⟵ α·A·x + β·y	none	α, A, β	packed upper/lower triangular	sspmv/dspmv
ssymv/dsymv	y ⟵ α·A·x + β·y	α, A, β	none	upper/lower triangular	ssymv/dsymv

Details (click to show)

decl

function cgbmv|zgbmv(
    trans: 'n' | 't' | 'c',
    m: number,
    n: number,
    kl: number,
    ku: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number,
    beta: Complex,
    y: FortranArr,
    incy: number
): void;

function chbmv|zhbmv(
    uplo: 'u' | 'l',
    n: number,
    k: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number,
    beta: Complex,
    y: FortranArr,
    incy: number
): void;

export function ssbmv|dsbmv(
    uplo: string,
    n: number,
    k: number,
    alpha: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number,
    beta: number,
    y: FortranArr,
    incy: number
): void;

function sgbmv|dgbmv(
    trans: string,
    m: number,
    n: number,
    kl: number,
    ku: number,
    alpha: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number,
    beta: number,
    y: FortranArr,
    incy: number
): void;

function stbmv | dtbmv(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    n: number,
    k: number,
    A: Matrix,
    lda: number,
    x: FortranArr,
    incx: number
): void; 

function chemv|zhemv(
    uplo: 'u' | 'l',
    n: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number,
    beta: Complex,
    y: FortranArr,
    incy: number
): void 

function sgemv|dgemv(
    trans: string,
    m: number,
    n: number,
    alpha: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number,
    beta: number,
    y: FortranArr,
    incy: number
): void;

function cgemv|zgemv(
    trans: 'n' | 't' | 'c',
    m: number,
    n: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number,
    beta: Complex,
    y: FortranArr,
    incy: number
): void;

function chpmv|zhpmv(
    uplo: 'u' | 'l',
    n: number,
    alpha: Complex,
    ap: FortranArr,
    x: FortranArr,
    incx: number,
    beta: Complex,
    y: FortranArr,
    incy: number
): void;

function sspmv|dspmv(
    uplo: 'u' | 'l',
    n: number,
    alpha: number,
    ap: FortranArr, // a symmetric matrix in packed form
    x: FortranArr,
    incx: number,
    beta: number,
    y: FortranArr,
    incy: number
): void

function ssymv|dsymv(
    uplo: 'u' | 'l',
    n: number,
    alpha: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number,
    beta: number,
    y: FortranArr,
    incy: number
): void

See: how to create fortranArr.

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');

const { 
 cgbmv, chbmv, dgbmv, dsbmv, sgbmv, ssbmv, zgbmv, zhbmv,
 cgemv, chemv, dgemv, sgemv, zgemv, zhemv,
 chpmv, dspmv, sspmv, zhpmv, dsymv, ssymv } = BLAS.level2;

The matrix-vector operation, x ⟵ A·x, or x ⟵ Aᵀ·x, or x ⟵ Aᴴ·x

Aᴴ is the complex conjugate transpose of matrix A

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

stbmv, dtbmv, ctbmv, dtpmv, ctpmv, ztpmv, strmv, dtrmv, ctrmv, ztrmv

subroutine	operation	complex	real	type of matrix A	blas ref link
stbmv/dtbmv	x ⟵ A·x, or x ⟵ Aᵀ·x	none	A, x	upper/lower band	stbmv/dtbmv
ctbmv/ztbmv	x ⟵ A·x, or x ⟵ Aᵀ·x, or x ⟵ Aᴴ·x	A, x	none	upper/lower band	ctbmv/ztbmv
stpmv/dtpmv	x ⟵ A·x, or x ⟵ Aᵀ·x	none	A, x	upper/lower triangular packed	stpmv/dtpmv
ctpmv/ztpmv	x ⟵ A·x, or x ⟵ Aᵀ·x, or x ⟵ Aᴴ·x	A, x	none	upper/lower triangular packed	ctpmv/ztpmv
strmv/dtrmv	x ⟵ A·x, or x ⟵ Aᵀ·x	none	A, x	upper/lower triangular	strmv/dtrmv
ctrmv/ztrmv	x ⟵ A·x, or x ⟵ Aᵀ·x, or x ⟵ Aᴴ·x	A, x	none	upper/lower triangular	ctrmv/ztrmv

Details (click to show)

decl

function stbmv|dtbmv(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    n: number,
    k: number,
    A: Matrix,
    lda: number,
    x: FortranArr,
    incx: number
): void;

function ctbmv|ztbmv(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    n: number,
    k: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number
): void;

function stpmv|zhbmv (
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    n: number,
    ap: FortranArr,
    x: FortranArr,
    incx: number
): void; 

function ctpmv|ztpmv(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    n: number,
    ap: FortranArr,
    x: FortranArr,
    incx: number
): void;

function strmv|dtrmv(
    uplo: 'u' | 'l',
    trans: 't' | 'c' | 'n',
    diag: 'u' | 'n',
    n: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number
): void;

function ctrmv|ztrmv(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    n: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number
): void;

See: how to create fortranArr.

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');

const { 
 stbmv, dtbmv, ctbmv, ztbmv, stpmv, dtpmv, ctpmv, ztpmv, strmv
 dtrmv, ctrmv, ztrmv } = BLAS.level2;

Solves a systems of equations A·x = b, or Aᵀ·x = b, or Aᴴ·x = b

Aᴴ is the complex conjugate transpose of matrix A

stbsv, dtbsv, ctbsv, ztbsv, stpsv, dtpsv, ctpsv, ztpsv, ctrsv, ztrsv, strs, dtrsv

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

subroutine	operation	complex	real	type of matrix A	blas ref link
stbsv/dtbsv	A·x = b, or Aᵀ·x = b	none	A, b, x	upper/lower band	stbsv/dtbsv
ctbsv/ztbsv	A·x = b, or Aᵀ·x = b, or Aᴴ·x = b	A, b, x	none	upper/lower band	ctbsv/ztbsv
stpsv/dtpsv	A·x = b, or Aᵀ·x = b	none	A, b, x	packed upper/lower triangular	stpsv/dtpsv
ctpsv/ztpsv	A·x = b, or Aᵀ·x = b, or Aᴴ·x = b	A, b, x	none	packed upper/lower triangular	ctpsv/ztpsv
ctrsv/ztrsv	A·x = b, or Aᵀ·x = b, or Aᴴ·x = b	A, b, x	none	upper/lower triangular	ctrsv/ztrsv
strsv/dtrsv	A·x = b, or Aᵀ·x = b	none	A, b, x	upper/lower triangular	strsv/dtrsv

Details (click to show)

decl

function stbsv|dtbsv(
    uplo: 'u' | 'l',
    trans: 't' | 'n' | 'c',
    diag: 'u' | 'n',
    n: number,
    k: number,
    A: Matrix,
    lda: number,
    x: FortranArr,
    incx: number
): void;

function ctbsv|ztbsv(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    n: number,
    k: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number
): void;
 
function stpsv|dtpsv(
    uplo: 'u' | 'l',
    trans: 't' | 'n' | 'c',
    diag: 'u' | 'n',
    n: number,
    ap: FortranArr,
    x: FortranArr,
    incx: number
): void;

function ctpsv|ztpsv(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    n: number,
    ap: FortranArr,
    x: FortranArr,
    incx: number
): void 

function ctrsv|ztrsv(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    n: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number
): void 

function strsv|dtrsv(
    uplo: 'u' | 'l',
    trans: 't' | 'c' | 'n',
    diag: 'u' | 'n',
    n: number,
    a: Matrix,
    lda: number,
    x: FortranArr,
    incx: number
): void

See: how to create fortranArr.

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');

const { 
    stbsv, dtbsv, ctbsv, ztbsv, stpsv, 
    dtpsv, ctpsv, ztpsv, ctrsv, ztrsv, 
    strsv, dtrsv } = BLAS.level2;

Level 3 Routines

Routines categorized as Level 2 perform Matrix-vector operations.

Hermitian rank 2k: C ⟵ α·A·Bᴴ + con( α )·B·Aᴴ + β·C or C ⟵ α·Aᴴ·B + con( α )·Bᴴ·A + β·C

con( α ) is the conjugate of α.

Aᴴ is the conjugate transpose of Matrix A.

Bᴴ isthe conjugate transpose of Matrix B.

cher2k, zher2k

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

subroutine	operation	complex	real	type of matrix C	blas ref link
cher2k/zher2k	C ⟵ α·A·Bᴴ + con( α )·B·Aᴴ + β·C or C ⟵ α·Aᴴ·B + con( α )·Bᴴ·A + β·C	α, A, B, C	β	upper/lower triangular	cher2k/zher2k

Details (click to show)

decl

function cher2k | zher2k(
    uplo: 'u' | 'l',
    trans: 'n' | 'c',
    n: number,
    k: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number,
    beta: number,
    c: Matrix,
    ldc: number
): void;

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { cher2k,  zher2k } = BLAS.level3;

Symmetric rank 2k operations C ⟵ α·A·Bᵀ + α·B·Aᵀ + β·C, or C ⟵ α·Aᵀ·B + α·Bᵀ·A + β·C

ssyr2k, dsyr2k, csyr2k, zsyr2k

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

subroutine	operation	complex	real	type of matrix C	blas ref link
ssyr2k/dsyr2k	C ⟵ α·A·Bᵀ + α·B·Aᵀ + β·C, or C ⟵ α·Aᵀ·B + α·Bᵀ·A + β·C	none	α, A, β, B, C	upper/lower triangular	cher2k/zher2k
csyr2k/zsyr2k	C ⟵ α·A·Bᵀ + α·B·Aᵀ + β·C, or C ⟵ α·Aᵀ·B + α·Bᵀ·A + β·C	α, A, β, B, C	none	upper/lower triangular	csyr2k/zsyr2k

Details (click to show)

decl

function ssyr2k|dsyr2k(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    n: number,
    k: number,
    alpha: number,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number,
    beta: number,
    c: Matrix,
    ldc: number
): void;

function csyr2k|zsyr2k(
    uplo: 'u' | 'l',
    trans: 'n' | 't',
    n: number,
    k: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number,
    beta: Complex,
    c: Matrix,
    ldc: number
): void

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { ssyr2k, dsyr2k, csyr2k, zsyr2k } = BLAS.level3;

Hermatian rank k operations C ⟵ α·A·Aᴴ + β·C, or C ⟵ α·Aᴴ·A + β·C

Aᴴ is the conjugate transpose of Matrix A.

cherk, zherk

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

subroutine	operation	complex	real	type of matrix C	blas ref link
cherk/zherk	C ⟵ α·A·Aᴴ + β·C, or C ⟵ α·Aᴴ·A + β·C	A, C	α, β	upper/lower triangular	cherk/zherk

Details (click to show)

decl

function cherk|zherk(
    uplo: 'u' | 'l',
    trans: 'n' | 'c',
    n: number,
    k: number,
    alpha: number,
    a: Matrix,
    lda: number,
    beta: number,
    c: Matrix,
    ldc: number
): void;

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { cherk, zherk } = BLAS.level3;

Symmetric rank k operations C ⟵ α·A·Aᵀ + β·C, or C ⟵ α·Aᵀ·A + β·C

ssyrk, dsyrk, csyrk, zsyrk

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

subroutine	operation	complex	real	type of matrix C	blas ref link
ssyrk/dsyrk	C ⟵ α·A·Aᵀ + β·C, or C ⟵ α·Aᵀ·A + β·C	none	α, A, β, C	upper/lower triangular	ssyrk/dsyrk
csyrk/zsyrk	C ⟵ α·A·Aᵀ + β·C, or C ⟵ α·Aᵀ·A + β·C	α, A, β, C	none	upper/lower triangular	csyrk/zsyrk

Details (click to show)

decl

function ssyrk|dsyrk(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    n: number,
    k: number,
    alpha: number,
    a: Matrix,
    lda: number,
    beta: number,
    c: Matrix,
    ldc: number
): void;

function csyrk|zsyrk(
    uplo: 'u' | 'l',
    trans: 'n' | 't' | 'c',
    n: number,
    k: number,
    alpha: number,
    a: Matrix,
    lda: number,
    beta: number,
    c: Matrix,
    ldc: number
): void;

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { ssyrk, dsyrk, csyrk, zsyrk } = BLAS.level3;

Matrix-matrix operations C ⟵ α·f(A)·h(B) + β·C or C ⟵ α·h(B)·f(A) + β·C

f(A) is the result of an operation on matrix A, like Aᵀ, Aᴴ, or A (no-op)

h(B) is an operation on matrix B, like Bᵀ, Bᴴ, or B (no-op)

S(A) is the set of all possible results of f(A) for a routine.

S(B) is the set of all possible results of h(B) for a routine.

sgemm, dgemm, cgemm, zgemm

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

subroutine	operation	S(A)	S(B)	real	complex	type of matrix C	blas ref link
sgemm/dgemm	C ⟵ α·f(A)·h(B) + β·C	Aᵀ, A	Bᵀ, B	α, A, β, B, C	none	m x n	sgemm/dgemm
cgemm/zgemm	C ⟵ α·f(A)·h(B) + β·C	Aᴴ, Aᵀ, A	Bᴴ, Bᵀ, B	none	α, A, β, B, C	m x n	cgemm/zgemm

Details (click to show)

decl

function sgemm|dgemm(
    transA: 'n' | 't' | 'c',
    transB: 'n' | 't' | 'c',
    m: number,
    n: number,
    k: number,
    alpha: number,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number,
    beta: number,
    c: Matrix,
    ldc: number
): void;

function cgemm|zgemm(
    transA: 'n' | 't' | 'c',
    transB: 'n' | 't' | 'c',
    m: number,
    n: number,
    k: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number,
    beta: Complex,
    c: Matrix,
    ldc: number
): void

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { sgemm, dgemm, cgemm, zgemm } = BLAS.level3;

Matrix-matrix operations C ⟵ α·A·B + β·C or C ⟵ α·B·A + β·C

chemm, zhemm, ssymm, dsymm, csymm, zsymm

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

subroutine	operation	real	complex	type of matrix C	blas ref link
chemm/zhemm	C ⟵ α·A·B + β·C or C ⟵ α·B·A + β·C	none	α, A, B, β, C	m x n	chemm/zhemm
ssymm/dsymm	C ⟵ α·A·B + β·C or C ⟵ α·B·A + β·C	α, A, B, β, C	none	m x n	ssymm/dsymm
csymm/zsymm	C ⟵ α·A·B + β·C or C ⟵ α·B·A + β·C	none	α, A, B, β, C	m x n	csymm/zsymm

Details (click to show)

decl

function chemm|zhemm(
    side: 'l' | 'r',
    uplo: 'u' | 'l',
    m: number,
    n: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number,
    beta: Complex,
    c: Matrix,
    ldc: number
): void;

function ssymm|dsymm(
    side: 'l' | 'r',
    uplo: 'u' | 'l',
    m: number,
    n: number,
    alpha: number,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number,
    beta: number,
    c: Matrix,
    ldc: number
): void

function csymm|zsymm(
    side: 'l' | 'r',
    uplo: 'u' | 'l',
    m: number,
    n: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number,
    beta: Complex,
    c: Matrix,
    ldc: number
): void

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { chemm, zhemm,  ssymm, dsymm, csymm, zsymm } = BLAS.level3;

Matrix-matrix operations B ⟵ α·f(A)·B or B ⟵ α·B·f(A)

f(A) is the result of an operation on matrix A, like Aᵀ, Aᴴ, or A (no-op)

S(A) is the set of all possible results of f(A) for a routine.

strmm, dtrmm, ctrmm, ztrmm

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

subroutine	operation	S(A)	real	complex	type of matrix B	blas ref link
strmm/dtrmm	B ⟵ α·f(A)·B or B ⟵ α·B·f(A)	A, Aᵀ	α, B	none	m x n	strmm/dtrmm
ctrmm/ztrmm	B ⟵ α·f(A)·B or B ⟵ α·B·f(A)	A, Aᵀ, Aᴴ	none	α, A, B	m x n	ctrmm/ztrmm

Details (click to show)

decl

function strmm|dtrmm(
    side: 'l' | 'r',
    uplo: 'u' | 'l',
    transA: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    m: number,
    n: number,
    alpha: number,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number
): void;

function ctrmm|ztrmm(
    side: 'l' | 'r',
    uplo: 'u' | 'l',
    transA: 'n' | 't' | 'c',
    diag: 'u' | 'n',
    m: number,
    n: number,
    alpha: Complex,
    a: Matrix,
    lda: number,
    b: Matrix,
    ldb: number
): void;

See: how to create Matrix.

Usage:

const BLAS = require('blasjs');
const { strmm, dtrmm,  ctrmm, ztrmm } = BLAS.level3;

Solves the matrix equations: f( A )·X = α·B, or X·f( A ) = α·B

f(A) is the result of an operation on matrix A, like Aᵀ, Aᴴ, or A (no-op)

S(A) is the set of all possible results of f(A) for a routine.

strsm, dtrsm, ctrsm, ztrsm

The naming in blasjs does not reflect the precision used, precision is determined by argument construction. The naming is maintained for compatibility with the reference implementation.

subroutine	operation	S(A)	real	complex	type of matrix B	blas ref link
strsm/dtrsm	f( A )·X = α·B, or X·f( A ) = α·B	A, Aᵀ	α, A, B	none	m x n	strsm/dtrsm
ctrsm/ztrsm	f( A )·X = α·B, or X·f( A ) = α·B	A, Aᵀ, Aᴴ	none	α, A, B	m x n	ctrsm/ztrsm

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BLASjs (Basic Linear Algebra Subprograms)

summary

Node and Web

Installation

node

web

Table of Contents

Language differences with FORTRAN/BLAS

Helper functions

Types

fpArray

FortranArr

Type Complex

Matrix

Float[32/64]Array Complex number storage for Matrix.

Handling FORTRAN matrices (multidimensional Arrays).

Performance

Creating new transformed Matrix instances from existing ones

Matrix.prototype.slice

Matrix.prototype.setLower

Matrix.prototype.setUpper

Matrix.prototype.upperBand

Matrix.prototype.lowerBand

Matrix.prototype.real

Matrix.prototype.imaginary

Packed Matrices

Matrix.prototype.packedUpper

Matrix.prototype.packedLower

Convert Matrix object to a JS array

Matrix.prototype.toArr

Summary: Full type declaration of Matrix

Matrix Examples

General Helpers

arrayrify

complex

each

map

muxCmplx

numberPrecision

Vector Constructors

fortranArrComplex32

fortranArrComplex64

Vector creation examples

Matrix Constructors

fortranMatrixComplex32

fortranMatrixComplex64

Matrix Creation Examples

A note on numeric precision

Mimicking FORTRAN OUT Arguments

Level 1 routines

Euclidean norm: √(xᴴ·x) or √(xᵀ·x)

scrnm2/dznrm2, snrm2/dnrm2

Construct a Givens plane rotation

srotg/drotg, crotg/zrotg

Construct the modified Givens rotation matrix H

srotmg/drotmg

Apply the modified Givens Transformation

srotm/drotm

Applies a plane rotation

srot/drot, csrot/zdrot

Scale a vector by a constant

sscal/dscal, cscal/zscal, csscal/zdscal

Takes the sum of the absolute values of the components of vector

sasum/dasum, scasum/dzasum

Interchanges 2 vectors

sswap/dswap, cswap/zswap

Dot product of two complex vectors

cdotu/cdotc, zdotu/zdotc

Dot product of two non complex vectors

sdot/ddot, sdsdot/dsdot

Finds the index of the first element having maximum absolut value.

isamax/idamax, icamax/izamax

Copy a vector x to a vector y

scopy/dcopy, ccopy/zcopy

`fpArray`

`FortranArr`

`Type Complex`

`Matrix`

`Matrix.prototype.slice`

`Matrix.prototype.setLower`

`Matrix.prototype.setUpper`

`Matrix.prototype.upperBand`

`Matrix.prototype.lowerBand`

`Matrix.prototype.real`

`Matrix.prototype.imaginary`

`Matrix.prototype.packedUpper`

`Matrix.prototype.packedLower`

`Matrix.prototype.toArr`

`arrayrify`

`complex`

`each`

`map`

`muxCmplx`

`numberPrecision`

`fortranArrComplex32`

`fortranArrComplex64`

`fortranMatrixComplex32`

`fortranMatrixComplex64`

Construct the modified Givens rotation matrix `H`