Mathematical methods 

Mathematical methods play an important role in the solution of technical problems in all fields of technology. However, they are excluded from patentability under Art. 52(2)(a) when claimed as such (Art. 52(3)).

The exclusion applies if a claim is directed to a purely abstract mathematical method and the claim does not require any technical means. For instance, a method for performing a Fast Fourier Transform on abstract data which does not specify the use of any technical means is a mathematical method as such. A purely abstract mathematical object or concept, e.g. a particular type of geometric object or of graph with nodes and edges, is not a method but is nevertheless not an invention within the meaning of Art. 52(1) because it lacks a technical character.

If a claim is directed either to a method involving the use of technical means (e.g. a computer) or to a device, its subject-matter has a technical character as a whole and is thus not excluded from patentability under Art. 52(2) and (3).

Merely specifying the technical nature of the data or parameters of the mathematical method may not be sufficient on its own to define an invention within the meaning of Art. 52(1). Even if the resulting method would not be considered a purely abstract mathematical method as such within the meaning of Art. 52(2)(a) and (3), it may still fall under the excluded category of methods for performing mental acts as such if no use of technical means is implied (Art. 52(2)(c) and (3); see G‑II, 3.5.1).

Once it is established that the claimed subject-matter as a whole is not excluded from patentability under Art. 52(2) and (3) and is thus an invention within the meaning of Art. 52(1), it is examined in respect of the other requirements of patentability, in particular novelty and inventive step (G‑I, 1).

For the assessment of inventive step, all features which contribute to the technical character of the invention must be taken into account (G‑VII, 5.4). When the claimed invention is based on a mathematical method, it is assessed whether the mathematical method contributes to the technical character of the invention.

A mathematical method may contribute to the technical character of an invention, i.e. contribute to producing a technical effect that serves a technical purpose, by its application to a field of technology and/or by being adapted to a specific technical implementation (T 2330/13). The criteria for assessing these two situations are explained below.

Technical applications 

When assessing the contribution made by a mathematical method to the technical character of an invention, it must be taken into account whether the method, in the context of the invention, serves produces a technical effect serving a technical purpose (T 1227/05, T 1358/09).

Examples of technical purposes which may be served by contributions of a mathematical method are:

controlling a specific technical system or process, e.g. an X-ray apparatus or a steel cooling process; 
determining from measurements a required number of passes of a compaction machine to achieve a desired material density; 
digital audio, image or video enhancement or analysis, e.g. de-noising, detecting persons in a digital image, estimating the quality of a transmitted digital audio signal; 
separation of sources in speech signals; speech recognition, e.g. mapping a speech input to a text output; 
encoding data for reliable and/or efficient transmission or storage (and corresponding decoding), e.g. error-correction coding of data for transmission over a noisy channel, compression of audio, image, video or sensor data; 
encrypting/decrypting or signing electronic communications; generating keys in an RSA cryptographic system; 
optimising load distribution in a computer network; 
determining the energy expenditure of a subject by processing data obtained from physiological sensors; deriving the body temperature of a subject from data obtained from an ear temperature detector; 
providing a genotype estimate based on an analysis of DNA samples, as well as providing a confidence interval for this estimate so as to quantify its reliability; 
providing a medical diagnosis by an automated system processing physiological measurements;.
simulating the behaviour of an adequately defined class of technical items, or specific technical processes, under technically relevant conditions (see G‑II, 3.3.2).

A generic purpose such as "controlling a technical system" is not sufficient to confer a technical character to the mathematical method. The technical purpose must be a specific one.

Furthermore, the mere fact that a mathematical method may serve a technical purpose is not sufficient, either. The claim is to be functionally limited to the technical purpose, either explicitly or implicitly. This can be achieved by establishing a sufficient link between the technical purpose and the mathematical method steps, for example, by specifying how the input and the output of the sequence of mathematical steps relate to the technical purpose so that the mathematical method is causally linked to a technical effect. See G‑VII, for a worked-out example.

Defining the nature of the data input to a mathematical method does not necessarily imply that the mathematical method contributes to the technical character of the invention (T 2035/11, T 1029/06, T 1161/04). Whether a technical purpose is served by the mathematical method is primarily determined by the direct technical relevance of the results it provides.

If steps of a mathematical method are used to derive or predict the physical state of an existing real object from measurements of physical properties, as in the case of indirect measurements, those steps make a technical contribution regardless of what use is made of the results.

Technical implementations 

A mathematical method may also contribute to the technical character of the invention independently of any technical application when the claim is directed to a specific technical implementation of the mathematical method and the mathematical method is particularly adapted for that implementation in that its design is motivated by technical considerations of the internal functioning of the computer system or network (T 1358/09, G 1/19). This may happen if the mathematical method is designed to exploit particular technical properties of the technical system on which it is implemented to bring about a technical effect such as efficient use of computer storage capacity or network bandwidth. For instance, the adaptation of a polynomial reduction algorithm to exploit wordsize shifts matched to the word size of the computer hardware is based on such technical considerations and can contribute to producing the technical effect of an efficient hardware implementation of said algorithm. Another example is assigning the execution of data-intensive training steps of a machine-learning algorithm to a graphical processing unit (GPU) and preparatory steps to a standard central processing unit (CPU) to take advantage of the parallel architecture of the computing platform. The claim should be directed to the implementation of the steps on the GPU and CPU for this mathematical method to contribute to the technical character.

Computational efficiency 

If the mathematical method does not serve a technical purpose and the claimed technical implementation does not go beyond a generic technical implementation, the mathematical method does not contribute to the technical character of the invention. In such a case, it is not sufficient that the mathematical method is algorithmically more efficient than prior-art mathematical methods to establish a technical effect (see also G‑II, 3.6).

However, if it is established that the mathematical method produces a technical effect due to having been applied to a field of technology and/or adapted to a specific technical implementation, the computational efficiency of the steps affecting that established technical effect is to be taken into account when assessing inventive step. See G‑II, 3.6.4 for examples where an improvement in computational efficiency qualifies as a technical effect.

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